Integrating the Wigner Distribution on subsets of the phase space, a Survey
IINTEGRATING THE WIGNER DISTRIBUTIONON SUBSETS OF THE PHASE SPACE,A SURVEY
NICOLAS LERNER
Abstract.
We review several properties of integrals of the Wigner distributionon subsets of the phase space. Along our way, we provide a theoretical proofof the invalidity of Flandrin’s conjecture, a fact already proven via numericalarguments in our joint paper [4] with B. Delourme and T. Duyckaerts. We usealso the J.G. Wood & A.J. Bracken paper [37], for which we offer a mathematicalperspective. We review thoroughly the case of subsets of the plane whose boundaryis a conic curve and show that Mehler’s formula can be helpful in the analysis ofthese cases, including for the higher dimensional case investigated in the paper[27] by E. Lieb and Y. Ostrover. Using the Baire Category Theorem, we showthat, generically, the Wigner distribution of a pulse in L ( R n ) does not belongto L ( R n ) , providing as a byproduct a large class of examples of subsets of thephase space R n on which the integral of the Wigner distribution is infinite. Westudy as well the case of convex polygons of the plane, with a rather weak estimatedepending on the number of vertices, but independent of the area of the polygon. Wednesday 17 th February, 2021, 01:55
Contents
1. Preliminaries & Definitions 21.1. The Wigner Distribution 21.2. Weyl quantization, Composition formulas, Positive quantizations 41.3. Examples 211.4. Integrals of the Wigner distribution on subsets of the phase space 242. Quantization of radial functions and Mehler’s formula 252.1. Basic formulas in one dimension 252.2. Higher dimensional questions 263. Conics with eccentricity smaller than 1 283.1. Indicatrix of a disc 283.2. Indicatrix of an Euclidean ball 353.3. Ellipsoids in the phase space 403.4. A conjecture on integrals of products of Laguerre polynomials 504. Parabolas 554.1. Preliminary remarks 554.2. Calculation of the kernel 574.3. The main result 594.4. Paraboloids, a conjecture 605. Conics with eccentricity greater than 1 635.1. The quarter-plane, a counterexample to Flandrin’s conjecture 645.2. Hyperbolic regions 725.3. Comments and further results 845.4. Numerics 87 a r X i v : . [ m a t h . SP ] F e b NICOLAS LERNER
6. Unboundedness is Baire generic 926.1. Preliminaries 926.2. An explicit construction 936.3. Most pulses give rise to non-integrable Wigner distribution 956.4. Consequences on integrals of the Wigner distribution 977. Convex polygons of the plane 997.1. Convex Cones 997.2. Triangles 1017.3. Convex Polygons 1047.4. Symbols supported in a half-space 1078. Open questions & Conjectures 1098.1. Anisotropic Ellipsoids & Paraboloids 1098.2. Balls for the (cid:96) p norm 1108.3. On generic pulses in L ( R n ) Preliminaries & Definitions
The Wigner Distribution.
Let u, v be given functions in L ( R n ) . The func-tion Ω , defined on R n × R n by(1.1.1) R n × R n (cid:51) ( z, x ) (cid:55)→ u ( x + z v ( x − z u, v )( x, z ) , belongs to L ( R n ) from the identity(1.1.2) (cid:90) R n | Ω( u, v )( x, z ) | dxdz = (cid:107) u (cid:107) L ( R n ) (cid:107) v (cid:107) L ( R n ) . We have also(1.1.3) sup x ∈ R n (cid:90) R n | Ω( x, z ) | dz ≤ n (cid:107) u (cid:107) L ( R n ) (cid:107) v (cid:107) L ( R n ) . We may then give the following definition.
Definition 1.1.
Let u, v be given functions in L ( R n ) . We define the joint Wignerdistribution W ( u, v ) as the partial Fourier transform with respect to z of the function For f ∈ S ( R N ) , we define its Fourier transform by ˆ f ( ξ ) = (cid:82) R N e − iπx · ξ f ( x ) dx and we obtainthe inversion formula f ( x ) = (cid:82) R N e iπx · ξ ˆ f ( ξ ) dξ . Both formulas can be extended to tempereddistributions: for T ∈ S (cid:48) ( R N ) , we define the tempered distribution ˆ T by(1.1.4) (cid:104) ˆ T , φ (cid:105) S (cid:48) ( R N ) , S ( R N ) = (cid:104) T, ˆ φ (cid:105) S (cid:48) ( R N ) , S ( R N ) . NTEGRALS OF THE WIGNER DISTRIBUTION 3 Ω defined in (1.1.1) . We have for ( x, ξ ) ∈ R nx × R nξ , using (1.1.3) , (1.1.6) W ( u, v )( x, ξ ) = (cid:90) R n e − iπz · ξ u ( x + z v ( x − z dz. The Wigner distribution of u is defined as W ( u, u ) . Lemma 1.2.
Let u, v be given functions in L ( R n ) . The function W ( u, v ) belongsto L ( R n ) and we have (1.1.7) (cid:107)W ( u, v ) (cid:107) L ( R n ) = (cid:107) u (cid:107) L ( R n ) (cid:107) v (cid:107) L ( R n ) . We have also (1.1.8) W ( u, v )( x, ξ ) = W ( v, u )( x, ξ ) , so that W ( u, u ) is real-valued.Proof. Note that the function W ( u, v ) is in L ( R n ) and satisfies (1.1.7) from (1.1.2)and the definition of W as the partial Fourier transform of Ω . Property (1.1.8) isimmediate and entails that W ( u, u ) is real-valued. (cid:3) Remark 1.3.
We note also that the real-valued function W ( u, u ) can take negativevalues, choosing for instance u ( x ) = xe − πx on the real line, we get W ( u , u )( x, ξ ) = 2 / e − π ( x + ξ ) (cid:0) x + ξ − π (cid:1) . In fact the real-valued function W ( u, u ) will take negative values unless u is a Gauss-ian function, thanks to a Theorem due to E. Lieb (see [25] and the books [11] and[29]). As a matter of fact, this range of W ( u, u ) intersecting R − for most “pulses” u in L ( R n ) makes rather weird the qualification of W ( u, u ) as a “quasi-probability”(anyhow the emphasis must be on quasi , not on probability ). Remark 1.4.
We have also by Fourier inversion formula, say for u ∈ S ( R n ) ,(1.1.9) u ( x + z u ( x − z x, z ) = (cid:90) W ( u, u )( x, ξ ) e iπz · ξ dξ, so that, with z = 2 x = y , we get the Reconstruction Formula,(1.1.10) u ( y )¯ u (0) = (cid:90) W ( u, u )( y , ξ ) e iπy · ξ dξ, as well as(1.1.11) | u ( x ) | = (cid:90) W ( u, u )( x, ξ ) dξ, | ˆ u ( ξ ) | = (cid:90) W ( u, u )( x, ξ ) dx, Note also that with this normalization, it is natural to introduce the operators D αx defined for α ∈ N N by(1.1.5) D αx u = D α x . . . D α n x N u, D x j u = ∂u iπ∂x j , so that (cid:100) D αx u = ξ α ˆ u ( ξ ) , with ξ α = ξ α . . . ξ α N N . NICOLAS LERNER the former formula following from (1.1.9) and the latter from(1.1.12) (cid:90) W ( u, u )( x, ξ ) dx = (cid:120) e − iπzξ u ( x + z u ( x − z dzdx = (cid:120) e − iπξ ( x − x ) u ( x )¯ u ( x ) dx dx = | ˆ u ( ξ ) | . Lemma 1.5.
Let u be a function in L ( R n ) which is even or odd. Then W ( u, u ) isan even function.Proof. Using the notation(1.1.13) ˇ u ( x ) = u ( − x ) , we check W ( u, v )( − x, − ξ ) = (cid:90) R n e iπz · ξ u ( − x + z v ( − x − z dz = (cid:90) R n e iπz · ξ ˇ u ( x − z v ( x + z dz = (cid:90) R n e − iπz · ξ ˇ u ( x + z v ( x − z dz = W (ˇ u, ˇ v )( x, ξ ) , so that if ˇ u = ± u , we get W ( u, u )( − x, − ξ ) = W ( u, u )( x, ξ ) . (cid:3) N.B.
This lemma is a very particular case of the symplectic covariance propertydisplayed below in (1.2.33).It turns out that most of the properties of the Wigner distribution (in particularLemma 1.5) are inherited from its links with the Weyl quantization introduced byH. Weyl in 1926 in the first edition of [36] and our next remarks are devised to stressthat link.1.2.
Weyl quantization, Composition formulas, Positive quantizations.
Weyl quantization.
The main goal of Hermann Weyl in his seminal paper [36]was to give a simple formula, also providing symplectic covariance, ensuring thatreal-valued Hamiltonians a ( x, ξ ) get quantized by formally self-adjoint operators.The standard way of dealing with differential operators does not achieve that goalsince for instance the standard quantization of the Hamiltonian xξ (indeed real-valued) is the operator xD x , which is not symmetric ( D x is defined in (1.1.5));H. Weyl’s choice in that case was xξ should be quantized by the operator
12 ( xD x + D x x ) , (indeed symmetric), and more generally, say for a ∈ S ( R n ) , u ∈ S ( R n ) , the quantization of the Hamil-tonian a ( x, ξ ) , denoted by Op w ( a ) , should be given by the formula ( Op w ( a ) u )( x ) = (cid:120) e iπ ( x − y ) · ξ a (cid:0) x + y , ξ (cid:1) u ( y ) dydξ. NTEGRALS OF THE WIGNER DISTRIBUTION 5
For v ∈ S ( R n ) , we may consider (cid:104) Op w ( a ) u, v (cid:105) L ( R n ) = (cid:121) a ( x, ξ ) e − iπz · ξ u ( x + z v ( x − z dzdxdξ = (cid:120) R n × R n a ( x, ξ ) W ( u, v )( x, ξ ) dxdξ, and the latter formula allows us to give the following definition. Definition 1.6.
Let a ∈ S (cid:48) ( R n ) . We define the Weyl quantization Op w ( a ) of theHamiltonian a , by the formula (1.2.1) ( Op w ( a ) u )( x ) = (cid:120) e iπ ( x − y ) · ξ a (cid:0) x + y , ξ (cid:1) u ( y ) dydξ, to be understood weakly as (1.2.2) (cid:104) Op w ( a ) u, ¯ v (cid:105) S (cid:48) ( R n ) , S ( R n ) = (cid:104) a, W ( u, v ) (cid:105) S (cid:48) ( R n ) , S ( R n ) . We note that the sesquilinear mapping S ( R n ) × S ( R n ) (cid:51) ( u, v ) (cid:55)→ W ( u, v ) ∈ S ( R n ) , is continuous so that the above bracket of duality (cid:104) a, W ( u, v ) (cid:105) S (cid:48) ( R n ) , S ( R n ) makessense. We note as well that a temperate distribution a ∈ S (cid:48) ( R n ) gets quantized by acontinuous operator Op w ( a ) from S ( R n ) into S (cid:48) ( R n ) . This very general frameworkis not really useful since we want to compose our operators Op w ( a ) Op w ( b ) . A firststep in this direction is to look for sufficient conditions ensuring that the operatorOp w ( a ) is bounded on L ( R n ) . Moreover, for a ∈ S (cid:48) ( R n ) and b a polynomial in C [ x, ξ ] , we have the composition formula,Op w ( a ) Op w ( b ) = Op w ( a(cid:93)b ) , (1.2.3) ( a(cid:93)b )( x, ξ ) = (cid:88) k ≥ iπ ) k (cid:88) | α | + | β | = k ( − | β | α ! β ! ( ∂ αξ ∂ βx a )( x, ξ )( ∂ αx ∂ βξ b )( x, ξ ) , (1.2.4)which involves here a finite sum. This follows from (2.1.26) in [21] where severalgeneralizations can be found (see in particular in that reference the integral formula(2.1.18) which can be given a meaning for quite general classes of symbols). Proposition 1.7.
Let a be a tempered distribution on R n . Then we have (1.2.5) (cid:107) Op w ( a ) (cid:107) B ( L ( R n )) ≤ min (cid:0) n (cid:107) a (cid:107) L ( R n ) , (cid:107) ˆ a (cid:107) L ( R n ) (cid:1) . Proof.
In fact we have from (1.2.2), u, v ∈ S ( R n ) , (cid:104) Op w ( a ) u, v (cid:105) L ( R n ) = (cid:121) a ( x, ξ ) u (2 x − y )¯ v ( y ) e − iπ ( x − y ) · ξ n dydxdξ, so that defining for ( x, ξ ) ∈ R n the operator σ x,ξ by(1.2.6) ( σ x,ξ u )( y ) = u (2 x − y ) e − iπ ( x − y ) · ξ , NICOLAS LERNER we see that the operator σ x,ξ (called phase symmetry ) is unitary and self-adjoint and(1.2.8) Op w ( a ) = 2 n (cid:120) a ( x, ξ ) σ x,ξ dxdξ, proving the first estimate of the proposition. As a consequence of (1.2.8), we obtainthat(1.2.9) ( Op w ( a )) ∗ = Op w (¯ a ) , so that for a real-valued, ( Op w ( a )) ∗ = Op w ( a ) .To prove the second estimate, we introduce the so-called ambiguity function A ( u, v ) as the inverse Fourier transform of the Wigner function W ( u, v ) , so that for u, v inthe Schwartz class, we have ( A ( u, v ))( η, y ) = (cid:120) W ( u, v )( x, ξ ) e iπ ( x · η + ξ · y ) dxdξ, i.e.(1.2.10) ( A ( u, v ))( η, y ) = (cid:90) u ( x + y v ( x − y e iπx · η dx, which reads as well as(1.2.11) ( A ( u, v ))( η, y ) = (cid:90) u ( y z v ( y − z e iπz · η dz − n = W ( u, ˇ v )( y , − η − n . Remark 1.8.
With Ω( u, v ) defined by (1.1.1), we have(1.2.12) W ( u, v ) = F (cid:0) Ω( u, v ) (cid:1) , where F stands for the Fourier transformation with respect to the second variable.Taking the Fourier transform with respect to the second variable in the previousformula gives, with F j (resp. F ) standing for the Fourier transform with respect tothe j th variable (resp. all variables), F W = C Ω , F W = F C Ω , A = CF W = F C Ω , where C (resp. C or C ) stands for the “check” operator C in R n × R n given by(1.1.13) (resp. with respect to the first or second variable), the latter formula being(1.2.10).Applying Plancherel formula on (1.2.2), we get(1.2.13) (cid:104) Op w ( a ) u, v (cid:105) L ( R n ) = (cid:104) ˆ a, A ( u, v ) (cid:105) S (cid:48) ( R n ) , S ( R n ) . We note that a consequence of (1.2.4) is that for a linear form L ( x, ξ ) , we have L(cid:93)L = L , and more generally L (cid:93)N = L N . Indeed we have(1.2.7) ( σ x,ξ u )( y ) = ( σ x,ξ u )(2 x − y ) e − iπ ( x − y ) · ξ = u (2 x − (2 x − y )) e − iπ ( x − (2 x − y )) · ξ e − iπ ( x − y ) · ξ = u ( y ) , so that σ x,ξ = Id . We have (cid:104) σ ∗ x,ξ u, v (cid:105) L ( R n ) = (cid:104) u, σ x,ξ v (cid:105) L ( R n ) = W ( v, u )( x, ξ ) = W ( u, v )( x, ξ ) = (cid:104) σ x,ξ u, v (cid:105) L ( R n ) , proving that σ ∗ x,ξ = σ x,ξ . NTEGRALS OF THE WIGNER DISTRIBUTION 7
As a result, considering for ( y, η ) ∈ R n , the linear form L η,y defined by(1.2.14) L η,y ( x, ξ ) = x · η + ξ · y, we see that(1.2.15) A ( u, v )( η, y ) = (cid:104) Op w ( e iπ ( x · η + ξ · y ) ) u, v (cid:105) L ( R n ) , and thus we get Hermann Weyl’s original formula(1.2.16) Op w ( a ) = (cid:120) (cid:98) a ( η, y ) e i Op w ( L η,y ) dydη, which implies the second estimate in the proposition. (cid:3) Proposition 1.9.
Let a ∈ S (cid:48) ( R n ) . The distribution kernel k a ( x, y ) of the operator Op w ( a ) is (1.2.17) k a ( x, y ) = (cid:98) a ( x + y , y − x ) . Let k ∈ S (cid:48) ( R n ) be the distribution kernel of a continuous operator A from S ( R n ) into S (cid:48) ( R n ) . Then the Weyl symbol a of A is (1.2.18) a ( x, ξ ) = (cid:90) e − πit · ξ k ( x + t , x − t dt, where the integral sign means that we take the Fourier transform with respect to t ofthe distribution k ( x + t , x − t ) on R n (see (1.1.4) in footnote 1 for the definitionof the Fourier transformation on tempered distributions).Proof. With u, v ∈ S ( R n ) , we have defined Op w ( a ) via Formula (1.2.2) and usingRemark 1.8, we get (cid:104) Op w ( a ) u, ¯ v (cid:105) S (cid:48) ( R n ) , S ( R n ) = (cid:104) a ( x, ξ ) , (cid:98) Ω [2] ( x, ξ ) (cid:105) S (cid:48) ( R n ) , S ( R n ) = (cid:104) (cid:98) a [2] ( t, z ) , u ( t + z v ( t − z (cid:105) S (cid:48) ( R n ) , S ( R n ) = (cid:104) (cid:98) a [2] ( x + y , y − x ) , u ( y )¯ v ( x ) (cid:105) S (cid:48) ( R n ) , S ( R n ) , proving (1.2.17). As a consequence, we find that k a ( x + t , x − t ) = (cid:98) a [2] ( x, − t ) , andby Fourier inversion, this entails a ( x, ξ ) = F ourier t (cid:0) k a ( x + t , x − t (cid:1) ( ξ ) = (cid:90) e − πit · ξ k a ( x + t , x − t dt, where the integral sign means that we perform a Fourier transformation with respectto the variable t . (cid:3) A particular case of Segal’s formula (see e.g. Theorem 2.1.2 in [21]) is with F standing for the Fourier transformation on R n ,(1.2.19) F ∗ Op w ( a ) F = Op w ( a ( ξ, − x )) . We define the canonical symplectic form σ on R n × R n by(1.2.20) (cid:104) σX, Y (cid:105) = (cid:2) X, Y (cid:3) = ξ · y − η · x, with X = ( x, ξ ) , Y = ( y, η ) . NICOLAS LERNER
The symplectic group Sp ( n, R ) is the subgroup of S ∈ Sl (2 n, R ) such that(1.2.21) ∀ X, Y ∈ R n , [ SX, SY ] = [
X, Y ] , i.e. S ∗ σS = σ, with(1.2.22) σ = (cid:18) I n − I n (cid:19) . The symplectic group is generated by(i) ( x, ξ ) (cid:55)→ ( T x, t T − ξ ) , T ∈ Gl ( n, R ) , (1.2.23) (ii) ( x k , ξ k ) (cid:55)→ ( ξ k , − x k ) , other coordinates unchanged,(1.2.24) (iii) ( x, ξ ) (cid:55)→ ( x, ξ + Qx ) , Q ∈ Sym ( n, R ) . (1.2.25)Now for S ∈ Sp ( n, R ) , the operator(1.2.26) Op w ( a ◦ S ) = M ∗ Op w ( a ) M , where M belongs to the metaplectic group, which is a group of unitary transfor-mations of L ( R n ) . Let us describe the generators of the metaplectic group corres-ponding to the symplectic transformations (i-iii) above. The metaplectic group isgenerated by (j) ( M u )( x ) = | det T | − / u ( T − x ) , (1.2.27) (jj) partial Fourier transformation with respect to x k ,(1.2.28) (jjj) multiplication by e iπ (cid:104) Qx,x (cid:105) . (1.2.29)We note also that for Y = ( y, η ) ∈ R n , the symmetry S Y is defined by S Y ( X ) =2 Y − X and is quantized by the phase symmetry σ Y as defined by (1.2.6) with theformula(1.2.30) Op w ( a ◦ S Y ) = σ ∗ Y Op w ( a ) σ Y = σ Y Op w ( a ) σ Y . Similarly, the translation T Y is defined on the phase space by T Y ( X ) = X + Y andis quantized by the phase translation τ Y ,(1.2.31) ( τ ( y,η ) u )( x ) = u ( x − y ) e iπ ( x − y ) · η , and we have(1.2.32) Op w ( a ◦ T Y ) = τ ∗ Y Op w ( a ) τ Y = τ − Y Op w ( a ) τ Y . Note also that the covariance formula (1.2.26) can be reformulated as the followingproperty of the Wigner distribution, thanks to (1.2.2),(1.2.33) W ( M u, M v ) = W ( u, v ) ◦ S − . Since the metaplectic group is continuous from S ( R n ) into itself, Segal’s Formula(1.2.26) is valid as well for a ∈ S (cid:48) ( R n ) . We note also that Sp (1 , R ) = Sl (2 , R ) . N.B.
Property (1.2.33) can be extended to the affine symplectic group and we havewith the phase translations defined in (1.2.31),(1.2.34) ∀ ( X, Y ) ∈ R n × R n , W ( τ Y u, τ Y v ) ( X ) = W ( u, v )( X − Y ) . NTEGRALS OF THE WIGNER DISTRIBUTION 9
Theorem 1 in E. Lieb’s classical article [25] gives a more precise version of (1.2.36),(1.2.37) and (1.2.38) below.
Theorem 1.10.
Let u, v be in L ( R n ) . Then W ( u, v ) is a uniformly continuousfunction belonging to L ( R n ) ∩ L ∞ ( R n ) and using the definitions (1.2.31) , (1.2.6) for the phase translations and phase symmetry, we have W ( u, v )( X ) = 2 n (cid:104) σ X u, v (cid:105) L ( R n ) = 2 n (cid:104) τ ∗ X u, τ X ˇ v (cid:105) L ( R n ) (1.2.35) = 2 n (cid:104) σ τ − X u, v (cid:105) L ( R n ) , (cid:107)W ( u, v ) (cid:107) L ( R n ) = (cid:107) u (cid:107) L ( R n ) (cid:107) v (cid:107) L ( R n ) , (1.2.36) ∀ p ∈ [1 , + ∞ ] , (cid:107)W ( u, v ) (cid:107) L ∞ ( R n ) ≤ n (cid:107) u (cid:107) L p ( R n ) (cid:107) v (cid:107) L p (cid:48) ( R n ) . (1.2.37) More generally, for q ≥ and r ∈ [ q (cid:48) , q ] , we have (1.2.38) (cid:107)W ( u, v ) (cid:107) L q ( R n ) ≤ n ( q − q (cid:107) u (cid:107) L r ( R n ) (cid:107) v (cid:107) L r (cid:48) ( R n ) . Moreover, we have (1.2.39) lim R n (cid:51) X, | X |→ + ∞ (cid:104) W ( u, v )( X ) (cid:105) = 0 . Proof.
We have with ˇ v ( x ) = v ( − x ) = ( σ v )( x ) , W ( u, v )( x, ξ ) = 2 n (cid:90) u ( x + z )¯ v ( x − z ) e − iπzξ dz = 2 n (cid:90) u ( z − ( − x )) e iπ ( z − − x )( − ξ ) ¯ˇ v ( z − x ) e − iπ ( z − x ) ξ e − iπzξ +2 iπ ( z − − x ) ξ +2 iπ ( z − x ) ξ dz = 2 n (cid:90) ( τ ( − x, − ξ ) u )( z )( τ ( x,ξ ) ˇ v )( z ) dz = 2 n (cid:104) τ ∗ ( x,ξ ) u, τ ( x,ξ ) ˇ v (cid:105) L ( R n ) , or for short(1.2.40) W ( u, v )( X ) = 2 n (cid:104) τ ∗ X u, τ X ˇ v (cid:105) L ( R n ) . As a consequence we find from (1.2.8) that (cid:104) Op w ( a ) u, v (cid:105) = (cid:90) a ( X )2 n (cid:104) σ τ ∗ X u, v (cid:105) dX, and since ( σ x,ξ u )( y ) = u (2 x − y ) e − iπ ( x − y ) · ξ , we can verify directly that(1.2.41) σ τ − X = σ X . Indeed, we have composing the translation of vector − X in R n with the symmetrywith respect to 0, Y (cid:55)→ Y − X (cid:55)→ X − Y = Y (cid:48) ,
12 ( Y + Y (cid:48) ) = X, that is the symmetry with respect to X . Quantifying this equality, we use ( τ ( − x, − ξ ) u )( z ) = u ( z + 2 x ) e iπ ( z − − x )( − ξ ) = u ( z + 2 x ) e − iπ ( z + x ) ξ , so that we obtain σ ( τ ( − x, − ξ ) u )( z ) = u ( − z + 2 x ) e − iπ ( − z + x ) ξ = ( σ x,ξ u )( z ) , We use the standard notation: for p ∈ [1 , + ∞ ] we define p (cid:48) by the equality p + p (cid:48) = 1 . which proves (1.2.41) and thus (1.2.35). Formula (1.2.36) is already proven in (1.1.7)and (1.2.37) follows from (1.2.35), Hölder’s inequality and the fact that τ X is anendomorphism of L p ( R n ) with norm 1 (cf. the expression (1.2.31)). To prove (1.2.38)we note that from the expression (1.2.12), the Hausdorff-Young’s inequality implies(1.2.42) (cid:107)W ( u, v ) (cid:107) L q ⊗ L q ≤ (cid:107) Ω( u, v ) (cid:107) L q ⊗ L q (cid:48) ≤ (cid:107)| u | q (cid:48) ∗ | v | q (cid:48) (cid:107) /q (cid:48) L q/q (cid:48) n q − q , and since Young’s inequality gives (cid:107)| u | q (cid:48) ∗ | v | q (cid:48) (cid:107) L q/q (cid:48) ≤ (cid:107)| u | q (cid:48) (cid:107) L a/q (cid:48) (cid:107)| v | q (cid:48) (cid:107) L b/q (cid:48) ,a, b ≥ q (cid:48) with − q (cid:48) q = 1 − q (cid:48) a + 1 − q (cid:48) b , i.e. q (cid:48) (cid:0) a + 1 b (cid:1) = 1 + q (cid:48) q , that is a + 1 b = 1 , so that (cid:107)| u | q (cid:48) ∗ | v | q (cid:48) (cid:107) L q/q (cid:48) ≤ (cid:107) u (cid:107) q (cid:48) L a (cid:107) v (cid:107) q (cid:48) L b , in such a way that (1.2.42) yields (cid:107)W ( u, v ) (cid:107) L q ⊗ L q ≤ n q − q (cid:107) u (cid:107) L a (cid:107) v (cid:107) L b , a, b ≥ q (cid:48) , a + 1 b = 1 , which is (1.2.38) . We are left with the proof of uniform continuity of W ( u, v ) . Wehave for X, Y ∈ R n , W ( u, v )( Y ) − W ( u, v )( X ) = 2 n (cid:104) ( σ Y − σ X ) u, v (cid:105) L ( R n ) , and since σ Y = Id (see the footnote 2 on page 6), we find W ( u, v )( Y ) − W ( u, v )( X ) = 2 n (cid:104) ( σ Y σ X − Id) σ X u, v (cid:105) L ( R n ) = 2 n (cid:104) σ X u, ( σ X σ Y − Id) v (cid:105) L ( R n ) . According to Formula (2.1.16) in [21], we have(1.2.44) σ X σ Y = τ X − Y e iπ [ Y,X ] , and this implies(1.2.45) |W ( u, v )( Y ) − W ( u, v )( X ) | ≤ n (cid:107) u (cid:107) L ( R n ) (cid:107) τ X − Y ) v (cid:107) L ( R n ) . We have from (1.2.30), τ z,ζ v ( x ) − v ( x ) = v ( x − z ) e iπ ( x − z ) ζ − v ( x )= (cid:0) v ( x − z ) − v ( x ) (cid:1) e iπ ( x − z ) ζ + v ( x ) (cid:0) e iπ ( x − z ) ζ − (cid:1) , and thus (cid:107) τ Z v − v (cid:107) L ( R n ) ≤ (cid:18)(cid:90) | v ( x − z ) − v ( x ) | dx (cid:19) / + (cid:18)(cid:90) | v ( x ) | | e iπ ( x − z ) ζ − | dx (cid:19) / . For p, q, r ∈ [1 , + ∞ ] with p (cid:48) + q (cid:48) = r (cid:48) , we have(1.2.43) (cid:107) f ∗ g (cid:107) L r ≤ (cid:107) f (cid:107) L p (cid:107) g (cid:107) L q . NTEGRALS OF THE WIGNER DISTRIBUTION 11
We have the classical result, due to the density in L of continuous compactly sup-ported functions, lim R n (cid:51) z → (cid:90) | v ( x − z ) − v ( x ) | dx = 0 , and moreover the Lebesgue Dominated Convergence Theorem implies lim ( z,ζ ) → (0 , (cid:90) | v ( x ) | (cid:124) (cid:123)(cid:122) (cid:125) ∈ L ( R n ) | e iπ ( x − z ) ζ − | (cid:124) (cid:123)(cid:122) (cid:125) ≤ dx = 0 , so that lim R n (cid:51) Z → (cid:107) τ Z v − v (cid:107) L ( R n ) = 0 . As a consequence (1.2.45) implies theuniform continuity of W ( u, v ) . Moreover, we have, for φ, ψ ∈ S ( R n ) , W ( u, v ) = W ( u − φ, v ) + W ( φ, v − ψ ) + W ( φ, ψ ) , so that |W ( u, v )( x, ξ ) | ≤ (cid:90) | ( u − φ )( x + z || v ( x − z | dz + (cid:120) | ( v − ψ )( x − z || φ ( x + z | dz + |W ( φ, ψ )( x, ξ ) |≤ n (cid:107) u − φ (cid:107) L ( R n ) (cid:107) v (cid:107) L ( R n ) + 2 n (cid:107) v − ψ (cid:107) L ( R n ) (cid:107) φ (cid:107) L ( R n ) + |W ( φ, ψ )( x, ξ ) | . We choose now sequences ( φ k ) , ( ψ k ) of S ( R n ) converging respectively in L ( R n ) towards u, v . We obtain for all k ∈ N ,(1.2.46) |W ( u, v )( x, ξ ) | ≤ n (cid:107) u − φ k (cid:107) L ( R n ) (cid:107) v (cid:107) L ( R n ) + 2 n (cid:107) v − ψ k (cid:107) L ( R n ) (cid:107) φ k (cid:107) L ( R n ) + |W ( φ k , ψ k )( x, ξ ) | , so that using that W ( φ k , ψ k ) belongs to S ( R n ) , we get lim sup R n (cid:51) X, | X |→ + ∞ (cid:104) |W ( u, v )( X ) | (cid:105) ≤ n (cid:107) u − φ k (cid:107) L ( R n ) (cid:107) v (cid:107) L ( R n ) + 2 n (cid:107) v − ψ k (cid:107) L ( R n ) (cid:107) φ k (cid:107) L ( R n ) , and thus, taking the limit when k → + ∞ , we obtain lim R n (cid:51) X, | X |→ + ∞ (cid:104) |W ( u, v )( X ) | (cid:105) = 0 , completing the proof of Theorem 1.10. (cid:3) Remark 1.11.
Let u be in L ( R n ) be an even function. We then have(1.2.47) W ( u, u )(0 ,
0) = 2 n (cid:107) u (cid:107) L ( R n ) = (cid:107)W ( u, u ) (cid:107) L ∞ ( R n ) . On the other hand if u is odd we have(1.2.48) W ( u, u )(0 ,
0) = − n (cid:107) u (cid:107) L ( R n ) = −(cid:107)W ( u, u ) (cid:107) L ∞ ( R n ) , showing that for odd functions the minimum of the Wigner distribution is negative(we assume u (cid:54) = 0 in L ( R n ) ) and attained at 0. Let us check for instance the (odd) function u of Remark 1.3. We have (cid:107) u (cid:107) L ( R ) = 2 (cid:90) x e − πx dx = 4 (cid:90) + ∞ t π e − t (2 π ) − / t − / dt = 2Γ(3 / π ) / = Γ(1 / π ) / = 12 / π = −W ( u , u )(0 , . On weak versions of the Wigner distribution.
Let u, v be in the space S (cid:48) ( R n ) of tempered distributions. Then we can define as above the tempered distribution Ω( u, v ) in R n : we set(1.2.49) (cid:104) Ω( u, v )( x, z ) , Φ( x, z ) (cid:105) S (cid:48) ( R n ) , S ( R n ) = (cid:104) u ( x ) ⊗ ¯ v ( x ) , Φ( x + x , x − x ) (cid:105) S (cid:48) ( R n ) , S ( R n ) , and then we define the Wigner distribution W ( u, v ) as the Fourier transform withrespect to z of Ω( u, v ) , meaning that(1.2.50) (cid:104)W ( u, v ) , Ψ (cid:105) S (cid:48) ( R n ) , S ( R n ) = (cid:104) Ω( u, v ) , F Ψ (cid:105) S (cid:48) ( R n ) , S ( R n ) , where ( F Ψ)( x, ξ ) = (cid:90) R n e − iπz · ξ Ψ( x, z ) dz. Of course W ( u, v ) is only a tempered distribution on R n and we have the inversionformula, using the notations of Remark 1.8,(1.2.51) Ω( u, v ) = F C W ( u, v ) . The above remarks show that there is no difficulty to extend the definition of the jointWigner distribution W ( u, v ) to the case where u, v are both tempered distributionson R n . Some properties are surviving from the L theory, in particular the inversionformula, but one should be reasonably cautious at avoiding to write brackets ofduality as integrals. Theorem 2 in [25] gives a more complete version of the followingresult. Theorem 1.12.
Let u ∈ L ( R n ) such that W ( u, u ) ∈ L ( R n ) . Then u belongs to L p ( R n ) for all p ∈ [1 , + ∞ ] and we have (cid:107) u (cid:107) L ( R n ) (cid:107) u (cid:107) L ∞ ( R n ) ≤ n (cid:107)W ( u, u ) (cid:107) L ( R n ) . N.B.
We refer the reader to our Section 6.3 and in particular to our Theorem 6.7on page 97 showing that the set of u in L ( R n ) such that W ( u, u ) belongs to L ( R n ) is meager.Proof. Thanks to Theorem 1.10, we have W ( u, u ) ∈ L p ( R n ) for all p ∈ [1 , + ∞ ] andwe have in a weak sense, u ( x + z u ( x − z (cid:90) e iπz · ξ W ( u, u )( x, ξ ) dξ, so that u ( x )¯ u ( x ) = (cid:90) e iπ ( x − x ) · ξ W ( u, u )( x + x , ξ ) dξ, NTEGRALS OF THE WIGNER DISTRIBUTION 13 and thus (cid:90) | u ( x ) || u ( x ) | dx ≤ (cid:120) (cid:12)(cid:12) W ( u, u )( x + x , ξ ) (cid:12)(cid:12) dξdx = 2 n (cid:107)W ( u, u ) (cid:107) L ( R n ) , i.e. (cid:107) u (cid:107) L ( R n ) (cid:107) u (cid:107) L ∞ ( R n ) ≤ n (cid:107)W ( u, u ) (cid:107) L ( R n ) , proving the lemma. (cid:3) Composition Formulas.
Lemma 1.13.
Let u, v, f, g be in L ( R n ) . Then (1.2.52) (cid:104) u, g (cid:105) L ( R n ) (cid:104) f, v (cid:105) L ( R n ) = (cid:120) W ( u, v )( x, ξ ) W ( f, g )( x, ξ ) dxdξ. In other words, the Weyl symbol of the rank-one operator u (cid:55)→ (cid:104) u, g (cid:105) L ( R n ) f is W ( f, g ) . In particular if f = g is a unit vector in L ( R n ) we find that W ( f, f ) is the Weyl symbol of the orthogonal projection onto C f .Proof. Both functions W ( u, v ) , W ( f, g ) belong to L ( R n ) , so that the integral onthe right-hand-side of (1.2.52) actually makes sense. Also W ( u, v ) is the partialFourier transform with respect to the variable z of ( x, z ) (cid:55)→ u ( x + z/ v ( x − z/ ,thus applying Plancherel formula we obtain that (cid:120) W ( u, v )( x, ξ ) W ( f, g )( x, ξ ) dxdξ = (cid:120) u ( x + z/ v ( x − z/ f ( x − z/ g ( x + z/ dxdz = (cid:104) u, g (cid:105) L ( R n ) (cid:104) f, v (cid:105) L ( R n ) . The last property follows from (1.2.2). (cid:3)
Using Section 2.1.5 in [21], we obtain that for a, b ∈ S ( R n ) Op w ( a ) Op w ( b ) = (cid:120) R n × R n a ( Y ) b ( Z )2 n σ Y σ Z dY dZ. We get Op w ( a ) Op w ( b ) = Op w ( a(cid:93)b ) with ( a(cid:93)b )( X ) = 2 n (cid:120) R n × R n e − iπ [ X − Y,X − Z ] a ( Y ) b ( Z ) dY dZ (1.2.54) = (cid:120) R n × R n e iπ (cid:104) Ξ ,Z (cid:105) a (cid:0) X + σ Ξ2 (cid:1) b ( Z + X ) d Ξ dZ, (1.2.55) = (cid:90) R n e iπ (cid:104) X, Ξ (cid:105) a (cid:0) X − σ Ξ2 (cid:1)(cid:98) b (Ξ) d Ξ , (1.2.56) We refer of course to the formula (cid:104) ˆ u, ˆ v (cid:105) L ( R n ) = (cid:104) u, v (cid:105) L ( R n ) , when using the complex Hilbertspace L ( R n ) . Note however that Formula (1.1.4) is using the real duality between S ( R n ) and S (cid:48) ( R n ) so that to check, with S ∗ ( R N ) standing for the anti-dual of S ( R N ) (i.e. continuousanti-linear forms on S ( R N ) ), we have also(1.2.53) (cid:104) (cid:98) T , (cid:98) φ (cid:105) S ∗ ( R N ) , S ( R N ) = (cid:104) (cid:98) T , (cid:98) φ (cid:105) S (cid:48) ( R N ) , S ( R N ) = (cid:104) T, (cid:98)(cid:98) φ (cid:105) S (cid:48) ( R N ) , S ( R N ) = (cid:104) T, φ (cid:105) S (cid:48) ( R N ) , S ( R N ) = (cid:104) T, φ (cid:105) S ∗ ( R N ) , S ( R N ) . where [ · , · ] is the symplectic form (1.2.20) and σ is (1.2.22). Formula (1.2.55) isinteresting since very close to the group J t defined in Formula (4.1.14) of [21].1.2.4. L boundedness. Theorem 1.14.
Let a be a semi-classical symbol on R n , i.e. a smooth function of ( x, ξ ) depending on h ∈ (0 , such that (1.2.57) ∀ l ∈ N , p l ( a ) = sup ( x,ξ ) ∈ R n ,h ∈ (0 , | α | + | β |≤ l | ( ∂ αx ∂ βξ a )( x, ξ, h ) | h − | α | + | β | < + ∞ . Then the operator Op w ( a ( x, ξ, h )) is bounded on L ( R n ) and such that (1.2.58) (cid:107) Op w ( a ( x, ξ, h )) (cid:107) B ( L ( R n )) ≤ c n p (cid:96) n ( a ) , where c n and (cid:96) n depend only on n .Proof. Theorem 1.2 in A. Boulkhemair’s article [3] is providing that result (andmore) with (cid:96) n = [ n/
2] + 1 . Note also that Theorem 1.1.4 in [21] is providing anelementary proof of the above result for the ordinary quantization of a given by(1.2.59) ( Op ( a ) u )( x ) = (cid:90) e iπx · ξ a ( x, ξ, h )ˆ u ( ξ ) dξ = (cid:120) e iπ ( x − y ) · ξ a ( x, ξ, h ) u ( y ) dydξ. (cid:3) N.B.
Formula (1.2.56) appears as(1.2.60) ( a(cid:93)b )( X ) = (cid:16) Op (cid:0) a (cid:0) X − σ Ξ2 (cid:1) b (cid:17) ( X ) , where Op ( · ) stands for the ordinary quantization in n dimensions.The following classical result is a consequence of Theorem 1.14. Theorem 1.15.
Let us define C ∞ b ( R n ) as the set of bounded smooth complex-valuedfunctions on R n such that all derivatives are bounded. Let a be in C ∞ b ( R n ) ; thenthe operator Op w ( a ) is bounded on L ( R n ) and the B ( L ( R n )) norm of Op w ( a ) isbounded above by a fixed semi-norm of a in the Fréchet space C ∞ b ( R n ) . On the Heisenberg Uncertainty Relations.
Let u ∈ S ( R ) . We have, using thenotations (1.1.5),(1.2.61) (cid:104) D x u, ixu (cid:105) L ( R ) = (cid:104) [ D x , ix ] u, u (cid:105) L ( R ) = 12 π (cid:107) u (cid:107) L ( R ) , implying in particular(1.2.62) (cid:107) D x u (cid:107) L ( R ) (cid:107) xu (cid:107) L ( R ) ≥ π (cid:107) u (cid:107) L ( R ) , which is an equality for u ( x ) = e − πx ; moreover we infer also from (1.2.61) that (cid:104) π ( D x + x ) u, u (cid:105) ≥ (cid:107) u (cid:107) L ( R ) , (1.2.63) NTEGRALS OF THE WIGNER DISTRIBUTION 15 and for(1.2.64) q µ ( x, ξ ) = (cid:88) ≤ j ≤ n µ j ( x j + ξ j ) , ≤ µ ≤ · · · ≤ µ n , the inequality(1.2.65) (cid:104) Op w (cid:0) πq µ ( x, ξ ) (cid:1) u, u (cid:105) L ( R n ) ≥ (cid:107) u (cid:107) L ( R n ) (cid:88) ≤ j ≤ n µ j (cid:124) (cid:123)(cid:122) (cid:125) defined as trace + ( q µ ) , which is an equality for u ( x ) = e − π | x | . Note that the above (optimal) inequality canbe reformulated as(1.2.66) (cid:120) R n πq µ ( x, ξ ) W ( u, u )( x, ξ ) dxdξ ≥ (cid:107) u (cid:107) L ( R n )
12 trace + ( q µ ) . Note also that with the symplectic matrix σ defined by (1.2.22), the so-called fun-damental matrix of q µ is defined by(1.2.67) F q µ = σ − Q µ = (cid:18) − II (cid:19) (cid:18) M M (cid:19) = (cid:18) − MM (cid:19) , with M = diag( µ , . . . , µ n ) , so that(1.2.68) Spectrum F q µ = {± iµ j } ≤ j ≤ n , trace + ( q µ ) = (cid:88) λ eigenvalue of F q µ with Im λ> λ/i. With the notations (cid:40) C j = D x j + ix j , creation operators ,C ∗ j = D x j − ix j , annihilation operators , (1.2.69)we see that π [ C ∗ j , C j ] = π [ D x j − ix j , D x j + ix j ] = I, and(1.2.70) Op w ( q µ ) = π (cid:88) ≤ j ≤ n µ j C j C ∗ j + 12 trace + ( q µ ) , which provides another proof of (1.2.65). Lemma 1.16 (Quantum Mechanics must deal with unbounded operators ) . Let H be a Hilbert space and let J, K ∈ B ( H ) ; then the commutator [ J, K ] (cid:54) = Id .Proof. Let
J, K be bounded operators with [ J, K ] = Id . Then for all N ∈ N ∗ , wehave(1.2.71) [ J, K N ] = N K N − . Thus QM must involve infinite dimensional Hilbert spaces and unbounded operators on them.
Indeed, this is true for N = 1 and if it holds for some N ≥ , we find that [ J, K N +1 ] = J K N K − K N +1 J = [ J, K N ] K + K N J K − K N +1 J = [ J, K N ] K + K N ( J K − KJ ) = [ J, K N ] K + K N = ( N + 1) K N , qed . Note that (1.2.71) implies that for all N ∈ N ∗ , we have K N (cid:54) = 0 : of course K (cid:54) = 0 since [ J, K ] = Id and if we had K N = 0 for some N ≥ , (1.2.71) would imply K N − = 0 and eventually K = 0 . As a consequence, we get from (1.2.71) that forall N ≥ , N (cid:107) K N − (cid:107) B ( H ) ≤ (cid:107) J (cid:107) B ( H ) (cid:107) K N (cid:107) B ( H ) ≤ (cid:107) J (cid:107) B ( H ) (cid:107) K (cid:107) B ( H ) (cid:107) K N − (cid:107) B ( H ) , implying since (cid:107) K N − (cid:107) B ( H ) > , that ∀ N ≥ , N ≤ (cid:107) J (cid:107)(cid:107) K (cid:107) , which is impossible and proves the lemma. (cid:3) Lemma 1.17 (Hardy’s inequality: the study of non-self-adjoint operators may beuseful to determine lowerbounds of self-adjoint operators) . Let n ∈ N , n ≥ ; let u in L ( R n ) such that ∇ u ∈ L ( R n ) , | x | − u ∈ L ( R n ) . Then we have (1.2.72) (cid:107)∇ u (cid:107) L ( R n ) ≥ (cid:0) n − (cid:1) (cid:107)| x | − u (cid:107) L ( R n ) . Proof.
We write first (cid:88) ≤ j ≤ n (cid:107) ( D x j − iφ j ) u (cid:107) L ( R n ) = (cid:104)| D | u, u (cid:105) L ( R n ) + (cid:104)| φ | u, u (cid:105) L ( R n ) − π (cid:104) (div φ ) u, u (cid:105) L ( R n ) , so that with φ ( x ) = νx π | x | , we get the operator inequality | D | + ν π | x | ≥ ν ( n − π | x | , so that − ∆ ≥ | x | − ν ( n − − ν ) (cid:124) (cid:123)(cid:122) (cid:125) largest at ν =( n − / , proving the lemma. (cid:3) N.B.
A modern approach to the Heisenberg Uncertainty Principle should certainlybegin with reading C. Fefferman’s article [6] as well as E. Lieb’s book [26].1.2.6.
Non-negative quantizations formulas.
Lemma 1.18.
Let χ be an even function in S ( R n ) with L ( R n ) norm equal to 1.We define (1.2.73) Γ χ = ¯ χ(cid:93)χ. Then the function Γ χ belongs to S ( R n ) , is real-valued even and is such that (cid:90) R n Γ χ ( X ) dX = 1 . Let u be in L ( R n ) . Then the convolution W ( u, u ) ∗ Γ χ is non-negative. As a result,the operator with Weyl symbol a ∗ Γ χ is a non-negative operator whenever a is anon-negative function. NTEGRALS OF THE WIGNER DISTRIBUTION 17
Proof.
According to the book [21], the composition formula (1.2.54) is bilinear con-tinuous from S ( R n ) into S ( R n ) and we have also a(cid:93)b = ¯ b(cid:93) ¯ a. so that Γ χ is indeed real-valued. Moreover, we have (cid:90) R n Γ χ ( X ) dX = 2 n (cid:121) ( R n ) e − iπ [ X − Y,Y − Z ] ¯ χ ( Y ) χ ( Z ) dY dZdX = (cid:90) | χ ( Y ) | dY = 1 , and Γ χ ( − X ) = 2 n (cid:120) R n × R n e − iπ [ − X − Y, − X − Z ] ¯ χ ( Y ) χ ( Z ) dY dZ = 2 n (cid:120) R n × R n e − iπ [ − X + Y, − X + Z ] ¯ χ ( Y ) χ ( Z ) dY dZ = Γ χ ( X ) . We have also (cid:0) W ( u, u ) ∗ Γ χ (cid:1) ( Y ) = (cid:90) R n W ( u, u )( Y − X )Γ χ ( X ) dX = (cid:90) R n W ( u, u )( Y + X )Γ χ ( X ) dX = (cid:90) R n W ( u, u )( T Y ( X ))Γ χ ( X ) dX = (cid:90) R n W ( τ − Y u, τ − Y u )( X )Γ χ ( X ) dX = (cid:90) R n W ( τ − Y u, τ − Y u )( X )( ¯ χ(cid:93)χ )( X ) dX = (cid:104) Op w ( ¯ χ(cid:93)χ ) τ − Y u, τ − Y u (cid:105) L ( R n ) = (cid:107) Op w ( χ ) τ − Y u (cid:107) L ( R n ) ≥ , proving the first statement of non-negativity. Let a be a non-negative function, sayin L ( R n ) ; we haveOp w ( a ∗ Γ χ ) = 2 n (cid:120) a ( Y )Γ χ ( X − Y ) σ X dY dX = (cid:90) a ( Y ) (cid:90) ( ¯ χ(cid:93)χ )( X − Y )2 n σ X dXdY = (cid:90) a ( Y ) (cid:90) ( ¯ χ(cid:93)χ )( T − Y ( X ))2 n σ X dXdY = (cid:90) a ( Y ) τ Y Op w ( ¯ χ(cid:93)χ ) τ − Y dY = (cid:90) a ( Y ) τ Y Op w ( ¯ χ ) Op w ( χ ) τ − Y dY = (cid:90) a ( Y ) (cid:2) Op w ( χ ) τ − Y (cid:3) ∗ (cid:2) Op w ( χ ) τ − Y (cid:3)(cid:124) (cid:123)(cid:122) (cid:125) non-negative operator dY ≥ , if a ( Y ) ≥ for all Y ∈ R n and this concludes the proof. (cid:3) We can write as well(1.2.74)Op w ( a ∗ Γ χ ) = (cid:90) R n a ( Y ) (cid:2) τ Y Op w ( χ ) τ − Y (cid:3) ∗ (cid:2) τ Y Op w ( χ ) τ − Y (cid:3) dY = (cid:90) R n a ( Y )Σ χ ( Y ) dY, with(1.2.75) Σ χ ( Y ) = (cid:2) τ Y Op w ( χ ) τ − Y (cid:3) ∗ (cid:2) τ Y Op w ( χ ) τ − Y (cid:3) = (cid:0) Op w ( χ ( · − Y )) (cid:1) ∗ Op w ( χ ( · − Y )) . Remark 1.19.
The Gaussian case in the previous lemma gives rise to the standardnon-negativity properties of Coherent States. In fact choosing χ ( X ) = 2 n e − π | X | ,we see that χ is even, belongs to the Schwartz space and (cid:107) χ (cid:107) L ( R n ) = 2 n (cid:90) R n e − π | X | dX = 2 n − n/ = 1 . We have also Γ χ ( X ) = 2 n (cid:120) ( R n ) e − iπ [ X − Y,X − Z ] e − π ( | Y | + | Z | ) dY dZ = 2 n (cid:90) R n e iπ [ Y,X ] e − π ( | X + Y | + | Y | ) dY = 2 n (cid:90) R n e iπ [ Y,X ] e − π ( | Y + X | + | Y − X | ) dY = 2 n e − π | X | (cid:90) R n e iπ [ Y,X ] e − π | Y | dY = 2 n e − π | X | − n e − π | X | = χ ( X ) . In that case we find that Op w ( χ ) is a rank-one orthogonal projection on the funda-mental state Ψ of the Harmonic Oscillator π ( | D x | + | x | ) . According to (9.1.31)the one-dimensional k -th Hermite function is(1.2.76) ψ k ( x ) = ( − k k √ k ! 2 / e πx (cid:18) d √ πdx (cid:19) k ( e − πx ) , so that Ψ ( x ) = 2 n/ e − π | x | . We calculate Γ( x, ξ ) = W (Ψ , Ψ )( x, ξ ) = 2 n/ (cid:90) R n e − π ( | x + z/ | + | x − z/ | ) e − iπzξ dz = 2 n/ e − π | x | (cid:90) R n e − πz / e − iπzξ dz = 2 n e − π | x | e − π | ξ | = χ ( x, ξ ) . The anti-Wick quantization of a symbol a is defined as (see e.g. M. Shubin’s book[32])(1.2.77) Op aw ( a ) = (cid:90) R n a ( Y )Σ Y dY, where Σ Y is the rank-one orthogonal projection given by(1.2.78) Σ y,η u = (cid:104) u, τ y,η Ψ (cid:105) τ y,η Ψ . Remark 1.20.
It is interesting to notice that to produce non-negativity of theoperator with Weyl symbol a ∗ Γ χ when a is a non-negative function, we do notuse the non-negativity of Γ χ as a function, which by the way does not always hold(except in the Gaussian cases), but we use the fact that the quantization of Γ χ isnon-negative, as it is defined as Op w ( ¯ χ(cid:93)χ ) = ( Op w ( χ )) ∗ Op w ( χ ) . Proposition 4.1.1 in [21] is useful to compute the Fourier transform of Gaussian functions andis a notable asset of the Fourier normalization given in footnote 1 page 2.
NTEGRALS OF THE WIGNER DISTRIBUTION 19
Remark 1.21.
Another important remark is concerned with the Taylor expansionof a ∗ Γ χ : we have ( a ∗ Γ χ )( X ) = (cid:90) a ( X − Y )Γ χ ( Y ) dY = (cid:90) a ( X + Y )Γ χ ( Y ) dY = (cid:90) (cid:16) a ( X ) + a (cid:48) ( X ) Y + (cid:90) (1 − θ ) a (cid:48)(cid:48) ( X + θY ) Y (cid:17) Γ χ ( Y ) dY = a ( X ) + (cid:90) (cid:90) (1 − θ ) a (cid:48)(cid:48) ( X + θY ) Y Γ χ ( Y ) dY. As a result the difference ( a ∗ Γ χ ) − a depends only on the second derivative of a . Iffor instance a is a semi-classical symbol, i.e. a smooth function of ( x, ξ ) dependingon h ∈ (0 , such that(1.2.79) ∀ ( α, β ) ∈ N n × N n , sup ( x,ξ ) ∈ R n ,h ∈ (0 , | ( ∂ αx ∂ βξ a )( x, ξ, h ) | h − | α | + | β | < + ∞ . then the difference Op aw ( a ) − Op w ( a ) is bounded on L ( R n ) with an O ( h ) operator-norm, so that if a happens also to be non-negative, we findOp w ( a ) = Op w ( a ) − Op w ( a ∗ Γ χ ) (cid:124) (cid:123)(cid:122) (cid:125) O ( h ) as an operator,cf. Theorem 1.14 + Op w ( a ∗ Γ χ ) (cid:124) (cid:123)(cid:122) (cid:125) ≥ as an operator , and we obtain a version of the so-called Sharp Gårding Inequality,(1.2.80) Op w ( a ) + Ch ≥ (as an operator) . Theorem 1.22.
Let χ be an even function in the Schwartz space S ( R n ) with L ( R n ) norm equal to 1 and let Γ χ be given by (1.2.73) . For a ∈ L ∞ ( R n ) , wedefine (1.2.81) Op ( χ, a ) = Op w ( a ∗ Γ χ ) . Then Op ( χ, a ) is a bounded operator in L ( R n ) and we have (1.2.82) (cid:107) Op ( χ, a ) (cid:107) B ( L ( R n )) ≤ (cid:107) a (cid:107) L ∞ ( R n ) . Moreover, if a is valued in some interval J of the real line, we have the operatorinequalities (1.2.83) inf J ≤ Op ( χ, a ) ≤ sup J. In particular if a ( x, ξ ) ≥ for all ( x, ξ ) ∈ R n , we have the operator-inequalityOp ( χ, a ) ≥ . N.B.
The non-negativity of the anti-Wick quantization (1.2.77) and its avatarsUsimi, Coherent States are particular cases of the above theorem. More informationon this topic is available in Section 2.4 of the book [21]. Another remark is thatthis result can easily be extended to matrix-valued symbols as in Remark 2 page 79of L. Hörmander’s [17] and even to symbols valued in B ( H ) , where H is a Hilbertspace. Proof.
We start with Formulas (1.2.74), (1.2.75), entailingOp ( χ, a ) = (cid:90) R n a ( Y )Σ χ ( Y ) dY, with Σ χ ( Y ) = (cid:2) Op w ( χ ( · − Y )) (cid:3) ∗ Op w ( χ ( · − Y )) = τ Y Op w ( ¯ χ(cid:93)χ ) τ − Y . We note thatOp ( χ,
1) = (cid:90) R n τ Y Op w ( ¯ χ(cid:93)χ ) τ − Y dY, so has Weyl symbol X (cid:55)→ (cid:82) R n Γ χ ( X − Y ) dY = 1 from Lemma 1.18 and thusOp ( χ,
1) = Id . We infer that for u, v ∈ S ( R n ) , (cid:104) Op ( χ, a ) u, v (cid:105) L ( R n ) = (cid:90) R n a ( Y ) (cid:104) Op w ( χ ( · − Y )) u, Op w ( χ ( · − Y )) v (cid:105) dY, so that with any ν > , |(cid:104) Op ( χ, a ) u, v (cid:105) L ( R n ) |≤ (cid:107) a (cid:107) L ∞ ( R n ) (cid:90) R n (cid:0) ν (cid:107) Op w ( χ ( · − Y )) u (cid:107) L ( R n ) + ν − (cid:107) Op w ( χ ( · − Y )) v (cid:107) L ( R n ) (cid:1) dY = (cid:107) a (cid:107) L ∞ ( R n ) (cid:0) ν (cid:104) Op ( χ, u, u (cid:105) L ( R n ) + ν − (cid:104) Op ( χ, v, v (cid:105) L ( R n ) (cid:1) = (cid:107) a (cid:107) L ∞ ( R n ) (cid:0) ν (cid:107) u (cid:107) L ( R n ) + ν − (cid:107) v (cid:107) L ( R n ) (cid:1) , and taking the infimum of the right-hand-side with respect to ν , we obtain |(cid:104) Op ( χ, a ) u, v (cid:105) L ( R n ) | ≤ (cid:107) a (cid:107) L ∞ ( R n ) (cid:107) u (cid:107) L ( R n ) (cid:107) v (cid:107) L ( R n ) , proving (1.2.82). To prove (1.2.83), it is enough to prove the last statement in thetheorem which follows immediately from (1.2.74), (1.2.75) since each operator Σ Y isnon-negative. The proof of the theorem is complete. (cid:3) It is nice to have examples of non-negative quantizations, but somehow more im-portantly, it is crucial to relate these quantizations to the mainstream quantization,that is to the Weyl quantization. This is what we do in the next theorem, dealingwith semi-classical symbols.
Theorem 1.23 (Sharp Gårding Inequality) . Let a be a function defined on R n × R n × (0 , such that a ( x, ξ, h ) is smooth for all h ∈ (0 , and such that (1.2.84) ∀ ( α, β ) ∈ N n × N n , sup ( x,ξ,h ) ∈ R n × R n × (0 , | ( ∂ αx ∂ βξ a )( x, ξ, h ) | h −| β | < + ∞ . Let us assume that the function a is valued in R + . Then, there exists a constant C such that (1.2.85) Op w ( a ) + Ch ≥ . Proof.
We have given a proof of this result in Remark 1.21 but with a differentdefinition for a semi-classical symbol (see (1.2.79)). Starting with our definitionabove in (1.2.84), we define(1.2.86) b ( x, ξ, h ) = a ( h / x, h − / ξ, h ) NTEGRALS OF THE WIGNER DISTRIBUTION 21 and we see that b satisfies the estimates (1.2.79) and is a non-negative function sothat, applying Remark 1.21, we can find a constant C such thatOp w ( b ) + Ch ≥ . We note now that Segal’s formula (1.2.26) applied to the symplectic mapping ( x, ξ ) (cid:55)→ ( h / x, h − / ξ ) , shows that Op w ( b ) is unitarily equivalent to Op w ( a ) , providing the sought result. (cid:3) N.B.
Several versions of the above theorem can be found in the literature, in par-ticular Theorem 18.1.14 in [17]. The first proof of this result was given in 1966 byL. Hörmander in [14] for scalar-valued symbols and a proof for systems was givenby P. Lax & L. Nirenberg in [19] on the same year. Far-reaching refinements ofthat inequality were given by C. Fefferman & D.H. Phong, who proved in 1978 in[7] that, under the same assumption as in Theorem 1.23 for scalar-valued symbols,they obtain the much stronger(1.2.87) Op w ( a ) + Ch ≥ . A thorough discussion of these questions is given in Section 18.6 of [17] and in Section2.5 of [21] (see also [1]).1.3.
Examples.
Hermite functions.
We can easily calculate the Wigner distribution of Her-mite functions and since the Wigner distributions respect tensor products as partialFourier transforms, it is enough to do so in one dimension. With ψ k given in (1.2.76),the Wigner distribution W ( ψ k , ψ k ) appears as the Weyl symbol of P k ;1 = P k as de-fined in (9.1.32). We find that the Weyl symbol of P n , following (9.2.3), is n e − π ( | x | + | ξ | ) . More generally, the paper [18] provides in one dimension(1.3.1) W ( ψ k , ψ k )( x, ξ ) = ( − k e − π ( x + ξ ) L k (cid:0) π ( x + ξ ) (cid:1) , where L k is the standard Laguerre polynomial with degree k (see (9.3.1)). As aresult, the Weyl symbol of P k ; n is equal to π k,n ( x, ξ ) with(1.3.2) π k,n ( x, ξ ) = ( − k n e − π ( | x | + | ξ | ) (cid:88) α ∈ N n , | α | = k (cid:89) ≤ j ≤ n L α j (cid:0) π ( x j + ξ j ) (cid:1) . Note that the leading term in the polynomial ( − k L k ( t ) is t k /k ! and this impliesthat the set { ( x, ξ ) ∈ R , W ( ψ k , ψ k )( x, ξ ) < } where W ( ψ k , ψ k ) is given by (1.3.1) is a relatively compact open subset of R . One-sided exponentials.
Let us define for a > ,(1.3.3) f a ( t ) = H ( t ) a / e − at/ . We have W ( f a , f a )( x, ξ ) = aH ( x ) (cid:90) | z |≤ x e − iπzξ e − a ( x + z/ e − a ( x − z/ dz = aH ( x ) e − xa (cid:90) | z |≤ x e − iπzξ dz = 2 aH ( x ) e − xa (cid:90) x cos ( z πξ ) dz = aH ( x ) e − xa sin (4 πxξ ) πξ . (1.3.4)We can check (cid:120) W ( f a , f a )( x, ξ ) dxdξ = aπ (cid:90) + ∞ x =0 e − ax (cid:90) sin (4 πxξ ) ξ dξdx = 1 = (cid:107) f a (cid:107) L ( R ) , and since(1.3.5) (cid:90) R sin tt dt = π, we verify (see Lemma 1.13 and (1.1.6)), (cid:120) W ( f a , f a )( x, ξ ) dxdξ = a π (cid:90) + ∞ x =0 e − ax (cid:90) sin (4 πxξ ) ξ dξdx = 1 = (cid:107) f a (cid:107) L ( R ) . On the other hand, the ambiguity function A ( f a , f a ) is the inverse Fourier transformof W and we have A ( f a , f a )( η, y ) = aπ (cid:120) H ( x ) e − x ( a − iπη ) sin ξξ e iπ y πx ξ dxdξ = a (cid:90) + ∞| y | / e − x ( a − iπη ) dx = ae − | y | ( a − iπη ) a − iπη , which corresponds to Formula (9) in [12] noting that with our notations, we have A ( f, f )( η, y ) = (cid:101) A ( f, f )( y, − η ) , where (cid:101) A ( f, f ) is the normalization chosen in [12]. Going back to the Wigner distri-bution, that simple example is interesting since we have (cid:8) ( x, ξ ) , W ( f a , f a )( x, ξ ) < (cid:9) = ∪ k ∈ N (cid:8) ( x, ξ ) ∈ (0 , + ∞ ) × R ∗ , k < x | ξ | < k (cid:9) , and we see that the Lebesgue measure of E k = (cid:8) ( x, ξ ) ∈ (0 , + ∞ ) × R ∗ , k < x | ξ | < k (cid:9) , is infinite since | E k | = 2 (cid:90) + ∞ dx x = + ∞ . NTEGRALS OF THE WIGNER DISTRIBUTION 23
Moreover the function W ( f a , f a )( x, ξ ) does not belong to L ( R ) since (cid:120) H ( x ) e − xa (cid:12)(cid:12)(cid:12)(cid:12) sin (4 πxξ ) πξ (cid:12)(cid:12)(cid:12)(cid:12) dxdξ ≥ (cid:120) (0 , + ∞ ) e − xa (cid:12)(cid:12)(cid:12)(cid:12) sin ηπη (cid:12)(cid:12)(cid:12)(cid:12) dxdη = + ∞ . As a consequence, we have, using the notation for α ∈ R ,(1.3.6) α ± = max( ± α, , (1.3.7) (cid:120) (cid:0) W ( f a , f a )( x, ξ ) (cid:1) + dxdξ = (cid:120) (cid:0) W ( f a , f a )( x, ξ ) (cid:1) − dxdξ = + ∞ , since the real-valued function W ( f a , f a ) does not belong to L ( R ) and is such that (cid:120) W ( f a , f a )( x, ξ ) dxdξ = (cid:107) f a (cid:107) L ( R ) = 1 . We shall see in Section 6.4 several important consequences of that phenomenon forthe quantization of the indicatrix of some subsets of R , such as(1.3.8) E ± = (cid:8) ( x, ξ ) , ±W ( f a , f a )( x, ξ ) > (cid:9) . Box functions.
We start with(1.3.9) β ( t ) = [ − , ] ( t ) , for which a straightforward calculation gives(1.3.10) W ( β , β )( x, ξ ) = [ − , ] ( x ) sin (cid:0) π (1 − | x | ) ξ (cid:1) πξ . More generally for real parameters a ≤ b , defining β = ( b − a ) − / [ a,b ] ( x ) e iπωx , we find W ( β, β )( x, ξ ) = [( b − a ) π ( ξ − ω )] − (cid:16) [ a, a + b ] ( x ) sin[4 π ( ξ − ω )( x − a )]+ [ a + b ,b ] ( x ) sin[4 π ( ξ − ω )( b − x )] (cid:17) . Checking now(1.3.11) β ( t ) = [ − , ] ( t ) sign t, we find after a simple (but this time a bit tedious) calculation(1.3.12) W ( β , β )( x, ξ ) = (cid:0) | x | ≤ (cid:1) π | x | ξ ) − sin (cid:0) π (1 − | x | ) ξ (cid:1) πξ + (cid:0) ≤ | x | ≤ (cid:1) sin (cid:0) π (1 − | x | ) ξ (cid:1) πξ . Integrals of the Wigner distribution on subsets of the phase space.Lemma 1.24.
Let E be a measurable subset with finite Lebesgue measure of thephase space R n × R n and let E be the indicatrix function of the set E . Thenthe operator with Weyl symbol E is bounded self-adjoint on L ( R n ) and for any u ∈ L ( R n ) , we have (1.4.1) (cid:104) Op w ( E ) u, u (cid:105) L ( R n ) = (cid:120) E W ( u, u )( x, ξ ) dxdξ. Proof.
It follows immediately from (1.2.2) and (1.2.5). (cid:3)
We shall see below several examples where the operator Op w ( E ) is bounded on L ( R n ) with an E having infinite Lebesgue measure. We may note in particular thatOp w ( R n ) = Id , and for a given non-zero linear form L ( x, ξ ) on R n and(1.4.2) E = { ( x, ξ ) ∈ R n , L ( x, ξ ) ∈ J } , where J is a subset of R , we may find affine symplectic coordinates ( y, η ) on R n such that L ( x, ξ ) = y , imply-ing with (1.2.26) that Op w ( E ) is unitarily equivalent to the orthogonal projection u (cid:55)→ u ( y ) J ( y ) . Although in that case, the quantization of the indicatrix of E given by (1.4.2) istrivial, we shall see below that in many cases, including some rather explicit ones, theWeyl quantization of the rough Hamiltonian E ( x, ξ ) could be far from a projectionand may have a rather complicated spectrum with a supremum which could bestrictly larger than 1 and an infimum which could be negative.In some sense, although we have the trivial identity E ( x, ξ ) = E ( x, ξ ) , we shallsee that the quantization process by the Weyl formula is destroying that property;to understand integrals of the Wigner distribution on subsets of the phase space,Formula (1.4.1) forces us to consider the Weyl quantization of the function E ( x, ξ ) and the Heisenberg Uncertainty Principle shows that non-commutation propertiesare governing operators and these properties are of course distorting the classicalidentities satisfied by classical Hamiltonians.We must point out as well that we do not have here at our disposal a semi-classical version of our quantization which could ensure some bridge between classicalproperties and operator-theoretic results as it is the case for the quantization ofnice smooth semi-classical symbols depending on a small parameter h such as a C ∞ function a ( x, ξ, h ) satisfying (1.2.84). In particular for a symbol a satisfying(1.2.84), we have the following result: if for all ( x, ξ, h ) ∈ R n × R n × (0 , we have a ( x, ξ, h ) ≤ , then there exists a semi-norm C of the symbol a such that(1.4.3) Id − Op w ( a ) + Ch ≥ i.e. Op w ( a ) ≤ Id + Ch , an inequality following from the Fefferman-Phong Inequality (cf. (1.2.87)) which im-plies as well the following lemma. NTEGRALS OF THE WIGNER DISTRIBUTION 25
Lemma 1.25.
Let a be a semi-classical symbol of order 0, i.e. a smooth functionsatisfying (1.2.84) such that for all ( x, ξ, h ) ∈ R n × R n × (0 , we have ≤ a ( x, ξ, h ) ≤ . Then there exists a semi-norm C of the symbol a such that − Ch ≤ Op w ( a ) ≤ Id + Ch . Quantization of radial functions and Mehler’s formula
This section and the following are essentially based upon the author’s paper [24].2.1.
Basic formulas in one dimension.
In this section, we work in one dimensionand consider a function F in the Schwartz class of R . We want to calculate somewhatexplicitly the Weyl quantization of F ( x + ξ ) and also extend that computation tothe case where F is merely L ∞ ( R ) . We have, say for F in the Wiener algebra W ( R ) = Fourier (cid:0) L ( R ) (cid:1) ,Op w ( F ( x + ξ )) = (cid:90) R ˆ F ( τ ) Op w ( e iπτ ( x + ξ ) ) dτ, as an absolutely converging integral of a function defined on R (equipped with theLebesgue measure) valued in B ( L ( R )) (bounded endomorphisms of L ( R ) ). In factapplying Mehler’s Formula (9.2.2), we findOp w ( e iπτ ( x + ξ ) ) (cid:124) (cid:123)(cid:122) (cid:125) operator with Weyl symbol e iπτ ( x ξ = cos(arctan τ ) e iπ (arctan τ ) Op w ( x + ξ ) (cid:124) (cid:123)(cid:122) (cid:125) exponential e iM , with M self-adjoint operator =2 π (arctan τ ) Op w ( x + ξ ) , so that, using the spectral decomposition (9.1.32) of the Harmonic OscillatorOp w ( π ( x + ξ )) , we get Op w ( F ( x + ξ )) = (cid:90) R ˆ F ( τ ) (cid:88) k ≥ e i (arctan τ )( k + ) P k dτ √ τ = (cid:88) k ≥ (cid:90) R ˆ F ( τ ) e i ( k + ) arctan τ dτ √ τ P k , where the use of Fubini theorem is justified by (cid:90) R | ˆ F ( τ ) | dτ √ τ < + ∞ , P k ≥ , (cid:88) k ≥ P k = Id . We have (cid:90) R ˆ F ( τ ) e i ( k + ) arctan τ dτ √ τ = (cid:90) R ˆ F ( τ ) (cid:0) cos(arctan τ (cid:1) + i sin(arctan τ ) (cid:1) k +1 dτ √ τ , and, using Section 9.7.1, we get (cid:90) R ˆ F ( τ ) e i ( k + ) arctan τ dτ √ τ = (cid:90) R ˆ F ( τ ) (cid:0) iτ (cid:1) k +1 dτ (1 + τ ) k +1 . We have proven the following lemma.
Lemma 2.1.
Let F be a tempered distribution on R such that ˆ F is locally integrableand such that (2.1.1) (cid:90) R | ˆ F ( τ ) | dτ √ τ < + ∞ . Then the operator Op w ( F ( x + ξ )) has the spectral decomposition Op w ( F ( x + ξ )) = (cid:88) k ≥ (cid:90) R ˆ F ( τ )(1 + iτ ) k +1 (1 + τ ) k +1 dτ P k (2.1.2) = (cid:88) k ≥ (cid:90) R ˆ F ( τ )(1 + iτ ) k (1 − iτ ) k +1 dτ P k , (2.1.3) where the orthogonal projections P k are defined in (9.1.32) . Higher dimensional questions.
We work now in n dimensions and consider afunction F in the Schwartz class of R . We want to calculate somewhat explicitly theWeyl quantization of F (cid:0)(cid:80) ≤ j ≤ n µ j ( x j + ξ j ) (cid:1) , where the µ j are positive parameters,denoted by Op w (cid:0) F ( (cid:88) ≤ j ≤ n µ j ( x j + ξ j )) (cid:1) , q µ ( x, ξ ) = (cid:88) ≤ j ≤ n µ j ( x j + ξ j ) , and also extend that computation to the case where F is merely L ∞ ( R ) . We have,say for F in the Wiener algebra W ( R ) = Fourier (cid:0) L ( R ) (cid:1) ,Op w (cid:0) F (cid:0) q µ ( x, ξ ) (cid:1)(cid:1) = (cid:90) R ˆ F ( τ ) Op w (cid:0) e iπτ (cid:80) ≤ j ≤ n µ j ( x j + ξ j ) (cid:1) dτ, as an absolutely converging integral of a function defined on R (equipped with theLebesgue measure) valued in B ( L ( R n )) (bounded endomorphisms of L ( R n ) ). Infact applying Mehler’s Formula (9.2.2), we find by tensorisation,(2.2.1)Op w (cid:0) e iπτ (cid:80) ≤ j ≤ n µ j ( x j + ξ j ) (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) operator with Weyl symbol e iπτqµ ( x,ξ ) = (cid:89) ≤ j ≤ n cos(arctan( τ µ j )) e iπ (arctan( τµ j )) Op w ( x j + ξ j ) (cid:124) (cid:123)(cid:122) (cid:125) exponential e iMj , with M j self-adjoint operator =2 π (arctan( τµ j )) Op w ( x j + ξ j ) , so that, using the spectral decomposition (9.1.36) of the Harmonic Oscillator we getOp w (cid:0) F ( q µ ( x, ξ )) (cid:1) = (cid:90) R ˆ F ( τ ) (cid:88) α ∈ N n (cid:89) ≤ j ≤ n e i (arctan( τµ j ))( α j + ) P α j (cid:112) τ µ j ) dτ = (cid:88) α ∈ N n (cid:90) R ˆ F ( τ ) (cid:89) ≤ j ≤ n e i ( α j + ) arctan( τµ j ) (cid:112) τ µ j ) dτ P α , NTEGRALS OF THE WIGNER DISTRIBUTION 27 where the use of Fubini theorem is justified by (cid:90) R | ˆ F ( τ ) | dτ √ τ < + ∞ , P α ≥ , (cid:88) α P α = Id . We have (cid:90) R ˆ F ( τ ) (cid:89) ≤ j ≤ n e i ( α j + ) arctan( τµ j ) (cid:112) τ µ j ) dτ = (cid:90) R ˆ F ( τ ) (cid:89) ≤ j ≤ n (cid:0) cos(arctan( µ j τ ) (cid:1) + i sin(arctan( µ j τ )) (cid:1) α j +1 (cid:112) τ µ j ) dτ and, using Section 9.7.1, we get (cid:90) R ˆ F ( τ ) (cid:89) ≤ j ≤ n e i ( α j + ) arctan( τµ j ) (cid:112) τ µ j ) dτ = (cid:90) R ˆ F ( τ ) (cid:89) ≤ j ≤ n (1 + iτ µ j ) α j +1 (1 + ( τ µ j ) ) α j + (cid:112) τ µ j ) dτ We have proven the following lemma.
Lemma 2.2.
Let F be a tempered distribution on R such that ˆ F is locally integrableand such that (2.2.2) (cid:90) R | ˆ F ( τ ) | dτ √ τ < + ∞ . Then the operator Op w (cid:0) F ( (cid:80) ≤ j ≤ n µ j ( x j + ξ j )) (cid:1) has the spectral decomposition (2.2.3) Op w (cid:0) F (cid:0) (cid:88) ≤ j ≤ n µ j ( x j + ξ j ) (cid:1)(cid:1) = (cid:88) α ∈ N n (cid:90) R ˆ F ( τ ) (cid:89) ≤ j ≤ n (1 + iτ µ j ) α j +1 (1 + τ µ j ) α j +1 dτ P α = (cid:88) α ∈ N n (cid:90) R ˆ F ( τ ) (cid:89) ≤ j ≤ n (1 + iτ µ j ) α j (1 − iτ µ j ) α j +1 dτ P α , where P α is the rank-one orthogonal projection onto Ψ α given by (9.1.33) . Lemma 2.3.
Let F be as in Lemma 2.3 and let us assume that all the µ j are equalto µ (positive). Then (2.2.4) Op w (cid:0) F (cid:0) µ (cid:88) ≤ j ≤ n ( x j + ξ j ) (cid:1)(cid:1) = (cid:88) k ≥ (cid:90) R ˆ F ( τ ) (1 + iτ µ ) k (1 − iτ µ ) k + n dτ P k ; n , with (2.2.5) P k ; n = (cid:88) α ∈ N n | α | = k P α , where P α is the rank-one orthogonal projection onto Ψ α given by (9.1.33) .Proof. With all the µ j equal to µ > , we find (cid:89) ≤ j ≤ n (1 + iτ µ j ) α j (1 − iτ µ j ) α j +1 = (cid:89) ≤ j ≤ n (1 + iτ µ ) α j (1 − iτ µ ) α j +1 = (1 + iτ µ ) | α | (1 − iτ µ ) | α | + n , which depends only on | α | , so that applying the previous lemma gives (cid:16) F (cid:0) µ (cid:88) ≤ j ≤ n ( x j + ξ j ) (cid:1)(cid:17) w = (cid:88) k ≥ (cid:90) R ˆ F ( τ ) (1 + iτ µ ) k (1 − iτ µ ) k + n dτ P k ; n , giving the sought result. (cid:3) Conics with eccentricity smaller than 1
Indicatrix of a disc.
Let us assume now that, with some a ≥ , F = [ − a π , a π ] , so that F ( x + ξ ) = { π ( x + ξ ) ≤ a } . According to Section 9.7.3, we have ˆ F ( τ ) = sin aτπτ , so that (2.1.1) holds true. We findin this case,(3.1.1) Op w ( F ( x + ξ )) = (cid:88) k ≥ F k ( a ) P k , F k ( a ) = (cid:90) R sin aτπτ (1 + iτ ) k (1 − iτ ) k +1 dτ, so that (note that F k ( a ) is real-valued since F is real-valued and thus the opera-tor Op w ( F ( x + ξ )) is self-adjoint), and for a > , using the result (9.7.4) of theAppendix page 146, we obtain F (cid:48) k ( a ) = 1 π (cid:90) R cos aτ (1 + iτ ) k (1 − iτ ) k +1 dτ = 12 π (cid:90) R e iaτ (cid:26) (1 + iτ ) k (1 − iτ ) k +1 + (1 − iτ ) k (1 + iτ ) k +1 (cid:27) dτ = 12 π (cid:90) R e iaτ (cid:26) i k ( τ − i ) k ( − i ) k +1 ( τ + i ) k +1 + ( − i ) k ( τ + i ) k i k +1 ( τ − i ) k +1 (cid:27) dτ = ( − k iπ (cid:90) R e iaτ (cid:26) − ( τ − i ) k ( τ + i ) k +1 + ( τ + i ) k ( τ − i ) k +1 (cid:27) dτ. We shall now calculate explicitly both integrals above: let < R be given and letus consider the closed path(3.1.2) γ R = [ − R, R ] ∪ { Re iθ } ≤ θ ≤ π (cid:124) (cid:123)(cid:122) (cid:125) γ R . We have iπ (cid:90) γ R e iaτ (cid:26) − ( τ − i ) k ( τ + i ) k +1 + ( τ + i ) k ( τ − i ) k +1 (cid:27) dτ = Res ( e iaτ ( τ + i ) k ( τ − i ) k +1 ; i )= 1 k ! ( ddτ ) k (cid:8) e iaτ ( τ + i ) k (cid:9) | τ = i , and we note that, for a > , lim R → + ∞ (cid:90) γ R e iaτ (cid:26) − ( τ − i ) k ( τ + i ) k +1 + ( τ + i ) k ( τ − i ) k +1 (cid:27) dτ = 0 , NTEGRALS OF THE WIGNER DISTRIBUTION 29 - - Figure 1. γ R = [ − R, R ] ∪ { Re iθ } ≤ θ ≤ π since for R ≥ , (cid:90) π | e iaRe iθ | (cid:12)(cid:12)(cid:12)(cid:12) − ( Re iθ − i ) k ( Re iθ + i ) k +1 + ( Re iθ + i ) k ( Re iθ − i ) k +1 (cid:12)(cid:12)(cid:12)(cid:12) | iRe iθ | dθ ≤ (cid:90) π e − aR sin θ (cid:12)(cid:12)(cid:12)(cid:12) − ( e iθ − iR − ) k ( e iθ + iR − ) k +1 + ( e iθ + iR − ) k ( e iθ − iR − ) k +1 (cid:12)(cid:12)(cid:12)(cid:12) dθ ≤ (cid:90) π e − aR sin θ dθ sup ≤ ρ ≤ / (cid:26) (1 + ρ ) k (1 − ρ ) k +1 + (1 + ρ ) k (1 − ρ ) k +1 (cid:27) . For a > , we obtain lim R → + ∞ (cid:82) π e − aR sin θ dθ = 0 by dominated convergence. As aresult, we get F (cid:48) k ( a ) = ( − k k ! ( ddτ ) k (cid:8) e iaτ ( τ + i ) k (cid:9) | τ = i = ( − k k ! ( d ia d(cid:15) ) k (cid:8) e − a − (cid:15) ( i + i (cid:15)a + i ) k (cid:9) | (cid:15) =0 , that is F (cid:48) k ( a ) = ( − k k ! e − a ( dd(cid:15) ) k (cid:8) e − (cid:15) (2 a + (cid:15) ) k (cid:9) | (cid:15) =0 . We note that F (cid:48) k belongs to L ( R + ) as the product of e − a by a polynomial. We havealso that(3.1.3) lim a → + ∞ F k ( a ) = 1 (see the Appendix page 147),and this yields F k ( a ) = 1 + (cid:90) a + ∞ F (cid:48) k ( b ) db = 1 − (cid:90) + ∞ a ( − k k ! e − b ( dd(cid:15) ) k (cid:8) e − (cid:15) (2 b + (cid:15) ) k (cid:9) | (cid:15) =0 db, so that(3.1.4) F k ( a ) = 1 − e − a P k ( a ) , with(3.1.5) P k ( a ) = ( − k k ! (cid:90) + ∞ e − t ( dd(cid:15) ) k (cid:8) e − (cid:15) ( a + t + (cid:15) ) k (cid:9) | (cid:15) =0 dt = ( − k k ! (cid:90) + ∞ e t ( dd(cid:15) ) k (cid:8) e − (cid:15) − t ( a + t + (cid:15) ) k (cid:9) | (cid:15) =0 dt = ( − k k ! (cid:90) + ∞ e t ( ddt ) k (cid:8) e − t ( a + t ) k (cid:9) dt. We see that P k is a polynomial with leading monomial k a k k ! (by a direct computa-tion) and P k (0) = 1 (since F k (0) = 1 − P k (0) ) and moreover, using Laguerrepolynomials (see e.g. (9.3.1) in our Section 9.3), we obtain P k ( a ) = ( − k k ! (cid:90) + ∞ e − t e t +2 a ( d dt ) k (cid:8) e − t − a (2 a + 2 t ) k (cid:9) dt (3.1.6) = ( − k (cid:90) + ∞ e − t L k (2 t + 2 a ) dt, (3.1.7)and this gives in particular(3.1.8) P (cid:48) k ( a ) = ( − k (cid:90) + ∞ e − t L (cid:48) k (2 t + 2 a ) dt = ( − k (cid:110) [ e − t L k (2 t + 2 a )] t =+ ∞ t =0 + (cid:90) + ∞ e − t L k (2 t + 2 a ) dt (cid:111) = ( − k +1 L k (2 a ) + P k ( a ) . Moreover we have from (3.1.5), for k ≥ , P (cid:48) k ( a ) = ( − k k ! (cid:90) + ∞ e t ( ddt ) k (cid:8) e − t k ( a + t ) k − (cid:9) dt = ( − k k ! (cid:90) + ∞ e t ddt ( ddt ) k − (cid:8) e − t k ( a + t ) k − (cid:9) dt = ( − k k ! (cid:110)(cid:2) e t ( ddt ) k − (cid:8) e − t k ( a + t ) k − (cid:9)(cid:3) t =+ ∞ t =0 − (cid:90) + ∞ e t ( ddt ) k − (cid:8) e − t k ( a + t ) k − (cid:9) dt (cid:111) = ( − k − ( k − ddt ) k − (cid:8) e − t ( a + t ) k − (cid:9) | t =0 + ( − k − ( k − (cid:90) + ∞ e t ( ddt ) k − (cid:8) e − t ( a + t ) k − (cid:9) dt = ( − k − ( k − e t +2 a ( d dt ) k − (cid:8) e − t − a (2 a + 2 t ) k − (cid:9) | t =0 + ( − k − ( k − (cid:90) + ∞ e t ( ddt ) k − (cid:8) e − t ( a + t ) k − (cid:9) dt = ( − k − L k − (2 a ) + P k − ( a ) , so that(3.1.9) ∀ k ≥ , P (cid:48) k ( a ) = ( − k − L k − (2 a ) + P k − ( a ) = ( − k +1 L k (2 a ) + P k ( a ) . NTEGRALS OF THE WIGNER DISTRIBUTION 31
This implies for N ≥ , (cid:88) ≤ k ≤ N P k ( a ) − (cid:88) ≤ k ≤ N ( − k L k (2 a ) = (cid:88) ≤ k ≤ N − P k ( a ) + (cid:88) ≤ k ≤ N − ( − k L k (2 a ) , yielding P N ( a ) − P ( a ) (cid:124) (cid:123)(cid:122) (cid:125) =1= L ( a ) = (cid:88) ≤ k ≤ N ( − k L k (2 a ) + (cid:88) ≤ k ≤ N − ( − k L k (2 a ) , and(3.1.10) P N ( a ) = (cid:88) ≤ k ≤ N ( − k L k (2 a ) + (cid:88) ≤ k ≤ N − ( − k L k (2 a ) . Note that the previous formula holds as well for N = 0 , since P = 1 = L .Although the function R + (cid:51) a (cid:55)→ F k ( a ) has no monotonicity properties, we provebelow that R + (cid:51) a (cid:55)→ P k ( a ) is indeed increasing. For that purpose, let us use (3.1.9),which implies P (cid:48) k ( a ) = ( − k − L k − (2 a ) + P k − ( a ) , k ≥ ,P k − ( a ) = P k − ( a ) + ( − k − L k − (2 a ) + ( − k − L k − (2 a ) , k ≥ ,P (cid:48) k ( a ) = 2( − k − L k − (2 a ) + ( − k − L k − (2 a ) + P k − ( a ) , k ≥ . We claim that that for k ≥ ,(3.1.11) P (cid:48) k ( a ) = 2 (cid:88) ≤ l ≤ k − ( − l L l (2 a ) . That property holds for k = 1 since P ( a ) = 1 + 2 a : we check P (cid:48) ( a ) = 2 . Moreoverwe have P (cid:48) k +1 ( a ) = ( − k L k (2 a ) + P k ( a ) (from the first equation in (3.1.9))(using (3.1.10)) = ( − k L k (2 a ) + (cid:88) ≤ l ≤ k ( − l L l (2 a ) + (cid:88) ≤ l ≤ k − ( − l L l (2 a )= 2 (cid:88) ≤ l ≤ k ( − l L l (2 a ) , qed. As a byproduct we find from (9.3.3)(3.1.12) ∀ a ≥ , P (cid:48) k ( a ) ≥ , which implies that for a ≥ , P k ( a ) ≥ P k (0) = 1 . We have proven the following Lemma 3.1.
The polynomial P k ( a ) = e a (cid:0) − F k ( a ) (cid:1) is increasing on R + , P k (0) = 1 . Let us take a look at the first P k : we have P ( a ) = 1 ,P ( a ) = 1 + 2 a,P ( a ) = 1 + 2 a ,P ( a ) = 1 + 2 a − a + 4 a ,P ( a ) = 1 + 4 a − a a ,P ( a ) = 1 + 2 a − a + 16 a − a + 4 a ,P ( a ) = 1 + 6 a − a + 14 a − a
15 + 4 a ,P ( a ) = 1 + 2 a − a + 12 a − a a − a a ,P ( a ) = 1 + 8 a − a + 44 a − a a − a
105 + 2 a ,P ( a ) = 1 + 2 a − a + 64 a − a a − a a − a
45 + 4 a ,P ( a ) = 1 + 10 a − a a − a a − a
315 + 58 a − a a ,P ( a ) = 1 + 2 a − a + 100 a − a a − a
45 + 1184 a − a
45+ 148 a − a a ,P ( a ) = 1 + 12 a − a + 190 a − a a − a
315 + 478 a − a a − a a . We note as well that(3.1.13) P k ( x ) = (cid:88) ≤ m ≤ k x m m ! (cid:88) m ≤ l ≤ k l ( − k − l (cid:18) kl (cid:19) , since from (3.1.5), P k ( a ) = ( − k k ! (cid:90) + ∞ e t ( ddt ) k (cid:8) e − t ( a + t ) k (cid:9) dt = ( − k (cid:88) ≤ m ≤ k (cid:90) + ∞ e − t ( − k − m ( k − m )! k !( k − m )! m ! ( a + t ) k − m dt = ( − k (cid:88) ≤ m ≤ k (cid:90) + ∞ e − t ( − k − m ( k − m )! k !( k − m )! m ! (cid:88) ≤ l ≤ k − m a l t k − l − m (cid:18) k − ml (cid:19) dt = ( − k (cid:88) ≤ m ≤ k ≤ l ≤ k − m ( − k − m ( k − m )! k !( k − m )! m ! a l ( k − l − m )! (cid:18) k − ml (cid:19) = (cid:88) ≤ l + m ≤ k ( − m k − m ( k − m )! k ! m ! a l l ! = (cid:88) ≤ l ≤ k a l l ! (cid:88) l ≤ m (cid:48) ≤ k ( − k − m (cid:48) m (cid:48) (cid:18) km (cid:48) (cid:19) , qed. NTEGRALS OF THE WIGNER DISTRIBUTION 33
Lemma 3.2.
With the polynomial P k defined by (3.1.7) , we have (3.1.14) (cid:40) P k ( a ) = 2 (cid:80) ≤ l ≤ k − ( − l L l (2 a ) + ( − k L k (2 a ) ,P (cid:48) k ( a ) = 2 (cid:80) ≤ l ≤ k − ( − l L l (2 a ) . Proof.
We may use the already proven (3.1.10), (3.1.11) but we may also prove thisdirectly by induction on k . (cid:3) Proposition 3.3.
Let F k be given by (3.1.4) with P k defined by (3.1.5) . We have F k ( a ) = 1 − e − a P k ( a ) ≤ − e − a = F ( a ) for a ≥ , (3.1.15) F (cid:48) k ( a ) = e − a (cid:0) P k ( a ) − P (cid:48) k ( a ) (cid:1) = e − a ( − k L k (2 a ) , (3.1.16) F (cid:48) k (0) = ( − k , lim a → + ∞ F (cid:48) k ( a ) = 0 + , F k (0) = 0 , lim a → + ∞ F k ( a ) = 1 − . (3.1.17) Proof.
We use (3.1.4), (3.1.11) and (3.1.10) for the three first equalities, Lemma 3.1for the first inequality. The fourth equality follows from L k (0) = 1 , while the fifth isdue to the fact that the leading monomial of ( − k L k (2 a ) is k a k /k ! . The two lastequalities are a consequence of the first line. (cid:3) Remark 3.4.
The zeroes of F (cid:48) k on the positive half-line are the positive zeroes of theLaguerre polynomial L k divided by . When k is even (resp. odd) the function F k ispositive increasing (resp. negative decreasing) near , then oscillates with changes ofmonotonicity at each a such that L k (2 a ) = 0 and when a is larger than the largestzero of L k , the function F k is increasing, smaller than 1, with limit 1 at infinity.Typically we have F l (0) = 0 , F (cid:48) l (0) = +1 , (3.1.18) < a , l < a < · · · < a l − , l < a l, l the zeroes of L l (2 a ) , F l vanishes simply at b = 0 and at b j ∈ ( a j , a j +1 ) for ≤ j ≤ l − , also at b l > a l : l + 1 zeroes with a positive (resp. negative) derivative at b , b , . . . , b l (resp. at b , b , . . . , b l − ).Moreover, we have F l +1 (0) = 0 , F (cid:48) l +1 (0) = − , (3.1.19) < a , l +1 < a , l +1 < · · · < a l, l +1 < a l +1 , l +1 , the zeroes of L l +1 (2 a ) , F l +1 vanishes simply at b = 0 and at b j ∈ ( a j , a j +1 ) for ≤ j ≤ l , also at b l +1 >a l +1 : l + 2 zeroes with a positive (resp. negative) derivative at b , b , . . . , b l +1 (resp. at b , b , . . . , b l ).We note as well that a consequence of the previous remark is that min a ≥ F l ( a ) = min ≤ j ≤ l { F l ( a j, l ) } , (3.1.20) min a ≥ F l +1 ( a ) = min ≤ j ≤ l { F l +1 ( a j +1 , l +1 ) } , (3.1.21)where ( a p,k ) ≤ p ≤ k are defined in (3.1.18), (3.1.19). Theorem 3.5.
Let a ≥ be given and let (3.1.22) D a = { ( x, ξ ) ∈ R , x + ξ ≤ a π } . Then we have (3.1.23) Op w ( D a ) = (cid:88) k ≥ F k ( a ) P k ≤ − e − a . Proof.
An immediate consequence of (3.1.1) and (3.1.15). Note that the inequalityin the above theorem is due to P. Flandrin in [9] (see also the related references [13],[10]). (cid:3)
Curves.
Let us display some curves of R + (cid:51) a (cid:55)→ F k ( a ) = 1 − e − a P k ( a ) . - . . . . . . F F Figure 2.
Functions F , F . NTEGRALS OF THE WIGNER DISTRIBUTION 35 - . . . . . . F F F F F F F Figure 3.
Functions F k Indicatrix of an Euclidean ball.
The following result displays an explicitspectral decomposition on the Hermite basis for the Weyl quantization of the char-acteristic function of Euclidean balls.
Theorem 3.6.
Let a ≥ be given and let (3.2.1) Q a,n = Op w ( { π ( | x | + | ξ | ) ≤ a } ) , be the Weyl quantization of the characteristic function of the Euclidean ball of R n with center and radius (cid:112) a/ (2 π ) . Then we have (3.2.2) Q a,n = (cid:88) k ≥ F k ; n ( a ) P k ; n , with P k ; n = (cid:80) α ∈ N n , | α | = k P α , where P α is the orthogonal projection onto Ψ α (definedin (9.1.33) ), with | α | = (cid:80) ≤ j ≤ n α j = k and (3.2.3) F k ; n ( a ) = (cid:90) R sin aτπτ (1 + iτ ) k (1 − iτ ) k + n dτ. The spectral decomposition of the previous theorem allows a simple recovery ofthe result of the article [27] by E. Lieb and Y. Ostrover.
Theorem 3.7.
Let a ≥ , Q a,n , F k ; n be defined above. Then we have (3.2.4) F k ; n ( a ) ≤ − n ) (cid:90) + ∞ a e − t t n − dt = 1 − Γ( n, a )Γ( n ) , and thus we have (3.2.5) Q a,n ≤ − Γ( n, a )Γ( n ) , where the incomplete Gamma function Γ( · , · ) is defined in (9.7.8) .Proof of Theorems 3.6 and 3.7. We use the results of (the previous) Section 3.1: Letus assume now that, with some a ≥ , F = [ − a π , a π ] , so that F ( | x | + | ξ | ) = { π ( | x | + | ξ | ) ≤ a } . According to Section 9.7.3, we have ˆ F ( τ ) = sin aτπτ , so that (2.1.1) holds true. We findin this case, following the results of Lemma 2.3, (cid:0) F ( | x | + | ξ | ) (cid:1) w = (cid:88) k ≥ F k ; n ( a ) P k ; n , P k ; n = (cid:88) α ∈ N n , | α | = k P α , (3.2.6) F k ; n ( a ) = (cid:90) R sin aτπτ (1 + iτ ) k (1 − iτ ) k + n dτ, (3.2.7)where P α is the orthogonal projection onto Ψ α (defined in (9.1.33)), with | α | = (cid:80) ≤ j ≤ n α j = k . This completes the proof of Theorem 3.6.We postpone the proof of Theorem 3.7 until after settling a couple of lemmas. NTEGRALS OF THE WIGNER DISTRIBUTION 37
Lemma 3.8.
Let ( k, n ) ∈ N × N ∗ . With F k ; n ( a ) given by (3.2.7) , we have F k ; n ( a ) = 1 − e − a P k,n ( a ) , where P k ; n is the polynomial (3.2.8) P k ; n ( a ) = ( − k + n − ( k + n − (cid:90) + ∞ e − t ( t + a ) n − (cid:110) e s ( dds ) n + k − (cid:2) s k e − s (cid:3)(cid:111) | s =2 t +2 a dt, (3.2.9) P k ; n ( a ) = ( − k + n − ( k + n − n − (cid:90) + ∞ ( t + a ) n − e t ( ddt ) n + k − (cid:8) ( t + a ) k e − t (cid:9) dt. (3.2.10) Proof of the lemma.
The lemma holds true for n = 1 from Proposition 3.3. We havefor a > , n ≥ , F (cid:48) k ; n ( a ) = 1 π (cid:90) R cos aτ (1 + iτ ) k (1 − iτ ) k + n dτ = 12 π (cid:90) R e iaτ (1 + iτ ) k (1 − iτ ) k + n dτ + 12 π (cid:90) R e iaτ (1 − iτ ) k (1 + iτ ) k + n dτ = i iπ (cid:90) R e iaτ i k ( τ − i ) k ( − i ) k + n ( τ + i ) k + n dτ + i iπ (cid:90) R e iaτ ( − i ) k ( τ + i ) k i k + n ( τ − i ) k + n dτ, so that F (cid:48) k ; n ( a ) = i − n ( − k Res (cid:0) e iaτ ( τ + i ) k ( τ − i ) k + n ; i (cid:1) = i − n ( − k ( k + n − ddτ ) k + n − (cid:8) e iaτ ( τ + i ) k (cid:9) | τ = i and thus F (cid:48) k ; n ( a ) = i − n ( − k ( k + n − d ia d(cid:15) ) k + n − (cid:8) e − a − (cid:15) ( i + i (cid:15)a + i ) k (cid:9) | (cid:15) =0 = i − n ( − k a n − i n − ( k + n − dd(cid:15) ) k + n − (cid:8) e − a − (cid:15) (2 a + (cid:15) ) k (cid:9) | (cid:15) =0 = e a ( − k + n − a n − ( k + n − d d(cid:15) ) k + n − (cid:8) e − a − (cid:15) (2 a + 2 (cid:15) ) k (cid:9) | (cid:15) =0 , that is F (cid:48) k ; n ( t ) = ( − k + n − ( k + n − e t t n − ( dds ) k + n − (cid:8) e − s s k (cid:9) | s =2 t = ( − k + n − ( k + n − n − e t t n − ( ddt ) k + n − (cid:8) e − t t k (cid:9) . We have also that lim a → + ∞ F k ; n ( a ) = 1 (following the arguments of Section 3.1) andthis yields F k ; n ( a ) = 1 − ( − k + n − ( k + n − n − (cid:90) + ∞ a e t t n − ( ddt ) k + n − (cid:8) e − t t k (cid:9) dt = 1 − e − a ( − k + n − ( k + n − n − (cid:90) + ∞ ( t + a ) n − e t ( ddt ) k + n − (cid:8) e − t ( t + a ) k (cid:9) dt, concluding the proof of the Lemma. (cid:3) Let us go back to Formula (3.2.9), written as ( − k + n − n − (cid:90) + ∞ e − t (cid:110) (2 t + 2 a ) n − ( k + n − dd(cid:15) − n + k − (cid:2) ( (cid:15) + 2 t + 2 a ) k (cid:3)(cid:111) | (cid:15) =0 dt = P k ; n ( a ) = ( − k + n − n − (cid:90) + ∞ e − t L − nk + n − (2 t + 2 a ) dt, (3.2.11)where the generalized Laguerre polynomial L − nk + n − is defined by (9.3.6) (note that − n + k + n − k which not negative). Lemma 3.9.
Let n ∈ N ∗ , k ∈ N and let P k ; n be the polynomial defined in Lemma3.8 (and thus in (3.2.11) ). Then we have P k ; n ( X ) − P (cid:48) k ; n ( X ) = ( − k + n − n − L − nk + n − (2 X ) , P k ; n (0) = 1 , (3.2.12) for n ≥ , P (cid:48) k ; n = P k ; n − . (3.2.13) Proof.
From (3.2.11), we find(3.2.14) P (cid:48) k ; n ( a ) = ( − k + n − n − (cid:90) + ∞ e − t L − nk + n − ) (cid:48) (2 t + 2 a ) dt = ( − k + n − n − (cid:110)(cid:2) e − t ( L − nk + n − )(2 t + 2 a ) (cid:3) t =+ ∞ t =0 + (cid:90) + ∞ e − t L − nk + n − (2 t + 2 a ) dt (cid:111) = ( − k + n n − L − nk + n − (2 a ) + P k ; n ( a ) , and since F k ; n (0) = 1 − P k ; n (0) , this proves (3.2.12). Using now (3.2.11) and(9.3.8), we find that P k ; n ( a ) = ( − k + n n − (cid:90) + ∞ ddt (cid:8) e − t (cid:9) L − nk + n − (2 t + 2 a ) dt = ( − k + n n − (cid:110)(cid:2) e − t L − nk + n − (2 t + 2 a ) (cid:3) t =+ ∞ t =0 − (cid:90) + ∞ e − t L − nk + n − ) (cid:48) (2 t + 2 a ) dt (cid:111) = ( − k + n n − (cid:110) − L − nk + n − (2 a ) + (cid:90) + ∞ e − t L − nk + n − )(2 t + 2 a ) dt (cid:111) = ( − k + n − n − L − nk + n − (2 a ) (cid:124) (cid:123)(cid:122) (cid:125) P k ; n ( a ) − P (cid:48) k ; n ( a ) from (3.2.12) + ( − k + n − n − (cid:90) + ∞ e − t L − nk + n − (2 t + 2 a ) dt (cid:124) (cid:123)(cid:122) (cid:125) P k ; n − ( a ) from (3.2.11) , so that for n ≥ , k ∈ N , we obtain (3.2.13), completing the proof of the lemma. (cid:3) Lemma 3.10.
Let k, n, P k ; n be as in Lemma 3.9. Then we have (3.2.15) ∀ j ∈ (cid:74) ..n − (cid:75) , (cid:18) ddX (cid:19) j P k ; n = P k ; n − j . Moreover, for all a ≥ and all k ∈ N , (3.2.16) P k ; n ( a ) ≥ P n ( a ) = 1( n − (cid:90) + ∞ e − t ( t + a ) n − dt = e a Γ( n, a )Γ( n ) . NTEGRALS OF THE WIGNER DISTRIBUTION 39
Proof.
Formula (3.2.15) follows immediately by induction from (3.2.13) since thelatter is proving (3.2.15) for j = 1 , n ≥ , k ∈ N . Assuming that (3.2.15) holds truefor some ≤ j < n , all k ∈ N , we have P ( j ) k ; n = P k,n − j and if j + 1 < n , we obtainfrom (3.2.13) that P k,n − j − = P (cid:48) k,n − j = P ( j +1) k ; n , proving (3.2.15). The property (3.2.16) holds true for n = 1 . From (3.2.13) and P k ; n +1 (0) = 1 , we find that P k ; n +1 ( a ) = 1 + (cid:82) a P k ; n ( s ) ds and assuming that (3.2.16)holds true for n , we obtain for a ≥ , P k ; n +1 ( a ) ≥ (cid:90) a n − (cid:90) + ∞ e − t ( t + s ) n − dtds = 1 + (cid:90) + ∞ e − t (cid:104) ( t + s ) n n ! (cid:105) s = as =0 dt = 1 + 1 n ! (cid:90) + ∞ e − t (cid:0) ( t + a ) n − t n (cid:1) dt = 1 n ! (cid:90) + ∞ e − t ( t + a ) n dt, completing the proof of the lemma (cid:3) We can now prove Theorem 3.7: since F k ; n ( a ) = 1 − e − a P k ; n ( a ) the estimate(3.2.15) implies indeed F k ; n ( a ) ≤ Γ( n,a )Γ( n ) , concluding the proof. (cid:3) Remark 3.11.
Our methods of proof in one and more dimensions are quite similar: • Using Mehler’s Formula, we diagonalize in the Hermite basis the quantizationof the indicatrix of the Euclidean ball D a ; n = { ( x, ξ ) ∈ R n , π (cid:0) | x | + | ξ | (cid:1) ≤ a } . • Once we get the diagonalizationOp w ( D a ; n ) = (cid:88) k ∈ N F k ; n ( a ) P k ; n , we study explicitly the functions F k ; n and prove that F k ; n ( a ) = 1 − e − a P k ; n ( a ) , where P k ; n is a polynomial given in terms of the generalized Laguerre poly-nomials P k ; n ( a ) = ( − k + n − n − (cid:90) + ∞ e − t L − nk + n − (2 t + 2 a ) dt. • Following the Flandrin paper [9], we use Feldheim inequality in [8] to tacklethe case n = 1 , and next we use an induction on n , made possible by therelationship between the standard and the generalized Laguerre polynomials.It is interesting to note that the functions F k ; n have no monotonicity proper-ties: with value 0 at 0, they have an oscillatory behavior for a ≤ a k,n and for a large enough, increase monotonically to 1 (see for instance Figures 2 and3 in the 1D case); the inequality F k ; n ( a ) ≤ − e − a holds true for all a ≥ inall dimensions. On the other hand the polynomials P k ; n are increasing andlarger than 1 on the positive half-line. The key ingredients are thus Mehler’s formula and Feldheim inequality, but it shouldbe pointed out that the arguments proving Feldheim inequality (Formula (6.8) andTheorem 12) in the R. Askey & G. Gasper’s article [2] are also based upon a versionof Mehler’s Formula which appears thus as the basic result for our investigation.The paper [27] by E. Lieb and Y. Ostrover has a slightly different line of argumentsand takes advantage of symmetry properties of the sphere. We shall go back to thisin a situation where the symmetry is absent, such as for some general ellipsoids.3.3.
Ellipsoids in the phase space.
Preliminaries.
We provide below a couple of remarks on Ellipsoids in higherdimensions. Let us first recall a particular case of Theorem 21.5.3 in [17].
Theorem 3.12 (Symplectic reduction of quadratic forms) . Let q be a positive-definite quadratic form on R n × R n equipped with the canonical symplectic form (1.2.20) . Then there exists S in the symplectic group Sp ( n, R ) of R n and µ , . . . , µ n positive such that for all X = ( x, ξ ) ∈ R n × R n , (3.3.1) q ( SX ) = (cid:88) ≤ j ≤ n µ j ( x j + ξ j ) . Note that an interesting consequence of this theorem is that, considering a generalellipsoid in R n (with center of gravity at ), E = { X ∈ R n , q ( X ) ≤ } where q is a positive definite quadratic form, we are able to find symplectic coordi-nates such that q is given by (3.3.1). Note however that no further simplification ispossible and that the µ j are symplectic invariants of E . In particular the volume of E is given by | E | n = π n n ! µ . . . µ n . Spectral decomposition for the quantization of the characteristic function of theellipsoid.
Let a , . . . , a n be positive numbers. We consider the ellipsoid E ( a , . . . , a n ) given by(3.3.2) E ( a ) = E ( a , . . . , a n ) = { ( x, ξ ) ∈ R n × R n , π (cid:88) ≤ j ≤ n x j + ξ j a j ≤ } . We define on R n the function F ( X , . . . , X n ) = [ − , ( 2 πa X + · · · + 2 πa n X n ) . Theorem 3.13.
Let a = ( a j ) ≤ j ≤ n be positive numbers and let E ( a ) be defined by (3.3.2) . Then we have (3.3.3) Op w ( E ( a ) ) = (cid:88) α ∈ N n F α ( a ) P α , NTEGRALS OF THE WIGNER DISTRIBUTION 41 where P α is defined in (9.1.36) and (3.3.4) F α ( a ) = 1 − K α ( a ) , with (3.3.5) K α ( a ) = (cid:90) (cid:80) t j /a j ≥ t j ≥ e − ( t + ··· + t n ) (cid:89) ≤ j ≤ n ( − α j L α j (2 t j ) dt, Remark 3.14.
For all α ∈ N n , the functions F α , K α are holomorphic on(3.3.6) U = { a ∈ C n , ∀ j ∈ (cid:74) ..n (cid:75) , Re a j > } . Indeed let K be a compact subset of U ; there exists ρ > such that ∀ ( a , . . . , a n ) ∈ K, min ≤ j ≤ n Re a j ≥ ρ, and as a result for a ∈ K , we have for s ∈ R n + | e − ( a s + ··· + a n s n ) (cid:89) ≤ j ≤ n ( − α j L α j (2 a j s j ) | ≤ e − ρ ( s + ··· + s n ) C K,α (1 + | s | ) | α | , so that (cid:90) (cid:80) s j ≥ s j ≥ sup a ∈ K | e − ( a s + ··· + a n s n ) (cid:89) ≤ j ≤ n ( − α j L α j (2 a j s j ) | ds ≤ (cid:90) (cid:80) s j ≥ s j ≥ e − ρ ( s + ··· + s n ) C K,α (1 + | s | ) | α | ds ≤ C K,α (cid:90) R n e − ρσ n | s | (1 + | s | ) | α | ds < + ∞ . Since we have K α ( a ) = (cid:90) (cid:80) s j ≥ s j ≥ e − ( a s + ··· + a n s n ) (cid:89) ≤ j ≤ n ( − α j L α j (2 a j s j ) dsa . . . a n , this proves the sought holomorphy. Proof of the theorem.
We haveOp w ( E ( a ) ) = (cid:0) F ( x + ξ , . . . , x n + ξ n ) (cid:1) w = (cid:90) R n ˆ F ( τ ) Op w (cid:0) e iπ (cid:80) j τ j ( x j + ξ j ) (cid:1) dτ = (cid:88) α ∈ N n (cid:90) R n ˆ F ( τ ) (cid:89) ≤ j ≤ n (1 + iτ j ) α j +1 (1 + τ j ) α j +1 dτ P α = (cid:88) α ∈ N n (cid:90) R n ˆ F ( τ ) (cid:89) ≤ j ≤ n (1 + iτ j ) α j (1 − iτ j ) α j +1 dτ P α , where P α is defined in (9.1.36). On the other hand we have ˆ F ( τ ) = (cid:90) e − iπτ · x [ − , ( 2 πa x + · · · + 2 πa n x n ) dx . . . dx n = a . . . a n (2 π ) − n (cid:90) e − i (cid:80) j τ j a j y j [ − , ( (cid:88) y j ) dy, so that, with M k defined in (9.3.4), using (9.3.5), we getOp w ( E ( a ) )= a . . . a n (cid:88) α ∈ N n (cid:120) R n × R n e − i π (cid:80) j τ j a j y j [ − , ( (cid:88) y j ) dy (cid:89) ≤ j ≤ n (1 + i πτ j ) α j (1 − i πτ j ) α j +1 dτ P α = a . . . a n (cid:88) α ∈ N n (cid:90) R n (cid:90) R n e − i π (cid:80) j τ j a j y j [ − , ( (cid:88) y j ) dy (cid:89) ≤ j ≤ n ˆ G α j ( τ j ) dτ P α = a . . . a n (cid:88) α ∈ N n (cid:90) R n [ − , ( (cid:88) y j ) (cid:89) ≤ j ≤ n G α j ( a j y j ) dy P α = (cid:88) α ∈ N n (cid:90) R n [ − , ( (cid:88) t j /a j ) (cid:89) ≤ j ≤ n ( − α j H ( t j ) e − t j L α j (2 t j ) dt P α , with F α ( a ) = (cid:90) R n (cid:0) − [1 , + ∞ ] ( (cid:88) t j /a j ) (cid:1) (cid:89) ≤ j ≤ n ( − α j H ( t j ) e − t j L α j (2 t j ) dt (3.3.7) = 1 − (cid:90) R n [1 , + ∞ ] ( (cid:88) t j /a j ) (cid:89) ≤ j ≤ n ( − α j H ( t j ) e − t j L α j (2 t j ) dt, where we have used that P k ;1 (0) = 1 (see page 30) , so that setting K α ( a ) = (cid:90) (cid:80) t j /a j ≥ t j ≥ e − ( t + ··· + t n ) (cid:89) ≤ j ≤ n ( − α j L α j (2 t j ) dt, we have F α ( a ) = 1 − K α ( a ) , concluding the proof of the theorem. (cid:3) Remark 3.15.
We have from (3.3.7)(3.3.8) F α ( a , . . . , a n ) = (cid:90) R n [0 , ( (cid:88) ≤ j ≤ n s j ) (cid:89) ≤ j ≤ n ( − α j H ( s j ) e − a j s j L α j (2 a j s j ) a j ds, and since the set { s ∈ R n + , (cid:80) ≤ j ≤ n s j ≤ } is compact, we obtain that F α is an entirefunction, as well as K α which is indeed given by (3.3.5) on the open subset U definedin (3.3.6) . Lemma 3.16.
With the notations of Theorem 3.13, we have with µ j = 1 /a j , (3.3.9) F α ( a ) = (cid:16) (cid:89) ≤ j ≤ n a j (cid:17) (cid:90) R sin τπτ (cid:16) (cid:89) ≤ j ≤ n ( a j + iτ ) α j ( a j − iτ ) α j +1 (cid:17) dτ = (cid:90) R sin τπτ (cid:16) (cid:89) ≤ j ≤ n (1 + iτ µ j ) α j (1 − iτ µ j ) α j +1 (cid:17) dτ. Proof.
Mehler’s formula implies in one dimension that(3.3.10) Op w ( e πiτ ( x + ξ ) ) = (1 + τ ) − / exp (cid:2) πi (arctan τ )( x + D x ) (cid:3) , and a simple tensorisation givesOp w ( e πiτ (cid:80) j µ j ( x j + ξ j ) ) = (cid:89) j (1 + ( τ µ j ) ) − / exp (cid:2) πi (cid:88) j (arctan( τ µ j ))( x j + D x j ) (cid:3) , NTEGRALS OF THE WIGNER DISTRIBUTION 43 so that we haveOp w (cid:0) F (cid:0)(cid:88) j µ j ( x j + ξ j ) (cid:1)(cid:1) = (cid:90) R ˆ F ( τ ) Op w (cid:0) e πiτ (cid:80) j µ j ( x j + ξ j ) (cid:1) dτ = (cid:90) R ˆ F ( τ ) (cid:89) j (1 + ( τ µ j ) ) − / exp (cid:2) πi (cid:88) j (arctan( τ µ j ))( x j + D x j ) (cid:3) dτ = (cid:88) α ∈ N n (cid:90) R ˆ F ( τ ) (cid:32)(cid:89) j (1 + ( τ µ j ) ) − / exp (cid:2) i (arctan( τ µ j ))( α j + 12 ) (cid:3)(cid:33) dτ P α = (cid:88) α ∈ N n (cid:90) R ˆ F ( τ ) (cid:32)(cid:89) j (1 + ( τ µ j ) ) − / (1 + iτ µ j ) α j +1 (1 + ( τ µ j ) ) α j + (cid:33) dτ P α = (cid:88) α ∈ N n (cid:90) R ˆ F ( τ ) (cid:16) (cid:89) ≤ j ≤ n (1 + iτ µ j ) α j (1 − iτ µ j ) α j +1 (cid:17) dτ P α , and for F ( t ) = [ − , (2 πt ) , we find ˆ F ( τ ) = sin τπτ and the sought result. (cid:3) Remark 3.17.
It is also possible to provide a direct checking for the above lemma,since with the notations (9.3.4), (9.3.5), we have (1 + iτ µ j ) α j (1 − iτ µ j ) α j +1 = ˇ (cid:100) G α j (cid:0) τ µ j / (2 π ) (cid:1) , and thus F α ( a ) = (cid:90) R ˆ F ( τ ) (cid:89) j ˇ (cid:100) G α j (cid:0) τ µ j / (2 π ) (cid:1) dτ = (cid:90) R ˆ F ( τ ) (cid:90) R n (cid:89) j ( − α j L α j (2 t j ) H ( t j ) e − t j e πiτµ j t j / (2 π ) dtdτ = (cid:90) R n (cid:89) j ( − α j L α j (2 t j ) H ( t j ) e − t j F (cid:0)(cid:88) j µ j t j / π (cid:1) dt. Now since we have F (cid:0)(cid:80) j µ j t j / π (cid:1) = [ − , ( (cid:80) j µ j t j ) , this fits with the expressionof F α in Theorem 3.13. Remark 3.18.
Another interesting remark is that the expression (3.3.9) dependsobviously only on | α | and a = a = · · · = a n in the case where all the a j are equal:indeed in that case, we have with µ = 1 /a , (cid:89) ≤ j ≤ n (1 + iτ µ j ) α j (1 − iτ µ j ) α j +1 = (1 + iτ µ ) | α | (1 − iτ µ ) | α | + n , and this gives another ( a posteriori ) justification of our calculations in the isotropiccase of Section 3.2. On the other hand, we get also the identity(3.3.11) F N n ( a , . . . , a n ) = (cid:90) R sin τπτ Re (cid:16) (cid:89) ≤ j ≤ n (1 − iτ µ j ) − (cid:17) dτ, where the explicit expression (3.3.13) is given for the left-hand-side. Lemma 3.19.
With the notations of Theorem . , the function K α ,...,α n ( a , . . . , a n ) is symmetric in the variables ( α , a ; . . . ; α n , a n ) , i.e. for a permutation π of { , . . . , n } ,we have (3.3.12) K α π (1) ,...,α π ( n ) ( a π (1) , . . . , a π ( n ) ) = K α ,...,α n ( a , . . . , a n ) . Proof.
Formula (3.3.5) yields K α ( a ) = (cid:90) (cid:80) s j ≥ s j ≥ (cid:89) ≤ j ≤ n (cid:0) e − a j s j a j ( − α j L α j (2 a j s j ) (cid:1) ds, and the domain of integration is invariant by permutation of the variables, entailingthe sought result. (cid:3) Lemma 3.20.
With the notations of Theorem 3.13, we have K α ,...,α n ( a , . . . , a n ) = e − a n P α n ( a n )+ (cid:90) a n ( − α n L α n (2 t n ) e − t n K α ,...,α n − (cid:0) a (1 − t n /a n ) , . . . , a n − (1 − t n /a n ) (cid:1) dt n = e − a n P α n ( a n )+ (cid:90) ( − α n L α n (2 a n θ ) e − θa n K α ,...,α n − (cid:0) a (1 − θ ) , . . . , a n − (1 − θ ) (cid:1) dθa n . Proof.
The domain of integration is the disjoint union (cid:26) t a + · · · + t n − a n − ≥ − t n a n , t j ≥ , ≤ t n a n ≤ (cid:27) (cid:116) (cid:26) t n a n > , t j ≥ , ≤ j ≤ n − (cid:27) , so that K α ,...,α n ( a , . . . , a n ) = e − a n P α n ( a n )+ (cid:90) a n ( − α n L α n (2 t n ) e − t n K α ,...,α n − (cid:0) a (1 − t n /a n ) , . . . , a n − (1 − t n /a n ) (cid:1) dt n = e − a n P α n ( a n )+ (cid:90) ( − α n L α n (2 a n θ ) e − θa n K α ,...,α n − (cid:0) a (1 − θ ) , . . . , a n − (1 − θ ) (cid:1) dθa n , which is the sought result. (cid:3) Lemma 3.21.
With the notations of Theorem 3.13, we have, assuming that the ( a j ) ≤ j ≤ n are positive distinct numbers, (3.3.13) K ,..., ( a , . . . , a n ) = (cid:88) ≤ j ≤ n e − a j (cid:81) k (cid:54) = j a k (cid:81) k (cid:54) = j ( a k − a j ) . NTEGRALS OF THE WIGNER DISTRIBUTION 45
Proof.
The latter formula is true for n = 1 since we have K ( a ) = e − a . We havealso K ∈ N n ( a , . . . , a n ) = e − a n + a n (cid:90) e − θa n K ∈ N n − (cid:0) a (1 − θ ) , . . . , a n − (1 − θ ) (cid:1) dθ = e − a n + a n (cid:90) e − θa n (cid:88) ≤ j ≤ n − e − a j (1 − θ ) (cid:81) k (cid:54) = j a k (cid:81) k (cid:54) = j ( a k − a j ) dθ = e − a n + a n (cid:88) ≤ j ≤ n − (cid:81) k (cid:54) = j a k (cid:81) k (cid:54) = j ( a k − a j ) (cid:90) e − θa n e − a j (1 − θ ) dθ = e − a n + (cid:88) ≤ j ≤ n − a n (cid:81) k (cid:54) = j a k (cid:81) k (cid:54) = j ( a k − a j ) e − a j (cid:90) e θ ( a j − a n ) dθ = e − a n + (cid:88) ≤ j ≤ n − a n (cid:81) k (cid:54) = j a k (cid:81) k (cid:54) = j ( a k − a j ) e − a j e a j − a n − a j − a n = e − a n + (cid:88) ≤ j ≤ n − a n (cid:81) k (cid:54) = j a k (cid:81) k (cid:54) = j ( a k − a j ) e − a n − e − a j ( a j − a n )= e − a n (cid:32) (cid:88) ≤ j ≤ n − a n (cid:81) k (cid:54) = j a k (cid:81) k (cid:54) = j ( a k − a j ) 1( a j − a n ) (cid:33) + (cid:88) ≤ j ≤ n − a n (cid:81) k (cid:54) = j a k (cid:81) k (cid:54) = j ( a k − a j ) e − a j ( a n − a j ) (cid:124) (cid:123)(cid:122) (cid:125) OK . We need to prove that (cid:32) (cid:88) ≤ j ≤ n − a n (cid:81) k (cid:54) = j, ≤ k ≤ n − a k (cid:81) k (cid:54) = j, ≤ k ≤ n − ( a k − a j ) 1( a j − a n ) (cid:33) = (cid:81) ≤ l ≤ n − a l (cid:81) ≤ l ≤ n − ( a l − a n ) . that is (cid:89) ≤ l ≤ n − a l = (cid:89) ≤ l ≤ n − ( a l − a n ) (cid:32) (cid:88) ≤ j ≤ n − a n (cid:81) k (cid:54) = j, ≤ k ≤ n − a k (cid:81) k (cid:54) = j, ≤ k ≤ n − ( a k − a j ) 1( a j − a n ) (cid:33) , which is (cid:89) ≤ l ≤ n − a l = (cid:89) ≤ l ≤ n − ( a l − a n ) + (cid:88) ≤ j ≤ n − a n (cid:81) k (cid:54) = j, ≤ k ≤ n − a k (cid:81) k (cid:54) = j, ≤ k ≤ n − ( a k − a j ) (cid:81) ≤ l ≤ n − ( a l − a n )( a j − a n ) , i.e. (cid:89) ≤ l ≤ n − a l = (cid:89) ≤ l ≤ n − ( a l − a n ) + (cid:88) ≤ j ≤ n − a n (cid:81) k (cid:54) = j, ≤ k ≤ n − a k ( a k − a n ) (cid:81) k (cid:54) = j, ≤ k ≤ n − ( a k − a j ) . (3.3.14)Let us reformulate (3.3.14) as an equality between polynomials (to be proven) with (cid:89) ≤ l ≤ n − ( a l − X ) + (cid:88) ≤ j ≤ n − X (cid:81) k (cid:54) = j, ≤ k ≤ n − a k ( a k − X ) (cid:81) k (cid:54) = j, ≤ k ≤ n − ( a k − a j ) − (cid:89) ≤ l ≤ n − a l = 0 , (3.3.15) and let us assume that the ( a j ) ≤ j ≤ n − are distinct and different from . The poly-nomial Q on the left-hand-side has degree less than n − and we have Q (0) = 0 , and ∀ j ∈ (cid:74) ..n − (cid:75) , Q ( a j ) = a j (cid:81) k (cid:54) = j, ≤ k ≤ n − a k ( a k − a j ) (cid:81) k (cid:54) = j, ≤ k ≤ n − ( a k − a j ) − (cid:89) ≤ l ≤ n − a l = 0 , so that Q has degree less than n − with n distinct roots and this proves the identity(3.3.15) when the ( a j ) ≤ j ≤ n − are distinct and all different from , proving (3.3.13)in that case; of course we may assume that all a j are positive and noting from (3.3.5)that K α is continuous on ( R ∗ + ) n , we get Formula (3.3.13) in all cases where all the a j are positive, concluding the proof of the lemma. (cid:3) Lemma 3.22.
With the notations of Theorem 3.13, we have, assuming < a ≤ · · · ≤ a n , the inequality (3.3.16) K ∈ N n ( a , . . . , a n ) ≥ (cid:88) ≤ j ≤ n e − a j (cid:81) ≤ l The above estimate is sharp in the sense that when all the a j areequal to the same a > , we have proven in (3.2.4) that K ( a ) = e − a ( n − (cid:90) + ∞ e − s ( s + a ) n − ds = e − a (cid:88) ≤ l ≤ n − a l ( n − − l )! l ! Γ( n − l )= e − a (cid:88) ≤ l ≤ n − a l l ! = e − a (cid:88) ≤ j ≤ n a j − ( j − (cid:88) ≤ j ≤ n e − a j (cid:81) ≤ l The property is true for n = 1 since K ( a ) = e − a . We check the case n = 2 with a < a , and we find K (0 , ( a , a ) = e − a + (cid:90) a e − t e − a (1 − t /a ) dt = e − a + e − a e a − a − a a − e − a + e − a a e a − a − a − a ≥ e − a + e − a a . NTEGRALS OF THE WIGNER DISTRIBUTION 47 Let us consider for some n ≥ , < a < · · · < a n and inductively, K ∈ N n ( a , . . . , a n )= e − a P ( a ) + (cid:90) a e − t K ∈ N n − (cid:0) a (1 − t /a ) , . . . , a n (1 − t /a ) (cid:1) dt = e − a P ( a ) + a (cid:90) e − a θ K ∈ N n − (cid:0) a (1 − θ ) , . . . , a n (1 − θ ) (cid:1) dθ ≥ e − a + a (cid:90) e − a θ (cid:88) ≤ j ≤ n e − a j (1 − θ ) (cid:81) ≤ l The reader may have noticed that it is not obvious on Formula(3.3.13) K ,..., ( a , . . . , a n ) = (cid:88) ≤ j ≤ n e − a j (cid:81) k (cid:54) = j a k (cid:81) k (cid:54) = j ( a k − a j ) , that K is an entire function. Let us start with taking a look at K , ( a , a ) = e − a a a − a + e − a a a − a = a e − a − a e − a a − a = e − ( a a a e − a + a − a e − a + a a − a = e − ( a a a (cosh a − a + sinh a − a ) − a (cosh a − a + sinh a − a ) a − a = e − ( a a (cid:104) cosh( a − a a + a ) sinh( a − a ) a − a (cid:105) = e − ( a a (cid:104) cosh( a − a ( a + a ) sinh( a − a ) a − a (cid:105) = e − ( a a (cid:104) cosh( a − a a + a ) shc( a − a (cid:105) , (3.3.17)where shc stands for the even entire function defined by(3.3.18) shc t = sinh tt . We have also from Lemma 3.16(3.3.19) F α ( a ) = (cid:90) R sin τπτ (cid:16) (cid:89) ≤ j ≤ n (1 + iτ µ j ) α j (1 − iτ µ j ) α j +1 (cid:17) dτ, and defining the function F α ( a, λ ) as the absolutely converging integral,(3.3.20) F α ( a, λ ) = (cid:90) R sin( λτ ) πτ (cid:16) (cid:89) ≤ j ≤ n (1 + iτ µ j ) α j (1 − iτ µ j ) α j +1 (cid:17) dτ, F α ( a ) = F α ( a, , we get ∂F α ∂λ ( a, λ ) = 1 π (cid:90) R cos( λτ ) (cid:16) (cid:89) ≤ j ≤ n (1 + iτ µ j ) α j (1 − iτ µ j ) α j +1 (cid:17) dτ = 12 π (cid:90) R e iλτ (cid:16) (cid:89) ≤ j ≤ n (1 + iτ µ j ) α j (1 − iτ µ j ) α j +1 (cid:17) dτ + 12 π (cid:90) R e iλτ (cid:16) (cid:89) ≤ j ≤ n (1 − iτ µ j ) α j (1 + iτ µ j ) α j +1 (cid:17) dτ = 12 π (cid:90) R e iλτ (cid:16) (cid:89) ≤ j ≤ n (1 + iτ µ j ) α j (1 − iτ µ j ) α j +1 + (cid:89) ≤ j ≤ n (1 − iτ µ j ) α j (1 + iτ µ j ) α j +1 (cid:17) dτ = i (cid:88) ≤ j ≤ n Res (cid:32) e iλτ (cid:89) ≤ j ≤ n (1 − iτ µ j ) α j (1 + iτ µ j ) α j +1 ; τ = i/µ j = ia j (cid:33) = i (cid:88) ≤ j ≤ n Res (cid:32) e iλτ (cid:89) ≤ j ≤ n ( − iµ j ) α j ( ia j + τ ) α j ( iµ j ) α j +1 ( − ia j + τ ) α j +1 ; τ = ia j (cid:33) = 1 i n − (cid:88) ≤ j ≤ n Res (cid:32) e iλτ (cid:89) ≤ j ≤ n ( − α j a j ( ia j + τ ) α j ( τ − ia j ) α j +1 ; τ = ia j (cid:33) , so that assuming that the a j are positive and distinct, we get ∂F α ∂λ ( a, λ ) = 1 i n − ( (cid:89) a k ) × (cid:88) ≤ j ≤ n α j ! (cid:18) ddτ (cid:19) α j (cid:32) e iλτ ( − α j ( ia j + τ ) α j (cid:89) ≤ k ≤ n,k (cid:54) = j ( − α k ( ia k + τ ) α k ( τ − ia k ) α k +1 (cid:33) | τ = ia j = 1 i n − ( (cid:89) ≤ k ≤ n a k ) (cid:88) ≤ j ≤ n α j ! × (cid:18) didσ (cid:19) α j (cid:32) e − λσ ( − α j ( ia j + iσ ) α j (cid:89) ≤ k ≤ n,k (cid:54) = j ( − α k ( ia k + iσ ) α k ( iσ − ia k ) α k +1 (cid:33) | σ = a j NTEGRALS OF THE WIGNER DISTRIBUTION 49 = ( − n − | α | ( (cid:89) ≤ k ≤ n a k ) (cid:88) ≤ j ≤ n α j ! × (cid:18) ddσ (cid:19) α j (cid:32) e − λσ ( a j + σ ) α j (cid:89) ≤ k ≤ n,k (cid:54) = j ( a k + σ ) α k ( σ − a k ) α k +1 (cid:33) | σ = a j = ( (cid:89) ≤ k ≤ n a k ) (cid:88) ≤ j ≤ n ( − α j α j ! (cid:18) ddσ (cid:19) α j (cid:32) e − λσ ( a j + σ ) α j (cid:89) ≤ k ≤ n,k (cid:54) = j ( a k + σ ) α k ( a k − σ ) α k +1 (cid:33) | σ = a j . Since F α ( a, + ∞ ) = 1 , thanks to Lemma 9.7, we find eventually that F α ( a ) = F α ( a, 1) = (cid:90) ∞ ∂F α ∂λ ( a, λ ) dλ + 1 = 1 − K α ( a ) K α ( a ) = ( (cid:89) ≤ k ≤ n a k ) (cid:88) ≤ j ≤ n ( − α j α j ! (cid:90) + ∞ (cid:18) ddσ (cid:19) α j (cid:32) e − λσ ( a j + σ ) α j (cid:89) ≤ k ≤ n,k (cid:54) = j ( a k + σ ) α k ( a k − σ ) α k +1 (cid:33) | σ = a j dλ = (cid:88) ≤ j ≤ n ( − α j α j ! (cid:90) + ∞ e − λa j (cid:18) ddσ − λ (cid:19) α j (cid:32) ( a j + σ ) α j a j (cid:89) ≤ k ≤ n,k (cid:54) = j ( a k + σ ) α k a k ( a k − σ ) α k +1 (cid:33) | σ = a j dλ. = (cid:88) ≤ j ≤ n ( − α j α j ! (cid:90) + ∞ e − λa j (cid:18) ddσ − λ (cid:19) α j (cid:32) ( a j + σ ) α j (cid:89) ≤ k ≤ n,k (cid:54) = j ( a k + σ ) α k ( a k − σ ) α k +1 (cid:33) | σ = a j dλ = (cid:88) ≤ j ≤ n ( − α j α j ! (cid:90) + ∞ a j e − t j (cid:18) dda j s − t j a j (cid:19) α j (cid:32) ( a j + a j s ) α j (cid:89) ≤ k ≤ n,k (cid:54) = j a k ( a k + a j s ) α k ( a k − a j s ) α k +1 (cid:33) | s =1 dt j = (cid:88) ≤ j ≤ n ( − α j α j ! (cid:90) + ∞ a j e − t (cid:18) dds − t (cid:19) α j (cid:32) (1 + s ) α j (cid:89) ≤ k ≤ n,k (cid:54) = j a k ( a k + a j s ) α k ( a k − a j s ) α k +1 (cid:33) | s =1 dt = (cid:88) ≤ j ≤ n ( − α j α j ! (cid:90) + ∞ a j e − t (cid:18) dds − (cid:19) α j (cid:32) ( t + s ) α j (cid:89) ≤ k ≤ n,k (cid:54) = j a k ( a k + a j s/t ) α k ( a k − a j s/t ) α k +1 (cid:33) | s = t dt = (cid:88) ≤ j ≤ n ( − α j α j ! (cid:90) + ∞ a j e − t (cid:18) dd ( s + t ) − (cid:19) α j (cid:32) ( t + s ) α j (cid:89) ≤ k ≤ n,k (cid:54) = j ta k ( t ( a k − a j ) + a j ( s + t )) α k ( t ( a k + a j ) − a j ( s + t )) α k +1 (cid:33) | s + t =2 t dt = (cid:88) ≤ j ≤ n ( − α j × (cid:90) + ∞ a j e − t (cid:18) dds − (cid:19) α j (cid:32) s α j α j ! (cid:89) ≤ k ≤ n,k (cid:54) = j ta k ( t ( a k − a j ) + a j s ) α k ( t ( a k + a j ) − a j s ) α k +1 (cid:33) | s =2 t dt = (cid:88) ≤ j ≤ n ( − α j e − a j (cid:90) + ∞ e − t × (cid:18) dds − (cid:19) α j (cid:32) s α j α j ! (cid:89) ≤ k ≤ n,k (cid:54) = j ( t + a j ) a k (cid:0) ( t + a j )( a k − a j ) + a j s (cid:1) α k (cid:0) ( t + a j )( a k + a j ) − a j s (cid:1) α k +1 (cid:33) | s =2 t +2 a j dt. We have also to deal with (cid:89) ≤ k ≤ n,k (cid:54) = j ( t + a j ) a k (cid:0) ( t + a j )( a k − a j ) + a j s (cid:1) α k (cid:0) ( t + a j )( a k + a j ) − a j s (cid:1) α k +1 and (cid:0) ( t + a j )( a k + a j ) − a j (2 t + 2 a j ) (cid:1) = a j ( a k + a j ) − a j + t ( a k − a j ) = ( t + a j )( a k − a j )( t + a j )( a k + a j ) − a j s = ( t + a j )( a k − a j ) + a j (2 t + 2 a j − s ) so that(3.3.21) K α ( a ) = (cid:88) ≤ j ≤ n ( − α j e − a j (cid:90) + ∞ e − t × (cid:18) dds − (cid:19) α j (cid:18) s α j α j ! (cid:89) ≤ k ≤ n,k (cid:54) = j ( t + a j ) a k (cid:0) ( t + a j )( a k + a j ) + a j ( s − t − a j ) (cid:1) α k (cid:0) ( t + a j )( a k − a j ) − a j ( s − t − a j ) (cid:1) α k +1 (cid:19) | s =2 t +2 a j dt. A conjecture on integrals of products of Laguerre polynomials. Weformulate in this section a conjecture on the behaviour of the functions K α ( a ) ; asdisplayed in the previous sections, we know several useful elements for the analysisof these functions, including some quite explicit expression. However, in the non-isotropic case, we were not able to prove the estimate F α ( a ) ≤ , equivalent to K α ( a ) ≥ , except for the case α = 0 . We are thus reduced to conjectural statements. Conjecture 3.25. Let n ≥ be an integer and let α = ( α , . . . , α n ) ∈ N n . For a = ( a , . . . , a n ) ∈ (0 , + ∞ ) n we define (3.4.1) K α ( a ) = (cid:90) t =( t ,...,t n ) ∈ R n + (cid:80) ≤ j ≤ n t j /a j ≥ e − ( t + ··· + t n ) (cid:89) ≤ j ≤ n ( − α j L α j (2 t j ) dt, where L k stands for the classical Laguerre polynomial (3.4.2) L k ( X ) = (cid:16) ddX − (cid:17) k X k k ! . NTEGRALS OF THE WIGNER DISTRIBUTION 51 Then we conjecture that, assuming < a ≤ · · · ≤ a n , we have (3.4.3) K α ( a ) ≥ (cid:88) ≤ j ≤ n e − a j (cid:81) ≤ l A slightly stronger and more symmetrical version of the above con-jecture is that for n, α, a, K α as above, we have(3.4.4) K α ( a ) ≥ K ( a ) . It is indeed stronger since we have proven in Lemma . that K ( a ) is greater thanthe right-hand-side of (3.4.3). Theorem 3.27. The previous conjecture is a proven theorem in the following cases. (1) When n = 1 . (2) For all n ≥ , when all the a j are equal. (3) For all n ≥ , when α = 0 N n . (4) When n = 2 and min( α , α ) = 0 .Proof. ( ) When n = 1 , we have proven above (in Proposition 3.3) that for α ∈ N , a > ,(3.4.5) K α ( a ) = e − a P α ( a ) ≥ e − a , which is indeed (3.4.4) in that case. With the notations of Theorem 3.5 (and inparticular where D a is defined in (3.1.22)) this implies(3.4.6) Op w ( D a ) ≤ − e − a , an inequality due to P. Flandrin in the 1988 paper [9]. ( ) Assuming that all the a j are equal to a > , we have proven in Theorem 3.7 thatfor α ∈ N n , | α | = (cid:80) ≤ j ≤ n α j ,(3.4.7) K α ( a, . . . , a ) ≥ Γ( n, a )Γ( n ) = e − a (cid:88) ≤ j ≤ n a j − ( j − K ( a, . . . , a ) since from (3.3.5), we have K ( a, . . . , a ) = (cid:90) (cid:80) t j ≥ at j ≥ e − ( t + ··· + t n ) dt = (cid:90) t n ≥ at j ≥ e − ( t + ··· + t n ) dt + (cid:90) a e − t n (cid:90) (cid:80) t j ≥ a − t n e − ( t + ··· + t n − ) dt (inductively) = e − a + (cid:90) a e − t n e − ( a − t n ) (cid:88) ≤ j ≤ n − ( a − t n ) j − ( j − dt n = e − a (cid:16) (cid:88) ≤ j ≤ n − a j j ! (cid:17) = e − a (cid:88) ≤ j ≤ n a j − ( j − , proving (3.4.4) in that case. With(3.4.8) D ( a ) = { ( x, ξ ) ∈ R n , π | x | + | ξ | a ≤ } , this implies that(3.4.9) Op w ( D ( a ) ) ≤ − e − a (cid:88) ≤ j ≤ n a j − ( j − , an inequality proven in the 2010 article [27] by E. Lieb and Y. Ostrover. ( ) When α = 0 N n , we have proven (3.4.3) in Lemma 3.22. ( ) When n = 2 , from the case n = 1 we have K α ( a ) = e − a P α ( a ) , so that fromLemma 3.20, we obtain K α ,α ( a , a ) = e − a P α ( a ) + a (cid:90) e − θa − (1 − θ ) a ( − α L α (2 θa ) P α ( a (1 − θ )) dθ, and if α =0, it means that K ,α ( a , a ) = e − a + a (cid:90) e − θa − (1 − θ ) a P α ( a (1 − θ )) dθ ≥ e − a + a (cid:90) e − θa − (1 − θ ) a dθ = K , ( a , a ) , and the reasoning is identical for α = 0 , concluding the proof of the theorem. (cid:3) We are interested in the Weyl quantization of the indicatrix of(3.4.10) D a ,...,a n = { ( x, ξ ) ∈ R n , π (cid:88) ≤ j ≤ n x j + ξ j a j ≤ } , a j > , and we have a weaker conjecture. Conjecture 3.28 (A weak form of Conjecture 3.25) . With n, α, a, K α as in Con-jecture 3.25, we conjecture that (3.4.11) K α ( a ) ≥ . Note that Inequality (3.4.11) is equivalent to (3.4.12) Op w ( D a ,...,an ) ≤ . Remark 3.29. In the first place, although the second conjecture is much weakerthan the first, there is no reason to believe that the weak conjecture should be easierto prove than the first: in particular, in the known cases, it is indeed the proof ofthe precise statement (3.4.3) which leads to (3.4.11) and we are not aware of a directproof of (3.4.11), even in one dimension. A summary of our knowledge on the functions K α . As proven in Remarks 3.14and 3.15, the functions K α are entire functions given on the open subset (3.3.6) byFormula (3.3.5) (see also Formula (3.3.17)). Moreover the function F α ( a ) = 1 − K α ( a ) can be expressed as a simple integral for a j > ,(3.4.13) F α ( a , . . . , a n ) = (cid:90) R sin τπτ (cid:16) (cid:89) ≤ j ≤ n (1 + iτ µ j ) α j (1 − iτ µ j ) α j +1 (cid:17) dτ, µ j = 1 a j , NTEGRALS OF THE WIGNER DISTRIBUTION 53 and we have an explicit expression of the function K α as a sum of simple integralsin (3.3.21). However, having an explicit expression does not mean much and forinstance, we do have several explicit expressions for the Laguerre polynomials butInequality (9.3.3) remains very hard work, requiring a deep understanding of thesepolynomials. We have also an induction formula in Lemma 3.20. As a furtherremark, we have the following Lemma 3.30. Let n, α, a, K α as in Conjecture . . Then we have lim a n → + ∞ K α ,...,α n − ,α n ( a , . . . , a n − , a n ) = K α ,...,α n − ( a , . . . , a n − ) , (3.4.14) lim a → + K α ,α ,...,α n ( a , a , . . . , a n ) = 1 . (3.4.15) Proof. Formula (3.3.5) and Lebesgue Dominated Convergence Theorem imply thefirst equality (3.4.14). Lemma 3.20, in which we may swap the variables a and a n gives for a > K α ,α ,...,α n ( a , a , . . . , a n ) = e − a P α ( a )+ a (cid:90) e − θa ( − α L α (2 a θ ) K α ,...,α n (cid:0) a (1 − θ ) , . . . , a n (1 − θ ) (cid:1) dθ, and since P α is a polynomial such that P α (0) = 1 , we get (3.4.15). (cid:3) Reasons to believe in the conjecture . This is true in one dimension, also in n dimensions for spheres and it is a quadratic problem in the sense that ellipsoids areconvex subsets of R n characterized by an inequality { X ∈ R n , p ( X ) ≤ } , where p is a polynomial of degree 2 with a positive-definite quadratic part. Weshall see below in this paper that convexity of a set A does not guarantee thatthe quantization Op w ( A ) is smaller than 1 as an operator and that Flandrin’sconjecture is not true, but it is hard to believe that such a phenomenon could occurfor ellipsoids. We must point out a specific feature of anisotropy related to Mehler’sformula (2.2.1): if all the µ j are equal to the same µ > (this is the isotropic case),then, with q µ ( x, ξ ) = µ ( | x | + | ξ | ) , we haveOp w ( e iπτq µ ( x,ξ ) ) = φ ( τ µ ) e i arctan( τµ ) (cid:80) ≤ j ≤ n π ( x j + D j ) , where φ ( τ µ ) is a scalar quantity. As a consequence, if we quantize F ( q µ ( x, ξ )) , weget Op w (cid:0) F (cid:0) q µ ( x, ξ ) (cid:1)(cid:1) = (cid:90) R ˆ F ( τ ) φ ( τ µ ) e i arctan( τµ ) µ π Op w ( q µ ) dτ, and thusOp w (cid:0) F (cid:0) q µ ( x, ξ ) (cid:1)(cid:1) = (cid:101) F ( Op w ( q µ )) , (cid:101) F ( λ ) = (cid:90) R ˆ F ( τ ) φ ( τ µ ) e iπ arctan( τµ ) µ λ dτ, and Op w (cid:0) F (cid:0) q µ ( x, ξ ) (cid:1)(cid:1) appears as a function of the self-adjoint operator Op w ( q µ ) .Following the same route in the anisotropic case, we get, with q µ ( x, ξ ) = (cid:88) ≤ j ≤ n µ j ( x j + ξ j ) , (3.4.16) Op w (cid:0) F (cid:0) q µ ( x, ξ ) (cid:1)(cid:1) = (cid:90) R ˆ F ( τ ) φ ( τ µ ) e iπ (cid:80) ≤ j ≤ n ( arctan( τµjµj ) µ j ( x j + D j ) dτ, (3.4.17)and since µ j arctan( τ µ j ) does depend on µ j (and not only on τ ), the operatorOp w (cid:0) F (cid:0) q µ ( x, ξ ) (cid:1)(cid:1) is not a function of the self-adjoint operator Op w ( q µ ) .As a final comment on the strongest form of the Conjecture (3.4.4), we would saythat it could be seen as a property of the Laguerre polynomials, known in the case n = 1 , where it stands as follows: we define for k ∈ N , the polynomial P k by(3.4.18) P k ( x ) = (cid:90) + ∞ e − t ( − k L k (2 x + 2 t ) dt, and we have P k (0) = 1 from (9.3.5). Moreover, we have the inequality (equivalentto (3.4.4) for n = 1 )(3.4.19) ∀ x ≥ , P k ( x ) ≥ P k (0) . We note that e − x P k ( x ) = (cid:82) + ∞ x e − s ( − k L k (2 s ) ds, so that the unique solution P k ofthe Initial Value Problem for the ODE(3.4.20) P k ( x ) − P (cid:48) k ( x ) = ( − k L k (2 x ) , P k (0) = 1 , does satisfy (3.4.19). We note that from Lemma 3.2, we have P (cid:48) k ( X ) = 2 (cid:88) ≤ l There are several classical results on products of Laguerre polyno-mials, in particular the article [5], On some expansions in Laguerre polynomials byA. Erdélyi and also the paper [28], Linearization of the products of the generalizedLauricella polynomials and the multivariate Laguerre polynomials via their integralrepresentations by Shuoh-Jung Liu, Shy-Der Lin, Han-Chun Lu and H. M. Srivas-tava. However it seems that the non-negativity of the polynomials P α ;1 , P (cid:48) α ;1 do notsuffice to tackle the conjecture in two dimensions and more.4. Parabolas Preliminary remarks. We start with a picture, demonstrating that the epi-graph of a parabola is an increasing union of ellipses. It is easy to see that theepigraph of a parabola, i.e. the set { ( x, ξ ) ∈ R , ξ > x } is a countable increasingunion of ellipses in the sense that(4.1.1) P = { ( x, ξ ) ∈ R , ξ > x } = ∪ k ≥ { ( x, ξ ) ∈ R , ξ > x + k − ξ } (cid:124) (cid:123)(cid:122) (cid:125) E k . Note that for k ≥ we have E k ⊂ E k +1 ⊂ P since x + k − ξ ≥ x + ( k + 1) − ξ > x , from the fact that ξ > on E k . Moreover, if ξ > x and k > ξ/ (cid:112) ξ − x , we get ( x, ξ ) ∈ E k . Figure 4. The epigraph of a parabola is an increasing union of ellipses. Remark 4.1. The ellipse E k is symplectically equivalent to a circle with area πk since x + k − ξ − ξ = x + k − (cid:0) ξ − k (cid:1) − k λ − y ) + k − (cid:0) λη − k (cid:1) − k λ − y + λ k − (cid:0) η − k λ (cid:1) − k , so that choosing λ such that λ − = λ k − , e.g. λ = √ k , we get x + k − ξ − ξ = k − (cid:0) y + ( η − k λ ) (cid:1) − k , and E k = { ( y, ζ ) ∈ R , y + ζ < k } , where ( y, ζ ) are the affine symplectic coordi-nates y = xk / , ζ = ξk − / − k / . Lemma 4.2. Let u ∈ S ( R ) . Then W ( u, u ) belongs to S ( R ) and with E , E k definedby (4.1.1) , we have (cid:120) ξ>x W ( u, u )( x, ξ ) dxdξ = lim k → + ∞ (cid:120) E k W ( u, u )( x, ξ ) dxdξ ≤ (cid:107) u (cid:107) L ( R ) . NTEGRALS OF THE WIGNER DISTRIBUTION 57 Proof. Since W ( u, u ) belongs to S ( R n ) ⊂ L ( R n ) , we may apply the LebesgueDominated Convergence Theorem and (4.1.1) to obtain the equality in the lemma.On the other hand Theorem 3.5 and Remark 4.1 imply (cid:120) E k W ( u, u )( x, ξ ) dxdξ = (cid:104) Op w ( E k ) u, u (cid:105) ≤ (1 − e − πk ) (cid:107) u (cid:107) L ( R ) ≤ (cid:107) u (cid:107) L ( R ) , and the sought result. (cid:3) Remark 4.3. Moreover, Theorem 3.5 and the expression of F ( a ) = 1 − e − a implythat with ψ defined in (9.1.31), we have (cid:120) E k W ( ψ , ψ )( x, ξ ) dxdξ = (cid:104) Op w ( E k ) ψ , ψ (cid:105) = (cid:107) ψ (cid:107) L ( R ) (1 − e − πk / ) , so that from Lemma 4.2, we have (cid:115) P W ( ψ , ψ )( x, ξ ) dxdξ = (cid:107) ψ (cid:107) L ( R ) , entailing(4.1.2) sup φ ∈ S ( R ) , (cid:107) φ (cid:107) L R ) =1 (cid:120) P W ( φ, φ )( x, ξ ) dxdξ = 1 . Remark 4.4. We want to study the operator with Weyl symbol H ( ξ − x ) ( H = R + is the Heaviside function) and since ξ − x is a polynomial with degree less than 2,see from (1.2.4) that Op w ( H ( ξ − x )) commutes with D x − x = e πix / D x e − πix / ,and the latter has (continuous) spectrum R : we expect thus that Op w ( H ( ξ − x )) should have continuous spectrum and be conjugated to a Fourier multiplier.4.2. Calculation of the kernel. The Weyl symbol of the operator Op w ( P ) is H ( ξ − x ) , ( P is defined in (4.1.1), H is the Heaviside function H = R + ), corresponding to thedistribution kernel k P ( x, y ) obtained from Proposition 1.9 by (we use freely integralsmeaning only Fourier transform in the distribution sense), k P ( x, y ) = (cid:90) e iπ ( x − y ) ξ H ( ξ − ( x + y ) dξ = (cid:90) e iπ ( x − y )( ξ +( x + y ) ) H ( ξ ) dξ = e iπ ( x − y )( x + y ) (cid:0) δ ( y − x ) + 1 iπ ( y − x ) (cid:1) = δ ( y − x )2 + e iπ ( x − y )( x + y ) iπ ( y − x ) . We have x − y )( x + y = ( x − y )( x + y ) = x − y + x y − y x = 43 ( x − y ) + 13 ( y − x ) , so that(4.2.1) k P ( x, y ) = e i π x (cid:32) δ ( y − x )2 + e i π ( y − x ) iπ ( y − x ) (cid:33) e − i π y , and the operator Op w ( P ) is unitarily equivalent to the operator with kernel(4.2.2) ˜ k ( x, y ) = δ ( y − x )2 + e i π ( y − x ) iπ ( y − x ) . We have proven the following result. Lemma 4.5. The operator with Weyl symbol R (cid:51) ( x, ξ ) (cid:55)→ R + ( ξ − x ) has thedistribution kernel k P ( x, y ) = e i π x (cid:32) δ ( y − x )2 + e i π ( y − x ) iπ ( y − x ) (cid:33) e − i π y , and is thus unitarily equivalent to (4.2.3) Id2 + convolution with ie − iπt / π pv t . Lemma 4.6. The distribution ie − iπt / π pv t has the Fourier transform (4.2.4) π (cid:90) sin(2 πasτ + s ) s ds, a = (2 /π ) / . The operator (4.2.3) is the Fourier multiplier ω ( D t ) with (4.2.5) ω ( τ ) = 12 (cid:32) π (cid:90) + ∞−∞ sin( sη + s ) s ds (cid:33) , η = 2 / π / τ. Proof. We calculate in the distribution sense ( t = as, a = (2 /π ) / ), (cid:90) e − iπtτ i e − iπt / πt dt = i π (cid:90) e − iπasτ e − iπa s / s ds = i π (cid:90) ( − i ) sin( s + 2 πasτ ) s ds = 12 π (cid:90) sin(2 πasτ + s ) s ds, so that with η = 2 πaτ , we get ω ( τ ) = 12 (cid:32) π (cid:90) + ∞−∞ sin( sη + s ) s ds (cid:33) = 12 (cid:0) − F ( η ) (cid:1) = G ( η ) , proving the lemma. (cid:3) Lemma 4.7. We have, with η = 2 / π / τ , ω ( τ ) = 12 (cid:32) π (cid:90) + ∞−∞ sin( sη + s ) s ds (cid:33) = G ( η ) , ω (0) = 23 = G (0) , (4.2.6) G (cid:48) ( η ) = 12 π (cid:90) R cos( sη + s ds = Re 12 π (cid:90) R exp i ( sη + s ds = Ai ( η ) , (4.2.7) G ( η ) = 23 + (cid:90) η Ai ( ξ ) dξ, (4.2.8) where Ai is the Airy function defined as the inverse Fourier transform of t (cid:55)→ e i (2 πt ) / . NTEGRALS OF THE WIGNER DISTRIBUTION 59 Proof. We have(4.2.9) π (cid:90) + ∞−∞ sin( s ) s ds = 1 π (cid:90) + ∞−∞ sin( σ )3 / σ / / σ − / dσ = 13 π (cid:90) + ∞−∞ sin σσ dσ = 13 , proving (4.2.6). We have also G ( η ) = 12 + Im (cid:110) Inverse Fourier Transform (cid:8) y (cid:55)→ e i (2 πy ) / pv ( 12 πy ) (cid:9)(cid:111) , and thus G (cid:48) ( η ) = Im (cid:110) Inverse Fourier Transform (cid:8) y (cid:55)→ e i (2 πy ) / i (cid:9)(cid:111) = Im (cid:18)(cid:90) e iπyη e i (2 πy ) / idy (cid:19) = Im (cid:18) π (cid:90) e itη e it / idt (cid:19) = Ai ( η ) , which is (4.2.7), implying (4.2.8). (cid:3) Lemma 4.8. With G defined in Lemma 4.7, we get that G is an entire function,real-valued on the real line such that (4.2.10) lim η → + ∞ G ( η ) = 1 , lim η →−∞ G ( η ) = 0 , and moreover with η the largest zero of the Airy function ( η ≈ − . ), thefunction G has an absolute minimum at η with G ( η ) ≈ − . , (4.2.11) ∀ η ∈ R , G ( η ) ≤ G ( η ) < . Proof. The first statements follow from Lemma 4.7 and (4.2.10) is implied by (4.2.8)and (9.6.34), (9.6.38). The strict inequality in (4.2.11) follows for η ≥ from (4.2.7)since Ai is positive on [0 , + ∞ ) so that G is strictly increasing there from G (0) = 2 / to G (+ ∞ ) = 1 . The other statements are proven in Section 9.6 of the Appendix. (cid:3) The main result. Collecting the results of Lemmas 4.5, 4.6, 4.7, 4.8 and ofSection 9.6 in the Appendix, we have proven the following theorem. Theorem 4.9. Let H ( ξ − x ) = { ( x, ξ ) ∈ R , ξ ≥ x } be the indicatrix of theepigraph of the parabola with equation ξ = x . Then the operator with Weyl symbol H ( ξ − x ) is unitary equivalent to the Fourier multiplier G (2 / π / τ ) where (4.3.1) G ( η ) = 23 + (cid:90) η Ai ( ξ ) dξ = (cid:90) η −∞ Ai ( ξ ) dξ, ( Ai is the Airy function) . The function G is entire on C , real valued on the real line and such that G ( R ) = [ G ( η ) , , where η is the largest zero of the Airy function (4.3.2) we have η ≈ − . , G ( η ) ≈ − . . The operator with Weyl symbol H ( ξ − x ) is self-adjoint bounded on L ( R ) with norm , with spectrum equal to [ G ( η ) , (continuous spectrum) and (4.3.3) ∀ u ∈ L ( R ) , G ( η ) (cid:107) u (cid:107) L ( R ) ≤ (cid:120) ξ ≥ x W ( u, u )( x, ξ ) dxdξ ≤ (cid:107) u (cid:107) L ( R ) . - - - - - - - Figure 5. The function G . More details on G are given in the Ap-pendix 9.6.4.4. Paraboloids, a conjecture. We are interested now in multi-dimensional ver-sions of the previous results, namely, we would like to find a bound for integrals ofthe Wigner distribution on paraboloids of R n for n ≥ . Let us start with recallingTheorem 21.5.3 in [17], a version of which was given in our Theorem 3.12 in thepositive-definite case.4.4.1. On non-negative quadratic forms. Theorem 4.10 (Symplectic reduction of quadratic forms, Theorem 21.5.3 in [17]) . Let q be a non-negative quadratic form on R n × R n equipped with the canonicalsymplectic form (1.2.20) . Then there exists S in the symplectic group Sp ( n, R ) of R n , r ∈ { , . . . , n } , µ , . . . , µ r positive, and s ∈ N such that r + s ≤ n , NTEGRALS OF THE WIGNER DISTRIBUTION 61 so that for all X = ( x, ξ ) ∈ R n × R n , (4.4.1) q ( SX ) = (cid:88) ≤ j ≤ r µ j ( x j + ξ j ) + (cid:88) r +1 ≤ j ≤ r + s x j . Definition 4.11. Let n ∈ N ∗ and let R n be equipped with the canonical symplecticform (1.2.20) . Let q be a non-negative quadratic form on R n with rank n − and T be a non-zero vector in R n such that q ( σT ) = 0 . A paraboloid P of R n withvertex and shape ( q, T ) is defined by (4.4.2) P = { X ∈ R n , q ( X ) ≤ [ X, T ] } . A paraboloid Q with vertex m ∈ R n and shape ( q, T ) is defined as (4.4.3) Q = P + m, where P is a paraboloid with vertex and shape ( q, T ) . Remark 4.12. We can find some symplectic coordinates such that q ( X ) − [ X, T ] = (cid:88) ≤ j ≤ r µ j ( x j + ξ j ) + (cid:88) r +1 ≤ j ≤ r + s x j + (cid:88) ≤ j ≤ n ( x j τ j − ξ j t j ) , with r + s = 2 n − . We can get rid of the linear terms x j τ j − ξ j t j when ≤ j ≤ r by writing µ j ( x j + ξ j ) + x j τ j − ξ j t j = µ j (cid:0) x j + τ j µ j (cid:1) + µ j (cid:0) ξ j − t j µ j (cid:1) − µ j ( t j + τ j ) , and also of x j τ j for r + 1 ≤ j ≤ r + s , since x j + x j τ j = ( x j + τ j − τ j . We are left with using affine symplectic coordinates ( y, η ) so that q ( X ) − [ X, T ] = (cid:88) ≤ j ≤ r µ j ( y j + η j ) + (cid:88) r +1 ≤ j ≤ r + s y j − (cid:88) r +1 ≤ j ≤ r + s η j t j + (cid:88) r + s +1 ≤ j ≤ n ( y j τ j − η j t j ) − a. Since we have r + s = 2 n − , we get r + s + 1 = 2 n − r : we cannot have r + s + 1 ≤ n since it would imply that n − r ≤ n and thus r ≥ n , which is incompatible with r + s = 2 n − , r, s ≥ . We get then that s = 2 l + 1 , r = n − − l and since r + s ≤ n, ≤ s , we have l = 0 , s = 1 , r = n − , and q ( X ) − [ X, T ] = (cid:88) ≤ j ≤ n − µ j ( y j + η j ) + y n − η n t n − a, and t n ∈ R ∗ . With y n = t / ˜ y n , η n = t − / ˜ η n , we get q ( X ) − [ X, T ] = (cid:88) ≤ j ≤ n − µ j ( y j + η j ) + t / (˜ y n − ˜ η n − at − / ) , and the inequality q ( X ) − [ X, T ] ≤ is equivalent to (cid:88) ≤ j ≤ n − t − / µ j ( y j + η j ) + ˜ y n ≤ ˜ η n + at − / . We can thus assume ab initio that our paraboloid is given by the inequality(4.4.4) (cid:88) ≤ j ≤ n − ν j ( x j + ξ j ) + x n ≤ ξ n . On the kernel for the paraboloid. We shall consider the paraboloid(4.4.5) P n = { ( x, ξ ) ∈ R n , x n + (cid:88) ≤ j ≤ n − ( x j + ξ j ) ≤ ξ n } . We have, with X (cid:48) = ( x (cid:48) ; ξ (cid:48) ) = ( x , . . . , x n − ; ξ , . . . , ξ n − ) , P = Op w (cid:0) H ( ξ n − x n − | X (cid:48) | ) (cid:1) = (cid:90) R ˆ H ( τ ) Op w ( e iπτ ( ξ n − x n ) ) Op w ( e − iπτ | X (cid:48) | ) dτ = (cid:88) k ≥ (cid:90) R ˆ H ( τ ) P k ; n − ⊗ Op w ( e iπτ ( ξ n − x n ) ) e − i (arctan τ )(2 k + n − (1 + τ ) − ( n − dτ = 12 Id+ 12 iπ (cid:88) k ≥ P k ; n − ⊗ (cid:90) R Op w ( e iπτ ( ξ n − x n ) ) 1 τ (cid:16) − iτ (1 + τ ) / (cid:17) k + n − (1 + τ ) − ( n − dτ = 12 Id + 12 (cid:88) k ≥ P k ; n − ⊗ (cid:90) R Op w ( e iπτ ( ξ n − x n ) ) (1 − iτ ) k iπτ (1 + iτ ) k + n − dτ. Let k ( x n , y n ) be the kernel of the operator in the integral: we have k ( x n , y n ) = e iπ ( x n − y n ) e − iπ ( x n − y n ) iπ ( x n − y n ) (1 + i ( x n − y n )) k (1 − i ( x n − y n )) k + n − . As a result, we find that P is unitarily equivalent to ˜ P , with(4.4.6) P = (cid:88) k ≥ P k ; n − ⊗ (cid:0) I n + convolution with ie − iπ x n πx n (1 + ix n ) k (1 − ix n ) k + n − (cid:1) . We define(4.4.7) ω k,n − ( τ ) = 12 + (cid:90) ie − iπ t πt (1 + it ) k (1 − it ) k + n − e − iπtτ dt = 12 + (cid:90) e iπ t iπt (1 − it ) k (1 + it ) k + n − e iπtτ dt, and we get that(4.4.8) ˜ P = (cid:88) k ≥ P k ; n − ⊗ ω k,n − ( D x n ) . We note that for n = 1 , the sum is reduced to k = 0 with P = I , so that werecover Formula (4.2.6) with ω , = ω . We find also that(4.4.9) ω (cid:48) k,n − ( τ ) = (cid:90) e iπ t (1 − it ) k (1 + it ) k + n − e iπtτ dt, NTEGRALS OF THE WIGNER DISTRIBUTION 63 in the sense that the inverse Fourier transform of t (cid:55)→ e iπ t (1 − it ) k (1+ it ) k + n − is the distribu-tion derivative of ω k,n − . Going back to the normalization of Lemma 4.7, we have,with η = 2 / π / τ , G k,n − ( η ) = ω k,n − ( τ ) , (4.4.10) G (cid:48) k,n − ( η ) = 2 − / π − / (cid:90) e iπ t (1 − it ) k (1 + it ) k + n − e − iπ tη dt, = (cid:124)(cid:123)(cid:122)(cid:125) t = π − s π (cid:90) e is (1 − iπ − / / s ) k (1 + iπ − / / s ) k + n − e isη ds := A k,n − ( η ) . (4.4.11)We have A , = Ai and A k,n − is an entire function, real-valued on the real line; wehave G k,n − ( η ) = (cid:90) η −∞ A k,n − ( ξ ) dξ, G k,n − (+ ∞ ) = 1 . Remark 4.13. We claim that the asymptotic properties of the functions A k,n − areanalogous to the properties of the standard Airy function and we have indeed from(4.4.9),(4.4.12) ω (cid:48) k,n − ( τ ) = (1 − iD ) k (1 + iD ) − k − n +1 F − ( e iπ t ) . We claim as well that − < inf k ≥ ,η ∈ R G ( η ) < , sup k ≥ ,η ∈ R G ( η ) = 1 , so that ˜ P is bounded on L ( R n ) and(4.4.13) (cid:90) ξ n ≥ x n + (cid:80) ≤ j ≤ n − ( x j + ξ j ) W ( u, u )( x, ξ ) dxdξ ≤ (cid:107) u (cid:107) L ( R n ) . Conics with eccentricity greater than 1 We want to consider now integrals of the Wigner distribution on “hyperbolic”convex subsets of the plane such as(5.0.1) C σ = { ( x, ξ ) ∈ R , xξ ≥ σ, x ≥ } , where σ is a non-negative parameter. It is convenient to start with the limit-casewhere σ = 0 and C = { ( x, ξ ) ∈ R , x ≥ , ξ ≥ } (we will label C as the quarter-plane). The indicatrix function of C is H ( x ) H ( ξ ) where H = R + is the Heavisidefunction. Acknowledgements. The author is grateful to Thomas Duyckaerts for sharp com-ments on a first version of this section. N.B. The reader will see a great similarity between our calculations below in thissection and the J.G. Wood & A.J. Bracken paper [37]. This article is very importantfor the problem at stake – Integrating the Wigner distribution on subsets of the phasespace – and was a wealthy source of information for us, although as a mathematician,the author has a quite rigid relationship with calculations, and feels the need to justify formal manipulations; for instance, we may point out that the test functionsused in [37] are homogeneous distributions of type x − + iω ± , ω ∈ R , which are not in L ( R ) (not even in L loc ), a situation which raises some difficulties,first when you try to normalize in L these test functions and also when tryingto give a non-formal meaning to their images under the operator with Weyl sym-bol H ( x ) H ( ξ ) , images which are not clearly defined. In our joint paper [4] withB. Delourme and T. Duyckaerts, proving that Flandrin’s conjecture is not true, wefollowed numerical arguments which were quite apart from the arguments of [37].However, in this article, we do follow many of the arguments of [37], avoiding formalcalculations.5.1. The quarter-plane, a counterexample to Flandrin’s conjecture. Preliminaries. We study in this section the operator(5.1.1) A = Op w ( H ( x ) H ( ξ )) where H = R + , that is the Weyl quantization of the characteristic function of thefirst quarter of the plane. Lemma 5.1. The operator A given by (5.1.1) is bounded self-adjoint on L ( R ) .Proof. Since the Weyl symbol of A is real-valued, A is formally self-adjoint andit is enough to prove that A is bounded on L ( R ) . Let us start with recalling theclassical formulas(5.1.2) ˆ H ( t ) = δ ( t )2 + 12 iπ pv (cid:18) t (cid:19) , (cid:100) sign = 1 iπ pv (cid:18) t (cid:19) , useful below. The kernel of A is(5.1.3) k ( x, y ) = H ( x + y ) ˆ H ( y − x ) = H ( x + y ) 12 (cid:16) δ ( y − x ) + 1 iπ pv y − x (cid:17) . For λ > , we define A ,λ = (cid:0) H ( x ) [0 ,λ ] ( ξ ) (cid:1) w , whose distribution-kernel is the L ∞ ( R n ) function k ,λ ( x, y ) = H ( x + y ) e iπ ( x − y ) λ sin( π ( x − y ) λ ) π ( x − y ) . There is no difficulty at defining the product S (cid:0) ( x + y ) / (cid:1) T ( x − y ) for S, T tempered distributionson the real line since we may use the tensor product with (cid:104) S ( x + y T ( x − y ) , Φ( x, y ) (cid:105) S (cid:48) ( R ) , S ( R ) = (cid:104) S ( x ) ⊗ T ( x ) , Φ( x + x , x − x (cid:105) S (cid:48) ( R ) , S ( R ) . However, we shall not use directly Formula (5.1.3), since want to avoid formal manipulation involv-ing for instance meaningless products such as H ( x ) H ( y ) k ( x, y ) . We refer the reader to footnote9 on page 65 and to Remark 5.2 for more details on this matter. NTEGRALS OF THE WIGNER DISTRIBUTION 65 We can thus notice that(5.1.4) k ,λ ( x, y ) = k (cid:91) ,λ ( x,y ) H ( x ) H ( y ) e iπ ( x − y ) λ sin( π ( x − y ) λ ) π ( x − y )+ H ( x + y ) (cid:0) H ( − x ) H ( y ) + H ( x ) H ( − y ) (cid:1) sin( π ( x − y ) λ ) π ( x − y ) e iπ ( x − y ) λk (cid:93) ,λ ( x,y ) , and the operator with distribution-kernel k (cid:91) ,λ is H Op w ( [0 ,λ ] ( ξ )) H, that is H [0 ,λ ] ( D ) H ,where H stands for the operator of multiplication by the Heaviside function H . Onthe other hand, the operator with distribution kernel k (cid:93) ,λ is such that | k (cid:93) ,λ ( x, y ) | ≤ H ( x + y ) H ( − x ) H ( y ) + H ( x ) H ( − y ) π | x − y | = H ( x + y ) H ( − x ) H ( y ) π ( y − x ) + H ( x + y ) H ( x ) H ( − y ) π ( x − y ) . According to Proposition 9.13 in our Appendix, the Hardy operator and the modifiedHardy operators are bounded on L ( R ) and we obtain that, for φ, ψ ∈ S ( R n ) , with H = H ( x ) , ˇ H = H ( − x ) ,(5.1.5) (cid:12)(cid:12)(cid:12) (cid:120) H ( x ) [0 ,λ ] ( ξ ) W ( φ, ψ )( x, ξ ) dxdξ (cid:12)(cid:12)(cid:12) ≤ (cid:107) Hφ (cid:107) L ( R ) (cid:107) Hψ (cid:107) L ( R ) + 12 (cid:107) Hφ (cid:107) L ( R ) (cid:107) ˇ Hψ (cid:107) L ( R ) + 12 (cid:107) ˇ Hφ (cid:107) L ( R ) (cid:107) Hψ (cid:107) L ( R ) , so that(5.1.6) |(cid:104) A φ, ψ (cid:105) S ∗ ( R ) , S ( R ) | = (cid:12)(cid:12)(cid:12) (cid:120) H ( x ) H ( ξ ) ∈ S ( R ) W ( φ, ψ )( x, ξ ) dxdξ (cid:12)(cid:12)(cid:12) = lim λ → + ∞ (cid:12)(cid:12)(cid:12) (cid:120) H ( x ) [0 ,λ ] ( ξ ) W ( φ, ψ )( x, ξ ) dxdξ (cid:12)(cid:12)(cid:12) ≤ (cid:107) Hφ (cid:107) L ( R ) (cid:107) Hψ (cid:107) L ( R ) + 12 (cid:107) Hφ (cid:107) L ( R ) (cid:107) ˇ Hψ (cid:107) L ( R ) + 12 (cid:107) ˇ Hφ (cid:107) L ( R ) (cid:107) Hψ (cid:107) L ( R ) , yielding the L -boundedness of the operator A , and this concludes the proof of thelemma. (cid:3) Remark 5.2. That cumbersome detour with the operator A ,λ is useful to ensurethat the operator A is indeed bounded on L ( R ) . The kernel k of A is a distributionof order 1 and the product H ( x ) H ( y ) k ( x, y ) is not a priori meaningful, even when k is a Radon measure . However with the proven L -boundedness of A , the products Even a wave-front-set approach, which would allow the product H ( x ) pv (1 / ( y − x )) , does notoffer a meaning for the product H ( x ) H ( y ) pv (1 / ( y − x )) since the wave-front-set of pv (1 / ( y − x )) islocated on the conormal of the first diagonal (i.e. { ( x, x ; ξ, − ξ ) } x ∈ R ,ξ ∈ R ∗ ), whereas the wave-frontset at (0 , of H ( x ) H ( y ) contains all directions and in particular is antipodal to the conormal ofthe diagonal at (0 , . of operators HA H , ˇ HA H , HA ˇ H , ˇ HA ˇ H make sense and for instance we mayapproximate in the strong-operator-topology the operator HA H by the operator χ ( · /ε ) Aχ ( · /ε ) , where χ is a smooth function supported in [1 , + ∞ ) and equal to on [2 , + ∞ ) . We have indeed HAH = (cid:0) H − χ ( · /ε ) (cid:1) AH + χ ( · /ε ) A (cid:0) H − χ ( · /ε ) (cid:1) + χ ( · /ε ) Aχ ( · /ε ) , so that for u ∈ L ( R ) , HAHu = lim ε → + χ ( · /ε ) Aχ ( · /ε ) u. The operator with kernel H ( x + y ) χ ( x/ε ) χ ( y/ε ) pv iπ ( y − x ) = χ ( x/ε ) χ ( y/ε ) pv iπ ( y − x ) , converges strongly towards the operator H (sign D ) H . Proposition 5.3. Let A = Op w ( H ( x ) H ( ξ )) be the operator with Weyl symbol H ( x ) H ( ξ ) , a priori sending S ( R ) into S (cid:48) ( R ) . Then A can be uniquely extendedto a self-adjoint bounded operator on L ( R ) with (5.1.7) (cid:107) A (cid:107) B ( L ( R )) ≤ √ ≈ . N.B. The bound above can be significantly improved (see Proposition 5.30 for optimalbounds) and moreover we will show below that the spectrum of A actually intersects (1 , + ∞ ) . In fact it is easier to start with the information that A is indeed boundedon L ( R ) .Proof. The L ( R ) -boundedness of A is given by Lemma 5.1. We are left withproving the bound (5.1.7): we note that (5.1.6) implies |(cid:104) A u, u (cid:105) L ( R ) | ≤ (cid:107) Hu (cid:107) L ( R ) + (cid:107) Hu (cid:107) L ( R ) (cid:107) ˇ Hu (cid:107) L ( R ) , proving the proposition, since the eigenvalues of the quadratic form R (cid:51) ( x , x ) (cid:55)→ x + x x are (1 ± √ / . (cid:3) We can do much better and actually diagonalize the operator A , using as inProposition 9.13 logarithmic coordinates on each half-line. We state a lemma on“diagonal” terms whose proof is already given above. Lemma 5.4 (Diagonal terms) . Let A be the operator with Weyl symbol H ( x ) H ( ξ ) .With H standing as well for the operator of multiplication by H ( x ) , we have HA H = HH ( D ) H = H (Id + sign D )2 H. (5.1.8) Lemma 5.5 (Off-diagonal terms) . Let B = 2 Re ˇ HA H = ˇ HA H + HA ˇ H . Thenwe have for all u ∈ L ( R ) , (5.1.9) |(cid:104) B u, u (cid:105) L ( R ) | ≤ (cid:107) Hu (cid:107) L ( R ) (cid:107) ˇ Hu (cid:107) L ( R ) . Proof of the Lemma. For u ∈ S ( R ) such that / ∈ supp u , we define for t ∈ R ,(5.1.10) φ ( t ) = u ( e t ) e t/ , φ ( t ) = u ( − e t ) e t/ , NTEGRALS OF THE WIGNER DISTRIBUTION 67 so that(5.1.11) (cid:107) Hu (cid:107) L ( R ) = (cid:107) φ (cid:107) L ( R ) , (cid:107) ˇ Hu (cid:107) L ( R ) = (cid:107) φ (cid:107) L ( R ) . We have (cid:104) B u, u (cid:105) L ( R ) = (cid:120) H ( x + y ) (cid:0) ˇ H ( x ) H ( y ) + H ( x ) ˇ H ( y ) (cid:1) iπ ( y − x ) u ( y )¯ u ( x ) dydx = (cid:120) H ( − e s + e t ) e s + t iπ ( e t + e s ) φ ( t ) ¯ φ ( s ) dsdt − (cid:120) H ( e s − e t ) e s + t iπ ( e t + e s ) φ ( t ) ¯ φ ( s ) dsdt = (cid:120) H ( t − s )4 iπ cosh( s − t ) φ ( t ) ¯ φ ( s ) dsdt − (cid:120) H ( s − t )4 iπ cosh( s − t ) φ ( t ) ¯ φ ( s ) dsdt so that (cid:104) B u, u (cid:105) L ( R ) = (cid:104) ˜ S ∗ φ , φ (cid:105) L ( R ) + (cid:104) S ∗ φ , φ (cid:105) L ( R ) , (5.1.12) ˜ S ( t ) = ˇ H ( t )4 iπ cosh( t/ , S ( t ) = iH ( t )4 π cosh( t/ . (5.1.13)We calculate (cid:90) + ∞ dt π cosh( t/ 2) = 12 π [arctan(sinh( t/ + ∞ = 14 = (cid:90) −∞ dt π cosh( t/ , so that(5.1.14) |(cid:104) B u, u (cid:105) L ( R ) | ≤ (cid:107) φ (cid:107) L ( R ) (cid:107) φ (cid:107) L ( R ) = 12 (cid:107) Hu (cid:107) L ( R ) (cid:107) ˇ Hu (cid:107) L ( R ) , proving the estimate of the lemma for u ∈ S ( R ) such that / ∈ supp u . We use nowthat we already know that B is a bounded self-adjoint operator on L ( R ) : let u be a function in L ( R ) and let ( φ k ) k ≥ be a sequence in S ( R ) such that each φ k vanishes in a neighborhood of 0 so that lim k φ k = u in L ( R ) . We find that |(cid:104) B u, u (cid:105) L ( R ) | ≤ |(cid:104) B ( u − φ k ) , u (cid:105) L ( R ) | + |(cid:104) B φ k , u − φ k (cid:105) L ( R ) | + |(cid:104) B φ k , φ k (cid:105) L ( R ) |≤ (cid:107) B (cid:107) B ( L ( R )) (cid:0) (cid:107) u − φ k (cid:107) L ( R ) (cid:107) u (cid:107) L ( R ) + (cid:107) u − φ k (cid:107) L ( R ) (cid:107) φ k (cid:107) L ( R ) (cid:1) + 12 (cid:107) Hφ k (cid:107) L ( R ) (cid:107) ˇ Hφ k (cid:107) L ( R ) , providing readily the result of the lemma since the multiplication by H and ˇ H arebounded operators on L ( R ) . (cid:3) Remark 5.6. The estimate (5.1.9) and Lemma 5.4 are already improving (5.1.7),since the eigenvalues of the quadratic form R (cid:51) ( x , x ) (cid:55)→ x + x x are (2 ±√ / , so that the right-hand-side of (5.1.7) can be replaced by (2 + √ / ≈ . .Anyhow, we shall provide below a diagonalization of A and optimal bounds. Such a sequence is easy to find: a first step is to find a sequence ( ˜ φ k ) k ≥ in the Schwartz spaceconverging in L ( R ) towards u , then consider with a given ω ∈ C ∞ ( R ; [0 , such that ω ( t ) = 0 for | t | ≤ and ω ( t ) = 1 for | t | ≥ , φ k ( x ) = ω ( kx ) ˜ φ k ( x ) . N.B. We shall be a little faster in the sequel on the “cumbersome” detours to avoidformal multiplication of kernels by Heaviside functions but the reader should keep inmind that it is an important point to secure L ( R ) -boundedness before any furthermanipulation of the kernels.5.1.2. An isometric isomorphism. Remark 5.7. The mapping Ψ defined by(5.1.15) Ψ : L ( R ) −→ L ( R ; C ) u (cid:55)→ (cid:16) ( Hu )( e t ) e t/ , ( ˇ Hu )( − e t ) e t/ (cid:17) is an isometric isomorphism of Hilbert spaces: indeed we have (cid:107) u (cid:107) L ( R ) = (cid:90) R | u ( e t ) | e t dt + (cid:90) R | u ( − e t ) | e t dt. Moreover if ( φ , φ ) ∈ L ( R ; C ) , we may define for x ∈ R ∗ u ( x ) = H ( x ) φ (ln x ) x − / + ˇ H ( x ) φ (ln | x | ) | x | − / , and we have Ψ( u )( t ) = (cid:0) φ ( t ) , φ ( t ) (cid:1) . Remark 5.8. Using Lemma 5.4 and Notations (5.1.10) we see that (cid:104) HA Hu, u (cid:105) L ( R ) = 12 (cid:107) φ (cid:107) L ( R ) + (cid:120) iπ pv e ( s + t ) / e t − e s φ ( t ) ¯ φ ( s ) dsdt = 12 (cid:107) φ (cid:107) L ( R ) + (cid:120) iπ pv t − s ) φ ( t ) ¯ φ ( s ) dsdt = (cid:90) R | ˆ φ ( τ ) | (cid:0) 12 + ˆ T ( τ ) (cid:1) dτ, (5.1.16)with(5.1.17) T ( t ) = t t/ pv iπt . We have ˆ T = sign ∗ ρ , with ρ ( τ ) = (cid:90) t t/ e − iπtτ dt, (5.1.18)and we note that the function ρ belongs to S ( R ) , as the Fourier transform of afunction in S ( R ) . Also we have (cid:90) ρ ( τ ) dτ = ˆ ρ (0) = 12 , and this yields with ddτ (cid:110) + ˆ T (cid:111) = 2 ρ (which follows from (5.1.18)) and(5.1.19) 12 + ˆ T ( τ ) = 1 − (cid:90) + ∞ τ ρ ( τ (cid:48) ) dτ (cid:48) , since ddτ (cid:26) 12 + ˆ T + (cid:90) + ∞ τ ρ ( τ (cid:48) ) dτ (cid:48) (cid:27) = 0 and lim τ → + ∞ (sign ∗ ρ )( τ ) = 12 . NTEGRALS OF THE WIGNER DISTRIBUTION 69 Theorem 5.9. Let A be the operator with Weyl symbol H ( x ) H ( ξ ) . The operator A is bounded self-adjoint on L ( R ) so that we may define, with Ψ defined in (5.1.15) , (5.1.20) (cid:101) A = Ψ A Ψ − . The operator (cid:101) A is the Fourier multiplier on L ( R ; C ) given by the matrix (5.1.21) M ( τ ) = + ˆ T ( τ ) ˆ S ( τ )ˆ S ( τ ) 0 , where T , S are defined respectively in (5.1.17) , (5.1.13) . In particular we have with Φ = ( φ , φ ) ∈ L ( R ; C ) , (5.1.22) (cid:104) (cid:101) A Φ , Φ (cid:105) L ( R ; C ) = (cid:90) R e iπtτ (cid:104)M ( τ ) ˆΦ( τ ) , ˆΦ( τ ) (cid:105) C dτ. Remark 5.10. As a consequence of Theorem 5.9, we find that the spectrum ofthe self-adjoint bounded operator A is the closure of the set of eigenvalues of thematrices M ( τ ) when τ runs on the real line.Proof. The proof follows readily from Remarks 5.7, 5.8 and Lemmas 5.4, 5.5. (cid:3) Lemma 5.11. Let N be a × Hermitian matrix N = (cid:32) a a a (cid:33) . Then the eigenvalues λ − ≤ λ + of N are such that (5.1.23) λ − < < < λ + , if and only if (5.1.24) a (cid:54) = 0 and | a | > − a . Proof. The characteristic polynomial of N is p ( λ ) = λ − a λ − | a | and since a is real-valued, has two real roots λ − ≤ λ + . If (5.1.24) holds true, the roots aredistinct and p (0) = −| a | < , p (1) = 1 − a − | a | < , implying (5.1.23). Conversely, if (5.1.23) is satisfied, then p (0) , p (1) are both nega-tive, implying (5.1.24), completing the proof of the lemma. (cid:3) Lemma 5.12. Let us define for ω ∈ R , (5.1.25) I ( ω ) = 14 π (cid:90) + ∞ sin( tω )cosh( t/ dt. Then we have (5.1.26) I ( ω ) = 14 πω + O ( ω − ) , | ω | → + ∞ . Proof. Indeed we have for ω ∈ R ∗ , I ( ω ) = − πω (cid:90) + ∞ ddt cos( tω )cosh( t/ dt = 14 πω (cid:16) − (cid:90) + ∞ cos( tω )(cosh( t/ 12 sinh( t/ dt (cid:17) = 14 πω (cid:0) g ( ω ) (cid:1) , with g ( ω ) = − (cid:90) + ∞ dωdt { sin( tω ) } sech( t/ 2) 12 tanh( t/ dt = 12 ω (cid:90) + ∞ sin( tω ) ddt (cid:8) sech( t/ 2) tanh( t/ (cid:9) dt = − ω (cid:90) + ∞ ddt (cid:8) cos( tω ) (cid:9) ddt (cid:8) sech( t/ 2) tanh( t/ (cid:9) dt = 12 ω (cid:26)(cid:90) + ∞ cos( tω ) d dt (cid:8) sech( t/ 2) tanh( t/ (cid:9) dt + 12 (cid:27) = O ( ω − ) , proving the lemma. (cid:3) Proposition 5.13. The matrix M ( τ ) defined in (5.1.21) is equal to (5.1.27) M ( τ ) = a ( τ ) a ( τ ) a ( τ ) 0 with (5.1.28) − a ( τ ) = (cid:90) + ∞ τ ρ ( τ (cid:48) ) dτ (cid:48) , a ( τ ) = i π (cid:90) + ∞ t/ e − iπτt dt. We have − a ( τ ) = O ( τ − N ) for any N when τ → + ∞ , (5.1.29) Re( a ( τ )) = 18 π τ + O ( τ − ) when τ → + ∞ . (5.1.30) Proof. Formulas (5.1.27), (5.1.28) follow from Theorem 5.9, (5.1.19) and (5.1.13).The estimates (5.1.29) follow from the fact that ρ belongs to the Schwartz classand (5.1.30) is a reformulation of Lemma 5.12. (cid:3) Theorem 5.14. Let A be the operator with Weyl symbol H ( x ) H ( ξ ) , where H isthe Heaviside function. Then A is a bounded self-adjoint operator on L ( R ) suchthat (5.1.31) inf (cid:0) spectrum ( A ) (cid:1) < < < sup (cid:0) spectrum ( A ) (cid:1) . Proof. Using Remark 5.10 and Proposition 5.13 we find that for τ large enough,Conditions (5.1.24) are satisfied, proving readily (5.1.31). (cid:3) NTEGRALS OF THE WIGNER DISTRIBUTION 71 Corollary 5.15 (A counterexample to Flandrin’s conjecture) . There exists a func-tion φ ∈ S ( R ) , with L ( R ) norm equal to 1 such that (5.1.32) (cid:120) x ≥ ,ξ ≥ W ( φ , φ )( x, ξ ) dxdξ > . There exists a > such that (cid:115) ≤ x ≤ a, ≤ ξ ≤ a W ( φ , φ )( x, ξ ) dxdξ > . Remark 5.16. On page 2178 of [9], we find the sentence “it is conjectured that (5.1.33) ∀ u ∈ L ( R ) , (cid:120) C W ( u, u )( x, ξ ) dxdξ ≤ (cid:107) u (cid:107) L ( R ) , is true for any convex domain C ” , a quite mild commitment for the validity of(5.1.33), although that statement was referred to later on as Flandrin’s conjecture inthe literature. The second part of the above corollary is providing a disproof of thatconjecture based upon an “abstract” argument used in the proof of Theorem 5.14;the result of that corollary was already known via a numerical analysis argumentafter our joint work [4] with B. Delourme and T. Duyckaerts. Proof. From Theorem 5.14, we find u ∈ L ( R ) such that (cid:107) u (cid:107) L ( R ) < (cid:104) A u , u (cid:105) . Let ψ ∈ S ( R ) : we have |(cid:104) A u , u (cid:105) − (cid:104) A ψ, ψ (cid:105)| = |(cid:104) A ( u − ψ ) , u (cid:105) + (cid:104) A ψ, u − ψ (cid:105)|≤ (cid:107) A (cid:107) B ( L ( R )) (cid:107) u − ψ (cid:107) L ( R ) (cid:0) (cid:107) u (cid:107) L ( R ) + (cid:107) ψ (cid:107) L ( R ) (cid:1) , and thus if ( ψ k ) k ≥ is a sequence of S ( R ) converging towards u in L ( R ) , we get (cid:107) u (cid:107) L ( R ) < (cid:104) A u , u (cid:105)≤ (cid:104) A ψ k , ψ k (cid:105) + (cid:107) A (cid:107) B ( L ( R )) (cid:107) u − ψ k (cid:107) L ( R ) (cid:0) (cid:107) u (cid:107) L ( R ) + (cid:107) ψ k (cid:107) L ( R ) (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) = σ k , goes to 0 when k → + ∞ . . There exists k ≥ such that for k ≥ k , we have ≤ σ k ≤ (cid:0) (cid:104) A u , u (cid:105) − (cid:107) u (cid:107) L ( R ) (cid:1) = ε , ε > . We obtain that for k ≥ k , (cid:107) u (cid:107) L ( R ) < (cid:104) A u , u (cid:105) ≤ (cid:104) A ψ k , ψ k (cid:105) + ε , and thus (cid:107) ψ k (cid:107) L ( R ) = (cid:107) ψ k (cid:107) L ( R ) − (cid:107) u (cid:107) L ( R ) (cid:124) (cid:123)(cid:122) (cid:125) = θ k , goes to 0 when k → + ∞ + (cid:107) u (cid:107) L ( R ) = θ k + (cid:104) A u , u (cid:105) − ε ≤ θ k + (cid:104) A ψ k , ψ k (cid:105) + ε − ε = (cid:104) A ψ k , ψ k (cid:105) + θ k − ε . Choosing now k ≥ k and k large enough to have θ k < ε / , we get (cid:107) ψ k (cid:107) L ( R ) ≤ (cid:104) A ψ k , ψ k (cid:105) − ε < (cid:104) A ψ k , ψ k (cid:105) , and since for ˜ φ = ψ k , the Wigner distribution W ( ˜ φ, ˜ φ ) belongs to S ( R ) , we have (cid:107) ˜ φ (cid:107) L ( R ) < (cid:104) A ˜ φ, ˜ φ (cid:105) = (cid:120) H ( x ) H ( ξ ) W ( ˜ φ, ˜ φ )( x, ξ ) dxdξ, and noting that this strict inequality above implies that ˜ φ (cid:54) = 0 , we may set φ =˜ φ/ (cid:107) ˜ φ (cid:107) and get the first statement in the corollary. N.B. The proof above is complicated by the fact that the identity (cid:104) a w u, u (cid:105) L ( R n ) = (cid:120) R n a ( x, ξ ) W ( u, u )( x, ξ ) dxdξ, is valid a priori for u ∈ S ( R n ) (and in that case W ( u, u ) belongs to S ( R n ) ), butcould be meaningless as a Lebesgue integral even for Op w ( a ) bounded on L ( R n ) and u ∈ L ( R n ) , since we shall have W ( u, u ) ∈ L ( R n ) but not in L ( R n ) (we shallsee in Section 6 that generically the Wigner distribution of a pulse u in L ( R n ) does not belong to L ( R n ) ).Since W ( φ, φ ) belongs to the Schwartz space of R , the Lebesgue DominatedConvergence Theorem provides the last statement in the Corollary. (cid:3) N.B. The reader will notice that the results of the incoming Section 5.2 in thespecial case σ = 0 imply the results of Section 5.1, which could be then erased,say at the second reading. However, as far as the first – and maybe only – readingis concerned, we checked that most of the computational arguments in the nextsection are much more involved and it seemed worth while to the author to avoidunnecessary complications for the disproof of Flandrin’s conjecture via the quarter-plane example and set apart the more involved examples of the hyperbolic regionstackled in Section 5.2.5.2. Hyperbolic regions. We consider in this section the (5.0.1) set C σ with anon-negative σ .5.2.1. A preliminary observation. We want to consider the operator A σ with Weylsymbol H ( x ) H ( xξ − σ ) and as in Section 5.1.1, we would like to secure the fact that A σ is bounded on L ( R ) . Claim 5.17. For all σ ≥ the operator A σ is bounded self-adjoint on L ( R ) .Proof of the Claim. Let us choose(5.2.1) χ ∈ C ∞ ( R ; [0 , with (cid:40) χ ( t ) = 0 , for t ≤ , χ ( t ) = 1 , for t ≥ . NTEGRALS OF THE WIGNER DISTRIBUTION 73 For φ, ψ ∈ S ( R ) , we have(5.2.2) (cid:104) ( A − A σ ) φ, ψ (cid:105) S ∗ ( R ) , S ( R ) = (cid:120) H ( x ) H ( ξ ) H ( σ − xξ ) W ( φ, ψ )( x, ξ ) (cid:124) (cid:123)(cid:122) (cid:125) ∈ S ( R ) dxdξ = lim (cid:15) → + (cid:120) χ ( x/(cid:15) ) H ( ξ ) H ( σ − xξ ) W ( φ, ψ )( x, ξ ) dxdξ. The kernel k σ,(cid:15) of the operator with Weyl symbol χ ( x/(cid:15) ) H ( ξ ) H ( σ − xξ ) is(5.2.3) (cid:96) σ,(cid:15) ( x, y ) = χ (cid:0) x + y (cid:15) (cid:1) e iπσ x − yx + y sin( πσ ( x − y ) x + y ) π ( x − y ) , and we have (cid:120) (cid:96) σ,(cid:15) ( x, y ) φ ( y ) ¯ ψ ( x ) dydx = (cid:120) χ (cid:0) x + y (cid:15) (cid:1) e iπσ x − yx + y sin( πσ ( x − y ) x + y ) π ( x − y ) φ ( y ) ¯ ψ ( x ) dxdy = (cid:120) χ (cid:0) x + y (cid:1) e iπσ x − yx + y sin( πσ ( x − y ) x + y ) π(cid:15) ( x − y ) φ ( (cid:15)y ) ¯ ψ ( (cid:15)x ) (cid:15) dxdy = (cid:120) χ (cid:0) x + y (cid:1) e iπσ x − yx + y sin( πσ ( x − y ) x + y ) π ( x − y ) (cid:124) (cid:123)(cid:122) (cid:125) m σ ( x,y ) φ ( (cid:15)y ) (cid:15) / (cid:124) (cid:123)(cid:122) (cid:125) φ (cid:15) ( y ) ¯ ψ ( (cid:15)x ) (cid:15) / (cid:124) (cid:123)(cid:122) (cid:125) ¯ ψ (cid:15) ( x ) dydx. (5.2.4)We note that, assuming as we may that σ > ,(5.2.5) | m σ ( x, y ) H ( x ) H ( y ) | = χ (cid:0) x + y (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin( πσ ( x − y ) x + y ) πσ ( x − y ) x + y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σH ( x ) H ( y ) x + y ≤ σH ( x ) H ( y ) x + y , and(5.2.6) | m σ ( x, y ) ˇ H ( x ) H ( y ) | = χ (cid:0) x + y (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin( πσ ( x − y ) x + y ) π ( x − y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˇ H ( x ) H ( y ) ≤ ˇ H ( x ) H ( y ) π ( y − x ) , as well as(5.2.7) | m σ ( x, y ) ˇ H ( y ) H ( x ) | = χ (cid:0) x + y (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin( πσ ( x − y ) x + y ) π ( x − y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˇ H ( y ) H ( x ) ≤ ˇ H ( y ) H ( x ) π ( x − y ) . As a consequence, since we have also m σ ( x, y ) ˇ H ( x ) ˇ H ( y ) ≡ , the inequalities (5.2.5),(5.2.6), (5.2.7), the identities (5.2.4), (5.2.2) and Proposition 9.13 imply that |(cid:104) ( A − A σ ) φ, ψ (cid:105) S ∗ ( R ) , S ( R ) | ≤ πσ (cid:107) Hφ (cid:15) (cid:107) L ( R ) (cid:124) (cid:123)(cid:122) (cid:125) (cid:107) Hφ (cid:107) L R ) (cid:107) Hψ (cid:15) (cid:107) L ( R ) + (cid:107) ˇ Hφ (cid:15) (cid:107) L ( R ) (cid:124) (cid:123)(cid:122) (cid:125) (cid:107) ˇ Hφ (cid:107) L R ) (cid:107) Hψ (cid:15) (cid:107) L ( R ) + (cid:107) Hφ (cid:15) (cid:107) L ( R ) (cid:107) ˇ Hψ (cid:15) (cid:107) L ( R ) , proving that A − A σ is bounded on L ( R ) ; with Proposition 5.3, this implies that A σ is also bounded on L ( R ) , proving the claim. (cid:3) N.B. With that important piece of information in Claim 5.17, we shall be less strictin our manipulations of the kernels and accept below some abuse of language inthese matters.The Weyl quantization of C σ has the kernel(5.2.8) k σ ( x, y ) = H ( x + y ) e iπσ ( x − yx + y ) (cid:16) δ ( y − x ) + 1 iπ pv y − x (cid:17) , a formula to be compared to (5.1.3). Using the Schwartz function φ of Corollary5.15, we get from the Lebesgue Dominated Convergence Theorem that for σ smallenough,(5.2.9) (cid:104) Op w ( C σ ) φ , φ (cid:105) L ( R ) = (cid:120) xξ ≥ σ,x> W ( φ , φ )( x, ξ ) dxdξ > . However, this argument does not work for large positive σ and we must go back toa direct calculation.5.2.2. Diagonal terms. Denoting by A σ the operator with kernel (5.2.8) (and Weylsymbol H ( xξ − σ ) H ( x ) ), we find that for u ∈ S ( R ) , u + = Hu , we have (cid:104) A σ Hu, Hu (cid:105) L ( R ) = (cid:120) e iπσ ( x − yx + y ) (cid:16) δ ( y − x ) + 1 iπ pv y − x (cid:17) u + ( y )¯ u + ( x ) dydx = 12 (cid:107) u + (cid:107) L ( R + ) + (cid:120) R e iπσ ( es − etes + et ) iπ pv e t − e s u + ( e t )¯ u + ( e s ) e s + t dsdt = 12 (cid:107) u + (cid:107) L ( R + ) + (cid:120) R e iπσ tanh( s − t ) iπ pv e ( s + t ) / e t − e s φ ( t ) ¯ φ ( s ) dsdt, with(5.2.10) φ ( t ) = u + ( e t ) e t/ , so that (cid:107) φ (cid:107) L ( R ) = (cid:107) u + (cid:107) L ( R + ) . We get (cid:104) A σ Hu, Hu (cid:105) L ( R ) = 12 (cid:107) u (cid:107) L ( R + ) + 14 iπ (cid:120) R e iπσ tanh( s − t ) sinh( t − s ) φ ( t ) ¯ φ ( s ) dsdt, NTEGRALS OF THE WIGNER DISTRIBUTION 75 and noting that sinh x = xC ( x ) , with C even such that /C ∈ S ( R ) , we find (cid:104) A σ Hu, Hu (cid:105) L ( R ) = 12 (cid:107) φ (cid:107) L ( R ) − iπ (cid:120) R e iπσ tanh( s − t ) ( s − t ) C ( s − t ) φ ( t ) ¯ φ ( s ) dsdt = 12 (cid:107) φ (cid:107) L ( R ) + (cid:104) T σ ∗ φ , φ (cid:105) L ( R ) = (cid:90) R | ˆ φ ( τ ) | (cid:0) 12 + ˆ T σ ( τ ) (cid:1) dτ, (5.2.11)with(5.2.12) T σ ( t ) = 14 te iπσ tanh( t ) sinh( t/ pv iπt . We note that ˆ T σ ( τ ) = sign ∗ ρ σ , with(5.2.13) ρ σ ( τ ) = 14 (cid:90) te iπσ tanh( t ) sinh( t/ e − iπtτ dt, ρ σ ∈ S ( R ) , (5.2.14)since the function R (cid:51) t (cid:55)→ te iπσ tanh( t sinh( t/ belongs to the Schwartz space . Note alsothat the function ρ σ is real-valued on the real line. This entails that(5.2.15) ddτ (cid:110) 12 + ˆ T σ (cid:111) = 2 ρ σ , and since ρ σ ( τ ) = 14 F (cid:110) t (cid:55)→ te iπσ tanh( t/ sinh( t/ (cid:111) , implying (cid:90) R ρ σ ( τ ) dτ = 12 , we get that(5.2.16) lim τ →±∞ ˆ T σ ( τ ) = ± . This yields that(5.2.17) 12 + ˆ T σ ( τ ) − (cid:90) τ + ∞ ρ σ ( τ (cid:48) ) dτ (cid:48) = − (cid:90) τ −∞ ρ σ ( τ (cid:48) ) dτ (cid:48) , where the last equality follows from (5.2.16): indeed we have for τ > , from (5.2.15),(5.2.18) 12 + ˆ T σ ( τ ) − (cid:90) τ + ∞ ρ σ ( τ (cid:48) ) dτ (cid:48) = − (cid:90) τ −∞ ρ σ ( τ (cid:48) ) dτ (cid:48) , and for τ < , 12 + ˆ T σ ( τ ) = (cid:90) τ −∞ ρ σ ( τ (cid:48) ) dτ (cid:48) = 1 + (cid:90) τ + ∞ ρ σ ( τ (cid:48) ) dτ (cid:48) . We note that(5.2.19) ∀ N ∈ N , sup τ ∈ R | τ | N (cid:12)(cid:12)(cid:12)(cid:12) 12 + ˆ T σ ( τ ) − H ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) < + ∞ . Indeed, the iterated derivatives of tanh are polynomials of tanh (check this by induction onthe order of derivatives) and thus bounded on the real line; since the function t (cid:55)→ t/ sinh( t/ belongs to the Schwartz space, this proves that the above product is in S ( R ) . Indeed for τ > , we have, using ρ σ ∈ S ( R ) , (cid:12)(cid:12)(cid:12)(cid:12) τ N (cid:90) τ + ∞ ρ σ ( τ (cid:48) ) dτ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) + ∞ τ | ρ σ ( τ (cid:48) ) | τ (cid:48) N dτ (cid:48) ≤ (cid:90) + ∞ | ρ σ ( τ (cid:48) ) | τ (cid:48) N dτ < + ∞ . Also, for τ < , we have (cid:12)(cid:12)(cid:12)(cid:12) τ N (cid:90) τ −∞ ρ σ ( τ (cid:48) ) dτ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) τ −∞ | ρ σ ( τ (cid:48) ) || τ (cid:48) | N dτ (cid:48) ≤ (cid:90) −∞ | ρ σ ( τ (cid:48) ) || τ (cid:48) | N dτ < + ∞ . This means that the Fourier multiplier + ˆ T σ ( τ ) is somehow “exponentially close”to H ( τ ) for large values of | τ | and in particular close to 1 for large positive valuesof τ . We have also(5.2.20) ˆ T σ ( τ ) = i π (cid:90) R e − iπτt e iπσ tanh( t ) sinh( t/ dt = 12 π (cid:90) + ∞ sin(2 πtτ − πσ tanh( t/ t/ dt. The next lemma provides more precise estimates than (5.2.19). Lemma 5.18. Let τ > , σ ≥ . Defining a ( τ, σ ) = + ˆ T σ ( τ ) as given by (5.2.12) ,we have (5.2.21) | − a ( τ, σ ) | ≤ e − π τ e πσ . Proof. Using (5.2.18) and Lemma 9.16, we find that for τ > , | − a ( τ, σ ) | ≤ (cid:90) + ∞ τ | ρ σ ( τ (cid:48) ) | dτ (cid:48) ≤ (cid:90) + ∞ τ | ρ σ ( τ (cid:48) ) | dτ (cid:48) ≤ e πσ (cid:90) + ∞ τ e − π τ (cid:48) dτ (cid:48) = e πσ π e − π τ , entailing the sought result. (cid:3) Off-diagonal terms. We want now to check the off-diagonal terms: we havewith u ∈ S ( R ) , u + = Hu, u − = ˇ Hu, (5.2.22) φ ( t ) = u + ( e t ) e t/ , φ ( t ) = u − ( − e t ) e t/ (5.2.23) (cid:104) A σ ˇ Hu, Hu (cid:105) L ( R ) = (cid:120) e iπσ ( x − yx + y ) H ( x + y ) ˇ H ( y ) H ( x )2 iπ pv y − x u − ( y )¯ u + ( x ) dydx = (cid:120) e iπσ ( es + etes − et ) H ( e s − e t )2 iπ pv − e t − e s φ ( t ) ¯ φ ( s ) e t + s dtds = (cid:120) e iπσ coth ( s − t ) iH ( s − t )4 π t − s ) φ ( t ) ¯ φ ( s ) dtds = i π (cid:120) e iπσ coth ( s − t ) H ( s − t ) 1cosh( s − t ) φ ( t ) ¯ φ ( s ) dtds = (cid:104) S σ ∗ φ , φ (cid:105) L ( R ) , (5.2.24) NTEGRALS OF THE WIGNER DISTRIBUTION 77 with(5.2.25) S σ ( t ) = i π H ( t ) e iπσ coth ( t ) cosh( t ) , and(5.2.26) ˆ S σ ( τ ) = i π (cid:90) H ( t ) e iπσ coth ( t ) cosh( t ) e − iπtτ dt = i π (cid:90) + ∞ cos(4 πσ coth( t/ − πtτ )cosh( t ) dt − π (cid:90) + ∞ sin(4 πσ coth( t/ − πtτ )cosh( t ) dt = i π (cid:90) + ∞ cos(2 πtτ − πσ coth( t/ t ) dt + 14 π (cid:90) + ∞ sin(2 πtτ − πσ coth( t/ t/ dt. Note that from (5.2.12), (5.2.14), we have ˆ T σ ( τ ) = i π (cid:90) e iπσ tanh( t ) sinh( t/ e − iπtτ dt = 12 π (cid:90) + ∞ sin(2 πtτ − πσ tanh( t/ t/ dt. An isometric isomorphism. Theorem 5.19. Let σ ≥ be given, let C σ be the set defined by (5.0.1) and let A σ be the operator with Weyl symbol C σ , (whose kernel is given by (5.2.8) ). Theoperator A σ is bounded self-adjoint on L ( R ) so that we may define, with Ψ definedin (5.1.15) , (5.2.27) (cid:101) A σ = Ψ A σ Ψ − . The operator (cid:101) A σ is the Fourier multiplier on L ( R ; C ) given by the matrix (5.2.28) M σ ( τ ) = + ˆ T σ ( τ ) ˆ S σ ( τ )ˆ S σ ( τ ) 0 , where T σ , S σ are defined respectively in (5.2.12) , (5.2.20) , (5.2.25) . In particular wehave with Φ = ( φ , φ ) ∈ L ( R ; C ) , (5.2.29) (cid:104) (cid:101) A σ Φ , Φ (cid:105) L ( R ; C ) = (cid:90) R e iπtτ (cid:104)M σ ( τ ) ˆΦ( τ ) , ˆΦ( τ ) (cid:105) C dτ. Proof. We havekernel ( HA σ H ) = e iπσ x − yx + y H ( x ) H ( y ) ˆ H ( y − x ) , kernel ( ˇ HA σ H + HA σ ˇ H ) = e iπσ x − yx + y H ( x + y ) (cid:0) ˇ H ( x ) H ( y ) + H ( x ) ˇ H ( y ) (cid:1) iπ ( y − x ) , ˇ HA σ ˇ H = 0 . Proposition 9.13 in our Appendix is readily giving the L -boundedness (and self-adjointness) of ˇ HA σ H + HA σ ˇ H . We find also that HA σ H − H has kernel e iπσ x − yx + y H ( x ) H ( y ) 12 iπ ( y − x ) , and thus it is enough to study the operator with kernel e iπσ es − etes + et e s + t iπ ( e t − e s ) = e iπσ tanh( s − t ) iπ sinh( t − s ) , which is a convolution operator by T σ ( t ) = e iπσ tanh( t ) t t ) pv iπt , given by (5.2.12). Formula (5.2.14) implies in particular that ˆ T σ is bounded (andreal-valued) on the real line, entailing eventually the boundedness and self-adjoint-ness of A σ . Formulas (5.2.11), (5.2.24) and (5.2.25) are providing (5.2.29), complet-ing the proof of the theorem. (cid:3) The main result on hyperbolic regions. Theorem 5.20. Let σ ≥ be given and let A σ be the operator defined in Theorem5.19. Then A σ is a bounded self-adjoint operator on L ( R ) such that (5.2.30) inf (cid:0) spectrum ( A σ ) (cid:1) < < < sup (cid:0) spectrum ( A σ ) (cid:1) . The spectrum of A σ is the closure of the set of eigenvalues of M σ ( τ ) for τ runningon the real line. Remark 5.21. It is enough to prove that, with a given σ ≥ , there exists τ ∈ R such that M σ ( τ ) satisfies (5.1.24) .Proof. We have from (5.2.28), (5.2.20), (5.2.26), M σ ( τ ) = + π (cid:82) + ∞ πtτ − πσ tanh( t/ t/ dt · i π (cid:82) + ∞ e − iπ ( tτ − σ tanh( t/ 2) ) cosh( t/ dt iπ (cid:82) + ∞ e iπ ( tτ − σ tanh( t/ 2) ) cosh( t/ dt · (5.2.31) = (cid:18) a ( τ, σ ) a ( τ, σ ) a ( τ, σ ) a ( τ, σ ) (cid:19) . On the other hand we have(5.2.32) a = a = 14 iπ (cid:90) + ∞ e iπ ( tτ − σ tanh( t/ ) cosh( t/ dt, so that(5.2.33) Re a ( τ, σ ) = 14 π (cid:90) + ∞ sin[2 π ( tτ − σ tanh( t ) )]cosh( t ) dt. We note that the function t (cid:55)→ e iπ ( tτ − σ tanh( t/ 2) ) cosh( t/ is holomorphic on C \ iπ Z , with simplepoles at (2 Z + 1) iπ (zeroes of cosh( t/ ) and essential singularities at Z iπ (zeroesof sinh( t/ ). We shall need a more explicit quantitative expression for a to obtaina precise asymptotic result which could be compared to the estimate (5.2.21). The NTEGRALS OF THE WIGNER DISTRIBUTION 79 next lemma is proven in [37]; we provide a proof here for the convenience of thereader. Lemma 5.22. Let τ > , σ ≥ be given and let a ( τ, σ ) be given by (5.2.32) . Wehave Re a ( τ, σ ) = e − π τ π (cid:26)(cid:90) π (cid:16) e π ( tτ − σ tan( t/ − t/ 2) + sinh( t/ − sin( t/ t/ 2) sin( t/ (cid:17) dt (5.2.34) + (cid:90) π − cos 2 π ( tτ − σ tanh( t/ t/ dt − (cid:90) + ∞ π cos 2 π ( tτ − σ tanh( t/ t/ dt (cid:27) . Proof of Lemma 5.22. Let < (cid:15) < π/ < π < R be given. We consider the closedpath γ (cid:15),R of C \ iπ Z with index γ (cid:15),R ( iπ Z ) ≡ ,(5.2.35) γ (cid:15),R = [ (cid:15), R ] ∪ [ R, R + iπ ] ∪ [ R + iπ, (cid:15) + iπ ] ∪ { iπ + (cid:15)e iθ } ≥ θ ≥− π/ ∪ i [ π − (cid:15), (cid:15) ] ∪ { (cid:15)e iθ } π/ ≥ θ ≥ , and we have(5.2.36) (cid:73) γ (cid:15),R e iπ ( zτ − σ tanh( z/ ) cosh( z/ dz = 0 . We note as well that(5.2.37) I = (cid:73) [ R,R + iπ ] e iπ ( zτ − σ tanh( z/ ) cosh( z/ dz = i (cid:90) π e iπ (( R + it ) τ − σ tanh( R + it ) cosh( R + it ) dt = ie iπRτ (cid:90) π e − πtτ e − iπσ e − R − it − e − R − it dte R + it (1 + e − R − it ) , so that | I | ≤ e − R/ (cid:90) π e πσ Im (cid:16) e − R − it − e − R − it (cid:17) dt | − e − R | , and since Im (cid:18) e − R − it − e − R − it (cid:19) = Im (1 + e − R − it )(1 − e − R + it ) | − e − R − it | = − e − R sin t | − e − R − it | ≤ , we get(5.2.38) | I | ≤ e − R/ π − e − R , where I is defined in (5.2.37) . Let us now check (5.2.40) I = − (cid:90) − π/ e iπ (( iπ + (cid:15)e iθ ) τ − σ coth( iπ + (cid:15)eiθ )) cosh iπ + (cid:15)e iθ i(cid:15)e iθ dθ = − e − π τ (cid:90) − π/ e iπ (cid:16) (cid:15)e iθ τ − σ tanh( (cid:15)eiθ ) (cid:17) i sinh (cid:15)e iθ i(cid:15)e iθ dθ, and since (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e iπ (cid:16) (cid:15)e iθ τ − σ tanh( (cid:15)eiθ ) (cid:17) i sinh (cid:15)e iθ i(cid:15)e iθ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | z |≤ π/ | z sinh z | e π τ e πσ sup | z |≤ π/ | sinh z cosh z | , the Lebesgue Dominated Convergence Theorem gives(5.2.41) lim (cid:15) → + I = − πe − π τ . Defining now(5.2.42) I = − (cid:90) π/ e iπ ( (cid:15)e iθ τ − σ coth( (cid:15)eiθ )) cosh (cid:15)e iθ i(cid:15)e iθ dθ, and noting that πσ Im coth( (cid:15)e iθ πσ Im 1 + e − (cid:15)e iθ − e − (cid:15)e iθ = 4 πσ Im (1 + e − (cid:15)e iθ )(1 − e − (cid:15)e − iθ ) | − e − (cid:15)e iθ | = 4 πσ Im e − (cid:15)e iθ − e − (cid:15)e − iθ | − e − (cid:15)e iθ | = 4 πσ Im e − (cid:15) cos θ ( e − i(cid:15) sin θ − e i(cid:15) sin θ ) | − e − (cid:15)e iθ | = 4 πσe − (cid:15) cos θ Im ( − i ) sin( (cid:15) sin θ ) | − e − (cid:15)e iθ | = − πσe − (cid:15) cos θ (cid:15) sin θ ) | − e − (cid:15)e iθ | ≤ , we get that | I | ≤ (cid:90) π/ e − π(cid:15)τ sin θ min | z |≤ π/ | cosh z | dθ(cid:15) ≤ (cid:15) π/ | z |≤ π/ | cosh z | , entailing(5.2.43) lim (cid:15) → + I = 0 . With(5.2.44) I = (cid:73) [ (cid:15),R ] e iπ ( zτ − σ tanh( z/ ) cosh( z/ dz, we have from (5.2.32)(5.2.45) lim (cid:15) → + R → + ∞ I = 4 iπa . Let us note for future reference the standard formulas(5.2.39) cosh (cid:0) iπ z (cid:1) = i sinh z, sinh (cid:0) iπ z (cid:1) = i cosh z, tanh (cid:0) iπ z (cid:1) = coth z. NTEGRALS OF THE WIGNER DISTRIBUTION 81 We define now I = − (cid:73) [ i(cid:15),i ( π − (cid:15) )] e iπ ( zτ − σ tanh( z/ ) cosh( z/ dz = − (cid:90) π − (cid:15)(cid:15) e iπ ( itτ − σ tanh( it/ ) cosh( it/ idt = − (cid:90) π − (cid:15)(cid:15) e − πtτ e − iπσi tan( t/ cos( t/ idt = − i (cid:90) π − (cid:15)(cid:15) e − πtτ e − πσ tan( t/ cos( t/ dt = − i (cid:90) π − (cid:15)(cid:15) e − π ( π − s ) τ e − πσ tan(( π − s ) / cos(( π − s ) / ds = − ie − π τ (cid:90) π − (cid:15)(cid:15) e πsτ e − πσ sin( s/ s/ sin( s/ ds, so that I = − ie − π τ (cid:90) π − (cid:15)(cid:15) e πsτ e − πσ tan( s/ sin( s/ ds. (5.2.46)We have also(5.2.47) I = (cid:73) [ R + iπ,(cid:15) + iπ ] e iπ ( zτ − σ tanh( z/ ) cosh( z/ dz = − (cid:90) R(cid:15) e iπ (( t + iπ ) τ − σ tanh(( t + iπ ) / ) cosh(( t + iπ ) / dt, so that using Formulas (5.2.39), we get I = − e − π τ (cid:90) R(cid:15) e iπ ( tτ − σ tanh( t/ i sinh( t/ dt, and(5.2.48) I + I = ie − π τ (cid:18)(cid:90) R(cid:15) e iπ ( tτ − σ tanh( t/ sinh( t/ dt − (cid:90) π − (cid:15)(cid:15) e πtτ e − πσ tan( t/ sin( t/ dt (cid:19) = ie − π τ (cid:26) (cid:90) π − (cid:15)(cid:15) (cid:16) e iπ ( tτ − σ tanh( t/ sinh( t/ − e π ( tτ − σ tan( t/ sin( t/ (cid:17) dt + (cid:90) Rπ − (cid:15) e iπ ( tτ − σ tanh( t/ sinh( t/ dt (cid:27) . From (5.2.36), (5.2.35), (5.2.37), (5.2.40), (5.2.42), (5.2.44), (5.2.46), (5.2.47), wefind that I = − I − ( I + I ) − I − I , so that taking the limit of both sides when (cid:15) → + , R → + ∞ we get, thanks to(5.2.45), (5.2.38), (5.2.48), (5.2.41), (5.2.43),(5.2.49) iπa = − ie − π τ (cid:26) (cid:90) π (cid:16) e iπ ( tτ − σ tanh( t/ sinh( t/ − e π ( tτ − σ tan( t/ sin( t/ (cid:17) dt + (cid:90) + ∞ π e iπ ( tτ − σ tanh( t/ sinh( t/ dt (cid:27) + πe − π τ , I , I , I , I , I + I do have limits when (cid:15) → + , R → + ∞ . implying that a = e − π τ π (cid:26) (cid:90) π (cid:16) − e iπ ( tτ − σ tanh( t/ sinh( t/ 2) + e π ( tτ − σ tan( t/ sin( t/ (cid:17) dt − (cid:90) + ∞ π e iπ ( tτ − σ tanh( t/ sinh( t/ dt (cid:27) − i e − π τ that is(5.2.50) a = e − π τ π (cid:90) π (cid:16) e π ( tτ − σ tan( t/ sin( t/ − cos 2 π ( tτ − σ tanh( t/ t/ (cid:17) dt − e − π τ π (cid:90) + ∞ π cos 2 π ( tτ − σ tanh( t/ t/ dt − i e − π τ π (cid:90) π sin 2 π ( tτ − σ tanh( t/ t/ dt − i e − π τ − i e − π τ π (cid:90) + ∞ π sin 2 π ( tτ − σ tanh( t/ t/ dt, yielding(5.2.51) Re a = e − π τ π (cid:90) π (cid:16) e π ( tτ − σ tan( t/ sin( t/ − cos 2 π ( tτ − σ tanh( t/ t/ (cid:17) dt − e − π τ π (cid:90) + ∞ π cos 2 π ( tτ − σ tanh( t/ t/ dt, completing the proof of Lemma 5.22. (cid:3) Remark 5.23. Formula (5.2.50) also yields Im a = − Im a = e − π τ π (cid:110)(cid:90) π sin 2 π ( tτ − σ tanh( t/ t/ dt + π + (cid:90) + ∞ π sin 2 π ( tτ − σ tanh( t/ t/ dt (cid:111) , and since from (5.2.31), we have(5.2.52) a = 12 + 12 π (cid:90) + ∞ sin(2 πtτ − πσ tanh( t/ t/ dt, this gives(5.2.53) Im a = e − π τ π (cid:0) π ( a − 12 ) + π (cid:1) = e − π τ a . To complete the proof of Theorem 5.20, it will be enough, according to Lemma5.11, to prove that, for τ → + ∞ , | a | (cid:29) − a . To achieve that, we note from(5.2.53) that the imaginary part of a is useless and we shall prove simply that (Re a ) (cid:29) − a . NTEGRALS OF THE WIGNER DISTRIBUTION 83 To get this we are going to use (5.2.21) and a precise asymptotic behavior for (Re a ) displayed in the next lemma and issued from the explicit formula (5.2.34). Lemma 5.24. Let τ ≥ , σ ≥ be given and let a ( τ, σ ) be given by (5.2.32) . Wehave then Re a ( τ, σ ) ≥ e − π √ τ √ σ π τ − π e − π τ . (5.2.54) Proof of the lemma. Since for t ≥ we have sinh( t/ − sin( t/ ≥ , we get from(5.2.34),(5.2.55) Re a ( τ, σ ) ≥ e − π τ π (cid:26)(cid:90) π e π ( tτ − σ tan( t/ − t/ dt − (cid:90) + ∞ π t/ dt (cid:27) = e − π τ π (cid:90) π e π ( tτ − σ tan( t/ − t/ dt − e − π τ π ln (cid:0) coth π (cid:1) . Let us define(5.2.56) ω = 2 πτ, κ = 2 πσ, ν = κ / ω − / , φ ν ( s ) = s − ν tan s. We have π (cid:0) tτ − σ tan( t/ πτ (cid:0) t − ν tan( t/ (cid:1) = 4 πτ (cid:0) t − ν tan t (cid:1) = 2 ωφ ν ( t/ . We have thus(5.2.57) Re a ( τ, σ ) ≥ e − πω π (cid:90) π/ e ωφ ν ( s ) − s ds − e − πω π ln (cid:0) coth π (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) ≈ . . Defining(5.2.58) ψ ν ( ω ) = e − πω π (cid:90) π/ e ωφ ν ( s ) − s ds, we can use (5.2.56), (5.2.57) and (9.5.22) to get whenever τ > , π Re a ( τ, σ ) ≥ e − π √ τ √ σ π τ (cid:16) − τ (cid:17) − e − π τ , so that for τ ≥ we find π Re a ( τ, σ ) ≥ e − π √ τ √ σ π τ − e − π τ , (5.2.59)yielding the lemma. (cid:3) We eventually go back to the proof of Theorem 5.20: let σ > be given. FromLemma 5.24 and (5.2.21), we have for τ ≥ , | − a ( τ, σ ) | ≤ e − π τ e πσ , Re a ( τ, σ ) ≥ e − π √ τ √ σ π τ − π e − π τ = e − π √ τ √ σ π τ (cid:16) − π τ e π √ τ √ σ e π τ (cid:17) . This entails that for τ ≥ τ ( σ ) , we have(5.2.60) Re a ( τ, σ ) ≥ e − π √ τ √ σ π τ , and thus a (cid:54) = 0 and(5.2.61) | a ( σ, τ ) | ≥ e − π √ τ √ σ π τ > | − a ( τ, σ ) | , where the last inequality above holds true (thanks to (5.2.21)) whenever e − π τ e πσ < e − π √ τ √ σ π τ , which is indeed true for τ ≥ τ ( σ ) . As a result for τ ≥ max(4 σ, , τ ( σ ) , τ ( σ )) , weobtain that (5.2.61) is satisfied so that Remark 5.21 implies the result of Theorem5.20, completing our proof. (cid:3) Remark 5.25. The functions τ ( σ ) , τ ( σ ) can be determined rather easily, the firstone by the condition τ ≥ τ ( σ ) = ⇒ π τ e π √ τ √ σ e π τ ≤ , whereas the second one should satisfy τ ≥ τ ( σ ) = ⇒ e πσ π τ e π √ τ √ σ < e π τ . Comments and further results. Qualitative explanations on the various computations. We would like to goback to our proofs that(5.3.1) | a ( τ, σ ) | (cid:29) | − a ( τ, σ ) | , τ → + ∞ , which is our key argument via Lemma 5.11 and give a couple of qualitative expla-nations which may enlighten the calculations. It is of course much simpler to beginwith the case σ = 0 : in that case, according to Proposition 5.13 and (5.1.18), wehave(5.3.2) − a ( τ, 0) = (cid:90) + ∞ τ ρ ( τ (cid:48) ) dτ, ρ ( τ ) = (cid:90) (cid:18) t/ t/ (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) = f ( t ) , f ∈ S ( R ) holomorphicon | Im t | < π. e − iπtτ ds, so that ρ ( τ ) = ˆ f ( τ ) . We get thus readily that ρ belongs to the Schwartz space,as the Fourier transform of a function in the Schwartz space and this implies inparticular that − a ( τ, has fast decay towards 0 when τ → + ∞ , as proven inProposition 5.13. We note also that (5.2.53) gives Im a ( τ, = e − π τ a ( τ, / , and since the limit of a is 1, we do not expect any help from the imaginary partof a to proving (5.3.1). Turning our attention to Re a in (5.1.30), we have,(5.3.3) π Re a ( τ, 0) = (cid:90) + ∞ sin (2 πtτ )cosh( t/ dt, which is the sine-Fourier transform of the function t (cid:55)→ H ( t ) sech( t/ 2) = g ( t ) ,which has a singularity at t = 0 : as a consequence, thanks to Lemma 9.1, theFourier transform ˆ g cannot be rapidly decreasing, cannot even belong to L ( R ) (that would imply that g is continuous). Moreover the sine-Fourier transform above NTEGRALS OF THE WIGNER DISTRIBUTION 85 is the Fourier transform of the odd part of g , g odd ( t ) = sech( t/ 2) sign t , which is alsosingular at 0, thus (cid:100) g odd cannot be rapidly decreasing and is an odd function, which isenough to prove, without more calculations, that (5.3.1) holds true. In Section 5.1,we used a more explicit argument, with providing an equivalent of (5.3.3) equal to / (2 πτ ) near + ∞ . Summing-up, (5.3.1) in the case σ = 0 follows from the existenceof a singularity of the function g above, which is discontinuous at 0.Let us now take a look at the case σ > , which turns out to be more computa-tionally involved. We have from (5.2.32) πia ( τ, σ ) = (cid:90) R H ( t ) sech( t/ e − i πσ coth( t/ e iπtτ dt = ˇ (cid:98) g σ ( τ ) , (5.3.4) g σ ( t ) = H ( t ) sech( t/ e − i πσ coth( t/ . (5.3.5)The single discontinuity at t = 0 of g σ when σ > is much wilder than for σ = 0 : inthe latter case, we had only a jump discontinuity with different limits on both sides,whereas when σ > , we have an essential discontinuity with an oscillatory behaviourin ( − , +1) when t → + for the real and imaginary parts of a . However, g σ belongsto all L p ( R ) , p ∈ [1 , + ∞ ] , so that its Fourier transform belongs to L p ( R ) , p ∈ [2 , + ∞ ] :we expect then that both sides of (5.3.1) have limit for τ → + ∞ and we mustprove that − a decays much faster than a . Looking at a slightly simplifiedmodel and using the notations (5.2.56), we define for ω, ν positive, a function α presumably close to πia , given by(5.3.6) α ( ω, ν ) = (cid:90) + ∞ e i ωµ ν ( s ) sech( s ) ds, µ ν ( s ) = s − ν s , µ (cid:48) ν ( s ) = 1 + ν s . Trying our hand with the stationary phase method, we look at α ( ω, ν ) = 12 iω (cid:90) + ∞ dds (cid:8) e i ωµ ν ( s ) (cid:9) sech( s ) µ (cid:48) ν ( s ) ds = 12 iω (cid:90) + ∞ dds (cid:8) e i ωµ ν ( s ) (cid:9) s sech( s ) s + ν ds = i ω (cid:90) + ∞ e i ωµ ν ( s ) dds (cid:26) s sech( s ) s + ν (cid:27) , since the boundary term vanishes. Iterating that computation shows that α ( ω, ν ) = O σ ( ω − N ) for all N when ω → + ∞ , meaning that the information of fast decay for − a will not suffice to get (5.3.1). Also, it is worth noticing that no fast decay ofthe function α occurs when ω → −∞ , otherwise Lemma 9.1 would give smoothnessfor the function s (cid:55)→ e − iκ/s H ( s ) sech s : in fact we see also that for σ > , τ = − λ , λ > , we have πia ( − λ, σ ) = (cid:90) + ∞ sech( s ) e − i πσ coth( s ) e − iπsλ ds, and the phase function is ˜ µ ( s ) = − iπ ( sλ + σ coth( s )) and we have dds (cid:8) sλ + σ coth( s ) (cid:9) = λ − σ (1 − tanh s )tanh s = ( λ + σ ) tanh s − σ tanh s , which does vanish at tanh s = σ/ ( λ + σ ) . As a result we could say that, for σ > ,the C ∞ wave-front-set (see e.g. Section 8.1 in [16]) of the function g σ is reduced to { } × ( −∞ , . It turns out that we can show that the Gevrey-2 wave-front-set of g σ is { } × R ∗ , and it is expressed via the lowerbound estimate (5.2.54); the route thatwe took for proving this was an explicit calculation of Re a , following the paper[37]. Finally the upper bound (5.2.21) can be improved as(5.3.7) | − a ( τ, σ ) | ≤ C σ,(cid:15) e − ( π − (cid:15) )2 πτ , (cid:15) > , and is expressing the fact the the function t (cid:55)→ te iπσ tanh( t sinh( t/ is analytic on the real line,with a radius of convergence on the real line bounded below by π (cf. Proposition9.2).5.3.2. More results and examples: (cid:96) p balls, corners. For a, φ like in Corollary 5.15,defining Ω p = { ( x, ξ ) ∈ R , | x − a | p + | ξ − a | p < (cid:0) a (cid:1) p } , since W ( φ , φ ) ∈ S ( R ) , we get lim p → + ∞ (cid:120) Ω p W ( φ , φ )( x, ξ ) dxdξ = (cid:120) [0 ,a ] W ( φ , φ )( x, ξ ) dxdξ > (cid:107) φ (cid:107) L ( R ) , proving that the spectrum of Op w ( Ω p ) intersects (1 , + ∞ ) for p large enough, show-ing that a counterexample to Flandrin’s conjecture can be a convex analytic openbounded set. Moreover, defining Q a = { ( x, ξ ) ∈ R , | x | + | ξ | ≤ a/ √ } , we note that Q a is obtained by rotation and translation of [0 , a ] so that we can find φ in the Schwartz space such that (cid:120) Q a W ( φ , φ )( x, ξ ) dxdξ > (cid:107) φ (cid:107) L ( R ) . Since we have lim p → (cid:120) | x | p + | ξ | p ≤ ( a/ √ p W ( φ , φ )( x, ξ ) dxdξ = (cid:120) Q a W ( φ , φ )( x, ξ ) dxdξ > (cid:107) φ (cid:107) L ( R ) , we get that for p − small enough we have(5.3.8) (cid:120) | x | p + | ξ | p ≤ ( a/ √ p W ( φ , φ )( x, ξ ) dxdξ > (cid:107) φ (cid:107) L ( R ) , proving that (cid:96) p balls are counterexamples to Flandrin’s conjecture for p − or /p small enough. Convex affine cones with aperture strictly less than π of R are translations androtations of(5.3.9) Σ θ = { ( x, ξ ) ∈ R \ ( R − ×{ } ) , arg( x + iξ ) ∈ (0 , θ ) } , for some θ ∈ (0 , π ) .The vertex of Σ θ and its rotations is defined as 0 and the vertex of the translation ofvector T of Σ θ is defined as T . We note that all convex affine cones with aperture NTEGRALS OF THE WIGNER DISTRIBUTION 87 strictly less than π are symplectically equivalent in R , since Σ θ is symplecticallyequivalent to (the interior of) the quarter plane Σ π/ : indeed let θ be in (0 , π ) ; thesymplectic matrix M θ defined by M θ = (cid:18) − cotan θ (cid:19) , is such that M θ (cid:18) (cid:19) = (cid:18) (cid:19) , M θ (cid:18) cos θ sin θ (cid:19) = (cid:18) θ (cid:19) , proving that M θ Σ θ = Σ π/ . The next result follows from Theorem 1.3 in [4] and shows that many counterexam-ples to Flandrin’s conjecture can be be obtained. Theorem 5.26. Let K be a subset of the closure of a convex affine cone with aperturestrictly less than π and vertex X such that K contains a neighborhood of the vertexin the cone . Then there exists λ > such that, with K λ = X + λ ( K − X ) , there exists φ ∈ S ( R ) such that (cid:120) K λ W ( φ, φ )( x, ξ ) dxdξ > (cid:107) φ (cid:107) L ( R ) . (5.3.10) N.B. Note that (5.3.10) implies that φ is not the zero function. Also, taking K convex produces another counterexample to Flandrin’s conjecture since K λ will bethen convex, but we do not need that assumption to proving the result.Proof. There is no loss of generality at assuming X = 0 and [0 , ρ ] ⊂ K ⊂ Σ π/ , ρ > . Using Corollary 5.15, we find φ ∈ S ( R ) (so that W ( φ , φ ) ∈ S ( R ) ) such that lim λ → + ∞ (cid:120) K λ W ( φ , φ )( x, ξ ) dxdξ = (cid:120) Σ π/ W ( φ , φ )( x, ξ ) dxdξ > (cid:107) φ (cid:107) L ( R ) , implying for λ large enough that (cid:115) K λ W ( φ , φ )( x, ξ ) dxdξ > (cid:107) φ (cid:107) L ( R ) , which is thesought result. (cid:3) Numerics.Definition 5.27. Let σ ≥ be given. With the × Hermitian matrix M σ givenby (5.2.31) , we define for τ ∈ R , λ + ( τ, σ ) = 12 (cid:18) a ( τ, σ )+ (cid:113) a ( τ, σ ) + 4 | a ( τ, σ ) | (cid:19) , (5.4.1) λ − ( τ, σ ) = 12 (cid:18) a ( τ, σ ) − (cid:113) a ( τ, σ ) + 4 | a ( τ, σ ) | (cid:19) . (5.4.2) We shall say that the set K has a corner. Remark 5.28. According to (5.2.53) , we have λ + ( τ, σ ) = 12 (cid:18) a ( τ, σ )+ (cid:113) a ( τ, σ ) (cid:0) e − π τ (cid:1) + 4(Re a ( τ, σ )) (cid:19) , (5.4.3) λ − ( τ, σ ) = 12 (cid:18) a ( τ, σ ) − (cid:113) a ( τ, σ ) (cid:0) e − π τ (cid:1) + 4(Re a ( τ, σ )) (cid:19) , (5.4.4) so that the knowledge of a and Re a suffices for expressing λ ± . An immediate consequence of Theorem 5.20 is Theorem 5.29. Let σ ≥ be given and let A σ be the self-adjoint operator boundedin L ( R ) defined in Theorem 5.20. With the notations of Definition 5.27, we have M σ := sup { spectrum ( A σ ) } = sup τ ∈ R λ + ( τ, σ ) , (5.4.5) m σ := inf { spectrum ( A σ ) } = inf τ ∈ R λ − ( τ, σ ) . (5.4.6) Moreover for all σ ≥ we have (5.4.7) m σ < < < M σ . The quarter-plane: σ = 0 . Of course, as shown by the respective calculationsof Sections 5.1 and 5.2, the case σ = 0 , dealing with the quarter-plane is muchsimpler than the cases where σ > . Nonetheless we know explicitly a spectraldecomposition of the operator with Weyl symbol H ( x ) H ( ξ ) from Theorem 5.19, butwe can calculate without difficulty numerical expressions of M , m as defined in(5.4.5), (5.4.6). Proposition 5.30. We have from (9.5.37) , (5.2.33) , a ( τ, 0) = 11 + e − π τ , Re a ( τ, 0) = 14 π (cid:90) + ∞ sin(2 πtτ ) sech( t/ dt, and we can use these formulas and (5.4.3) , (5.4.4) , (5.4.5) , (5.4.6) to calculate nu-merically M ≈ . , ( λ + ( τ, at τ ≈ . , (5.4.8) m ≈ − . , ( λ − ( τ, at τ ≈ − . . (5.4.9)5.4.2. On hyperbolic regions. We want now to tackle the case σ > . In order touse the expressions (9.5.37), (5.2.34) respectively for a and a , we need first toevaluate the residue term in (9.5.37). The mapping z (cid:55)→ tanh z is a biholomorphism NTEGRALS OF THE WIGNER DISTRIBUTION 89 Figure 6. The function τ (cid:55)→ λ + ( τ, near its maximum, well above 1. - - - - Figure 7. The functions τ (cid:55)→ λ + ( τ, , λ − ( τ, .of neighborhoods of in the complex plane, so that we have for z near the origin, ζ = tanh z, dζ = (1 − ζ ) dz, z = arcth ζ = 12 ln (cid:18) ζ − ζ (cid:19) , (5.4.10) e iωz − iκ coth z cosh z dz = (cid:18) ζ − ζ (cid:19) iω e − i κζ (cid:16) ζ − ζ (cid:17) / + (cid:16) − ζ ζ (cid:17) / dζ (1 − ζ )= (1 + ζ ) − + iω (1 − ζ ) − − iω e − i κζ dζ, (5.4.11)so that(5.4.12) Res (cid:18) e iωz − iκ coth z cosh z , (cid:19) = Res (cid:16) (1 + ζ ) − + iω (1 − ζ ) − − iω e − i κζ , (cid:17) . Proposition 5.31. Let σ ≥ be given. Then for any τ ∈ R , with the notations ω = 2 πτ , κ = 2 πσ , we have, for any ρ ∈ (0 , , (5.4.13) a ( τ, σ ) = 11 + e − πω + e − πω e − πω ρ π Im (cid:90) π − π exp (cid:18) iω Log (cid:16) ρe iθ − ρe iθ (cid:17)(cid:19) e − iκe − iθρ e iθ (cid:112) − ρ e iθ dθ . (5.4.14) Re a ( τ, σ ) = e − πω π (cid:26) (cid:90) π/ e ( sω − κ tan s ) sinh( sω − κ tan s )sin s ds + ln (cid:0) coth π (cid:1) + 2 (cid:90) π/ sin ( sω − κ tanh s )sinh s ds − (cid:90) + ∞ π/ cos 2( sω − κ tanh s )sinh s ds (cid:27) , Im a ( τ, σ ) = e − πω a ( τ, σ ) . (5.4.15) Proof. Formula (5.4.13) follows from (5.4.12) and (9.5.37) whereas (5.4.14) is (5.2.34)after a change of variable t = 2 s , where the second integral term inside the bracketsis evaluated (cf. Lemma 9.14); Formula (5.4.15) is a reminder of (5.2.53). (cid:3) N.B. Our choice for ρ in the numerical calculations of (5.4.13) is ρ = 3 / , whichis a good compromise between using a value of ρ clearly away from (to avoidsingularities coming from small denominators in the Log term) and minimize theoscillations and size coming from the term exp( − iκρ − e − iθ ) ; note that the modulusof the latter is exp( − κρ − sin θ ) , which is a smooth function of ρ (flat at 0) when θ ∈ [0 , π ] , but is unbounded for ρ → + when θ ∈ ( − π, . There is no surprise here since although the residuedoes not depend on the choice of ρ ∈ (0 , , we cannot get the value of that residueby letting ρ go to 0 because of the part of the path in the lower half-plane. Theargument of exp( − iκρ − e − iθ ) is − κρ − cos θ and taking ρ too small would be de-vastating for the calculations because of the strong oscillations triggered by the term exp( − iκρ − cos θ ) all over the circle. Of course for the evaluation of Log (cid:16) ρe iθ − ρe iθ (cid:17) iseasier for ρ small, but we have to take into account the constraints in that directionmentioned above. Remark 5.32. It seems easier numerically for the evaluation of a to use (5.4.13)rather than any other expression (see e.g. Lemma 5.18, (5.2.31), (9.5.23)). Howeverthe following formula could be interesting, theoretically and numerically: recallingthat sinc x = sin xx , we have from (5.2.31)(5.4.16) a ( τ, σ ) = 12 + 2 ωπ (cid:90) + ∞ sinc(2 ωs ) s sinh s cos(2 κ tanh s ) ds − κπ (cid:90) + ∞ sinc(2 κs ) 1cosh s cos(2 ωs ) ds, NTEGRALS OF THE WIGNER DISTRIBUTION 91 but it turns out that numerical calculations involving (5.4.16) seem to be less reliablethan the methods using (5.4.13).We can also take a look at the following curves. λ ( τ ,1 / π ) λ ( τ ,2 / π ) λ ( τ ,3 / π ) Figure 8. Functions λ + ( τ, κ/ π ) with κ = 1 , , : their maxima arestrictly greater than 1. Remark 5.33. In the above figure, in order to put the three curves on the samepicture, we have used three different logarithmic scales on the vertical axis, namelywe have drawn τ (cid:55)→ α j Log (cid:0) λ + ( τ, σ j ) (cid:1) , ≤ j ≤ , σ j = j/ π, α = 20 , α = 100 , α = 500 . Of course we have α j Log (cid:0) λ + ( τ, σ j ) (cid:1) > ⇐⇒ Log (cid:0) λ + ( τ, σ j ) (cid:1) > ⇐⇒ λ + ( τ, σ j ) > , so that the piece of curves in Figure 8 which are above 1 are indeed correspondingto curves of τ (cid:55)→ λ + ( τ, σ j ) which go strictly above the threshold 1. We have also max τ λ + ( τ, σ ) ≈ × − at τ ≈ . , max τ λ + ( τ, σ ) ≈ × − at τ ≈ . , max τ λ + ( τ, σ ) ≈ − at τ ≈ . .We are glad to have a theoretical proof of Theorem 5.20 since the numerical analysisof cases where σ is large, say larger than 10, seem to be very difficult to achieve, at least through a standard use of Mathematica . The reason for that is quite clearsince using our Lemma 5.11, we did study the function β defined by(5.4.17) β ( τ, σ ) = | a ( τ, σ ) | + a ( τ, σ ) − , and proved that for each σ ≥ there exists T ( σ ) such that for all τ ≥ T ( σ ) wehave β ( τ, σ ) > and a ( τ, σ ) (cid:54) = 0 . Thanks to Lemma 5.18 and (5.2.60) we knewthat for τ ≥ T ( σ ) , we had | − a | ≤ e − π τ e πσ (cid:28) e − π √ τ √ σ π τ ≤ (Re a ) ≤ | a | , where the second inequality (cid:28) is in fact comparing for σ fixed two exponentialdecays. The numerical analysis of that inequality is certainly quite difficult when σ and τ are large since both sides are converging to zero quite fast for σ fixed and τ → + ∞ ; of course taking the logarithm of both sides looks quite reasonable, butin practice does not seem really easy numerically. When σ = 0 , the situation ismuch better, since we had to compare (cf. Subsection 5.3.1) an exponential decay | − a | ≤ e − π τ to a polynomial decay | Re a | ∼ π τ , τ → + ∞ , and this could be an a posteriori explanation for which our numerical argument in [4]worked smoothly to disprove Flandrin’s conjecture. So to pick-up the quarter-plane((5.0.1) with σ = 0 ) to produce a counterexample to that conjecture was indeeda very wise choice: if you choose instead C σ for σ large, our Theorem 5.20 showsthat it is also a counterexample to Flandrin’s conjecture , but we have a theoreticalproof for that Theorem and if we were depending on a numerical analysis, it is quitelikely that checking numerically the positivity of the function β defined in (5.4.17)could be rather difficult, even say for σ = 10 .6. Unboundedness is Baire generic In this section we show that for plenty of subsets E of the phase space R n , theoperator Op w ( E ) is not bounded on L ( R n ) .6.1. Preliminaries.Lemma 6.1. Let u, v ∈ L ( R n ) and let W ( u, u ) , W ( v, v ) , be their Wigner distribu-tions. Then we have (cid:107)W ( u, u ) − W ( v, v ) (cid:107) L ( R n ) ≤ (cid:107) u − v (cid:107) L ( R n ) (cid:0) (cid:107) u (cid:107) L ( R n ) + (cid:107) v (cid:107) L ( R n ) (cid:1) . As a consequence if a sequence ( u k ) is converging in L ( R n ) , then the sequence ( W ( u k , u k )) converges in L ( R n ) towards W ( u, u ) . As a convex subset of the plane on which the integral of the Wigner distribution of somenormalized pulse is > . NTEGRALS OF THE WIGNER DISTRIBUTION 93 Proof. We have by sesquilinearity W ( u, u ) − W ( v, v ) = W ( u − v, u ) + W ( v, u − v ) , so that (cid:107)W ( u, u ) − W ( v, v ) (cid:107) L ( R n ) ≤ (cid:107)W ( u − v, u ) (cid:107) L ( R n ) + (cid:107)W ( v, u − v ) (cid:107) L ( R n ) = (1.1.7) (cid:107) u − v (cid:107) L ( R n ) (cid:0) (cid:107) u (cid:107) L ( R n ) + (cid:107) v (cid:107) L ( R n ) (cid:1) , proving the lemma. (cid:3) Lemma 6.2. Let ( u k ) be a converging sequence in L ( R n ) with limit u . Let usassume that there exists C ≥ such that (6.1.1) ∀ k ∈ N , (cid:120) |W ( u k , u k )( x, ξ ) | dxdξ ≤ C . Then we have (cid:115) |W ( u, u )( x, ξ ) | dxdξ ≤ C . Proof. Let R > be given. We check (cid:120) | x | + | ξ | ≤ R |W ( u, u )( x, ξ ) − W ( u k , u k )( x, ξ ) | dxdξ ≤ (cid:120) | x | + | ξ | ≤ R |W ( u − u k , u )( x, ξ ) | dxdξ + (cid:120) | x | + | ξ | ≤ R |W ( u k , u − u k )( x, ξ ) | dxdξ ≤ (cid:112) | B n | R n (cid:0) (cid:107)W ( u − u k , u ) (cid:107) L ( R n ) + (cid:107)W ( u k , u − u k ) (cid:107) L ( R n ) (cid:1) = (cid:112) | B n | R n (cid:107) u − u k (cid:107) L ( R n ) (cid:0) (cid:107) u (cid:107) L ( R n ) + (cid:107) u k (cid:107) L ( R n ) (cid:1) , and thus (cid:120) | x | + | ξ | ≤ R |W ( u, u )( x, ξ ) | dxdξ ≤ (cid:120) | x | + | ξ | ≤ R |W ( u k , u k )( x, ξ ) | dxdξ + (cid:112) | B n | R n (cid:107) u − u k (cid:107) L ( R n ) (cid:0) (cid:107) u (cid:107) L ( R n ) + (cid:107) u k (cid:107) L ( R n ) (cid:1) ≤ C + (cid:112) | B n | R n (cid:107) u − u k (cid:107) L ( R n ) (cid:0) (cid:107) u (cid:107) L ( R n ) + (cid:107) u k (cid:107) L ( R n ) (cid:1) , implying for all R > , (cid:120) | x | + | ξ | ≤ R |W ( u, u )( x, ξ ) | dxdξ ≤ C , and thus the sought result. (cid:3) An explicit construction. We just calculate W ( v , v ) for(6.2.1) v = [ − / , / . Remark 6.3. When u is supported in a closed convex set J , we have in the integral (1.1.6) defining W , x ± z ∈ J = ⇒ x ∈ J, so that supp W ( u, u ) ⊂ J × R n . We have W ( v , v )( x, ξ ) = (cid:90) − / ≤ x + z/ ≤ / − / ≤ x − z/ ≤ / e iπzξ dz, and the integration domain is − min(1 − x, x ) = max( − − x, x − ≤ z ≤ min(1 − x, x ) , which is empty unless − x, x ≥ i.e. x ∈ [ − / , +1 / , and moreover wehave the equivalence − x ≤ x ⇐⇒ x ≥ , so that(6.2.2) W ( v , v )( x, ξ ) = H ( x ) (cid:90) − x − (1 − x ) e iπzξ dz + H ( − x ) (cid:90) x − (1+2 x ) e iπzξ dz = H ( x ) e iπξ (1 − x ) − e − iπξ (1 − x ) iπξ + H ( − x ) e iπξ (1+2 x ) − e − iπξ (1+2 x ) iπξ = [0 , / ( x ) sin(2 πξ (1 − x )) πξ + [ − / , sin(2 πξ (1 + 2 x )) πξ . More generally for a, b, ω real numbers with a < b and(6.2.3) u a,b,ω ( x ) = ( b − a ) − / [ a,b ] ( x ) e iπωx , we have(6.2.4) W ( u a,b,ω , u a,b,ω )( x, ξ )= (cid:0) [ a, a + b ] ( x ) sin[4 π ( ξ − ω )( x − a )] + [ a + b ,b ] ( x ) sin[4 π ( ξ − ω )( b − x )] (cid:1) ( b − a ) π ( ξ − ω ) . We check now, using (6.2.2), for N > , (cid:120) |W ( v , v )( x, ξ ) | dxdξ ≥ (cid:90) ≤ x ≤ / (cid:90) N (cid:12)(cid:12)(cid:12)(cid:12) sin(2 πξ (1 − x )) πξ (cid:12)(cid:12)(cid:12)(cid:12) dξdx = (cid:90) ≤ x ≤ / (cid:90) N π (1 − x )0 (cid:12)(cid:12)(cid:12)(cid:12) sin ηπη (cid:12)(cid:12)(cid:12)(cid:12) dηdx ≥ (cid:90) ≤ x ≤ / (cid:90) Nπ (cid:12)(cid:12)(cid:12)(cid:12) sin ηπη (cid:12)(cid:12)(cid:12)(cid:12) dηdx = 14 (cid:90) Nπ (cid:12)(cid:12)(cid:12)(cid:12) sin ηπη (cid:12)(cid:12)(cid:12)(cid:12) dη, so that(6.2.5) (cid:120) |W ( v , v )( x, ξ ) | dxdξ = + ∞ . Proposition 6.4. Let a, b, ω be real numbers with a < b and let us define u a,b,ω by (6.2.3) . Then we have (6.2.6) (cid:120) |W ( u a,b,ω , u a,b,ω )( x, ξ ) | dxdξ = + ∞ . NTEGRALS OF THE WIGNER DISTRIBUTION 95 N.B. Since u a,b,ω is a normalized L ( R ) function, we also have from (1.1.7) , (1.1.11) that the real-valued W ( u a,b,ω , u a,b,ω ) does satisfy (cid:90) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) W ( u a,b,ω , u a,b,ω )( x, ξ ) dx (cid:12)(cid:12)(cid:12)(cid:12) dξ = (cid:90) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) W ( u a,b,ω , u a,b,ω )( x, ξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12) dx = (cid:107) u a,b,ω (cid:107) L ( R ) = 1 , (cid:120) W ( u a,b,ω , u a,b,ω )( x, ξ ) dxdξ = (cid:107) u a,b,ω (cid:107) L ( R ) = 1 . We shall see in the next sections that most of the time in the Baire Category sense,we have for u ∈ L ( R n ) , (cid:115) | W ( u, u )( x, ξ ) | dxdξ = + ∞ . Proof. The proof is already given above for v = u − / , / , . Moreover we have with α = 1 b − a , β = b + a a − b ) , the formula v ( y ) = e − iπω ( y − β ) α − u a,b,ω (cid:0) ( y − β ) α − (cid:1) α − / , so that u a,b,ω = M v ,where M is an affine metaplectic transformation (cf. Section 1.2.1) and the covari-ance property (1.2.33) shows that the already proven (6.2.5) implies (6.2.6). (cid:3) Most pulses give rise to non-integrable Wigner distribution. In thesequel, n is an integer ≥ . Lemma 6.5. Let us define (6.3.1) Φ = { u ∈ L ( R n ) , (cid:90) R n | u ( x ) | dx < + ∞} . Then Φ is an F σ of L ( R n ) with empty interior.Proof. We have Φ = ∪ N ∈ N Φ N with Φ N = { u ∈ L ( R n ) , (cid:90) R n | u ( x ) | dx ≤ N } . The set Φ N is a closed subset of L ( R n ) since if ( u k ) k ≥ is a sequence in Φ N whichconverges in L ( R n ) with limit u , we get for R ≥ , (cid:90) | x |≤ R | u ( x ) | dx ≤ (cid:90) | x |≤ R | u ( x ) − u k ( x ) | dx + (cid:90) | x |≤ R | u k ( x ) | dx ≤ (cid:107) u − u k (cid:107) L ( R n ) ( | B n | R n ) / + N, implying (cid:82) | x |≤ R | u ( x ) | dx ≤ N and this for any R , so that we obtain u ∈ Φ N . The set Φ N is also symmetric and convex; as a result if the interior of Φ N were not empty, would be an interior point of Φ N so that there would exist ρ > such that (cid:107) u (cid:107) L ( R n ) ≤ ρ = ⇒ (cid:90) R n | u ( x ) | dx ≤ N, and thus for any non-zero u ∈ L ( R n ) , we would have (cid:90) R n | u ( x ) | dx (cid:107) u (cid:107) − L ( R n ) ρ ≤ N and thus (cid:107) u (cid:107) L ( R n ) ≤ N ρ − (cid:107) u (cid:107) L ( R n ) , implying as well L ( R n ) ⊂ L ( R n ) which is untrue proving that the interior of Φ N is actually empty. Now the Baire Category Theorem implies that the F σ set Φ is asubset of L ( R n ) with empty interior. (cid:3) Lemma 6.6. Let us define F = { u ∈ L ( R n ) , (cid:120) R n × R n |W ( u, u )( x, ξ ) | dxdξ < + ∞} . Then F is an F σ of L ( R n ) with empty interior, thus F is a set of first category.Proof. We have F = ∪ N ∈ N F N with(6.3.2) F N = { u ∈ L ( R n ) , (cid:120) R n × R n |W ( u, u )( x, ξ ) | dxdξ ≤ N } . The set F N is a closed subset of L ( R n ) since if ( u k ) k ≥ is a sequence in F N whichconverges in L ( R n ) with limit u , we have ∀ k ≥ , (cid:120) R n × R n |W ( u k , u k )( x, ξ ) | dxdξ ≤ N, so that we may apply Lemma 6.2 with C = N , and readily get that u belongs to F N . We have obtained that F is an F σ subset of L ( R n ) , but we have not yet proventhat F has an empty interior. For that purpose, we shall use the ReconstructionFormula (1.1.10) and note that, in a weak sense, we have u ( x + z u ( x − z (cid:90) W ( u, u )( x, ξ ) e iπzξ dξ, so that u ( x )¯ u ( x ) = (cid:90) W ( u, u )( x + x , ξ ) e iπ ( x − x ) ξ dξ, entailing | u ( x ) || u ( x ) | ≤ (cid:90) |W ( u, u )( x + x , ξ ) | dξ, and for φ, ψ compactly supported with | φ | bounded above by 1, we have (cid:120) | φ ( x ) || u ( x ) || u ( x ) || ψ ( x ) | dx dx ≤ (cid:121) |W ( u, u )( x + x , ξ ) || ψ ( x ) | dξdx dx = 2 n (cid:120) |W ( u, u )( x, ξ ) | dξdx (cid:107) ψ (cid:107) L ( R n ) , so that(6.3.3) (cid:107) uφ (cid:107) L ( R n ) (cid:107) uψ (cid:107) L ( R n ) ≤ n (cid:107)W ( u, u ) (cid:107) L ( R n ) (cid:107) ψ (cid:107) L ( R n ) . Let us then assume that u ∈ F , u (cid:54) = 0 and let us choose a compact set K suchthat (cid:107) uψ (cid:107) L ( R n ) > , ψ = K (Note that (cid:107) uψ (cid:107) L ( R n ) < + ∞ since u and ψ both We have (1 + | x | ) − n +12 ∈ L ( R n ) \ L ( R n ) . NTEGRALS OF THE WIGNER DISTRIBUTION 97 belong to L ( R n ) ). For R > , we choose φ R = {| x |≤ R } so that we get from (6.3.3)that for any R > , (cid:107) uφ R (cid:107) L ( R n ) (cid:107) uψ (cid:107) L ( R n ) ≤ n (cid:107)W ( u, u ) (cid:107) L ( R n ) (cid:107) ψ (cid:107) L ( R n ) , and thus (cid:107) u (cid:107) L ( R n ) (cid:107) uψ (cid:107) L ( R n ) ≤ n (cid:107)W ( u, u ) (cid:107) L ( R n ) (cid:107) ψ (cid:107) L ( R n ) , so that u ∈ Φ (given by (6.3.1)), proving the inclusion F ⊂ Φ , and since Φ has an empty interior, the interior of F is empty as well. (cid:3) Theorem 6.7. Defining (6.3.4) G = F c = { u ∈ L ( R n ) , (cid:120) R n × R n |W ( u, u )( x, ξ ) | dxdξ = + ∞} , we obtain that the set G is a dense G δ subset of L ( R n ) .Proof. It follows immediately from Lemma 6.6 and formula (cid:8) ˚ A (cid:9) c = A c , yielding for F N defined in (6.3.2), L ( R n ) = (cid:8) interior ( ∪ N F N ) (cid:9) c = ∩ N F cN . (cid:3) Consequences on integrals of the Wigner distribution.Lemma 6.8. Let G be defined in (6.3.4) and let u ∈ G . Then the positive andnegative part of the real-valued W ( u, u ) are such that (6.4.1) (cid:120) W ( u, u ) + ( x, ξ ) dxdξ = (cid:120) W ( u, u ) − ( x, ξ ) dxdξ = + ∞ . Proof. For h ∈ (0 , , we define the symbol(6.4.2) a ( x, ξ, h ) = e − h ( x + ξ ) , and we see that it is a semi-classical symbol in the sense (1.2.57). Let us start a reductio ad absurdum and assume that (cid:115) W ( u, u ) − ( x, ξ ) dxdξ < + ∞ , (which impliessince u ∈ G , (cid:115) W ( u, u ) + ( x, ξ ) dxdξ = + ∞ ). We note that (cid:104) Op w ( a ( x, ξ, h )) u, u (cid:105) L ( R n ) = (cid:120) a ( x, ξ, h ) (cid:124) (cid:123)(cid:122) (cid:125) ∈ L ( R n ) W ( u, u )( x, ξ ) (cid:124) (cid:123)(cid:122) (cid:125) ∈ L ( R n ) dxdξ, and thanks to Theorem 1.14 we have also sup h ∈ (0 , |(cid:104) Op w ( a ( x, ξ, h )) u, u (cid:105) L ( R n ) | ≤ σ n (cid:107) u (cid:107) L ( R n ) , so that (cid:120) e − h ( x + ξ ) W ( u, u )( x, ξ ) dxdξ + (cid:120) e − h ( x + ξ ) W ( u, u ) − ( x, ξ ) dxdξ = (cid:120) e − h ( x + ξ ) W ( u, u ) + ( x, ξ ) dxdξ, and thus with θ h ∈ [ − , , we have(6.4.3) θ h σ n (cid:107) u (cid:107) L ( R n ) + (cid:120) e − h ( x + ξ ) W ( u, u ) − ( x, ξ ) dxdξ = (cid:120) e − h ( x + ξ ) W ( u, u ) + ( x, ξ ) dxdξ. Choosing h = 1 /m, m ∈ N ∗ , we note that e − m ( x + ξ ) W ( u, u ) + ( x, ξ ) ≤ e − m +1 ( x + ξ ) W ( u, u ) + ( x, ξ ) . From the Beppo-Levi Theorem (see e.g. Theorem 1.6.1 in [22]) we get that lim m → + ∞ (cid:120) e − m ( x + ξ ) W ( u, u ) + ( x, ξ ) dxdξ = (cid:120) W ( u, u ) + ( x, ξ ) dxdξ = + ∞ . However the left-hand-side of (6.4.3) is bounded above by σ n (cid:107) u (cid:107) L ( R n ) + (cid:120) W ( u, u ) − ( x, ξ ) dxdξ, which is finite,triggering a contradiction. We may now study the case where (cid:120) W ( u, u ) + ( x, ξ ) dxdξ < + ∞ , (cid:120) W ( u, u ) − ( x, ξ ) dxdξ = + ∞ . The identity (6.4.3) still holds true with a left-hand-side going to + ∞ when h goes to0 whereas the right-hand side is bounded. This concludes the proof of the lemma. (cid:3) N.B. A shorter heuristic argument would be that the identity (cid:115) W ( u, u )( x, ξ ) dxdξ = (cid:107) u (cid:107) L ( R n ) and (cid:115) |W ( u, u )( x, ξ ) | dxdξ = + ∞ should imply the lemma, but the for-mer integral is not absolutely converging, so that argument fails to be completelyconvincing since we need to give a meaning to the first integral in this N.B. Theorem 6.9. Defining G by (6.3.4) we find that the set G is a dense G δ set in L ( R n ) and for all u ∈ G , we have (cid:120) W ( u, u ) + ( x, ξ ) dxdξ = (cid:120) W ( u, u ) − ( x, ξ ) dxdξ = + ∞ , (6.4.4) Defining (6.4.5) E ± ( u ) = { ( x, ξ ) ∈ R n , ±W ( u, u )( x, ξ ) > } , we have for all u ∈ G , (6.4.6) (cid:120) E ± ( u ) W ( u, u )( x, ξ ) dxdξ = ±∞ , and both sets E ± ( u ) are open subsets of R n with infinite Lebesgue measure.Proof. The first statements follow from Theorem 6.7 and Lemma 6.8. As far as(6.4.6) is concerned, we note that W ( u, u ) > (resp. < ) on E + ( u ) (resp. E − ( u ) ),so that Theorem 6.7 implies (6.4.6). Moreover E ± ( u ) are open subsets of R n since,thanks to Theorem 1.10, the function W ( u, u ) is continuous; also, both subsets haveinfinite Lebesgue measure from (6.4.4) since W ( u, u ) belongs to L ( R n ) . (cid:3) Note that W ( u, u ) is real-valued. Thanks to Theorem 1.10, the function W ( u, u ) is a continuous function, so it makes sense toconsider its pointwise values. NTEGRALS OF THE WIGNER DISTRIBUTION 99 Remark 6.10. As a consequence of the previous theorem, we could say that forany generic u in L ( R n ) (i.e. any u ∈ G ), we can find open sets E + , E − such thatthe real-valued ±W ( u, u ) is positive on E ± and (cid:120) E ± W ( u, u )( x, ξ ) dxdξ = ±∞ . We shall see in the next section some results on polygons in the plane and forinstance, we shall be able to prove that there exists a “universal number” µ +3 > such that for any triangle T in the plane, we have(6.4.7) ∀ u ∈ L ( R ) , (cid:120) T W ( u, u )( x, ξ ) dxdξ ≤ µ +3 (cid:107) u (cid:107) L ( R ) . Note in particular that we will show that (6.4.7) holds true regardless of the area ofthe triangle (which could be infinite according to our definition of a triangle in ourfootnote 19). Although that type of result may look pretty weak, it gets enhancedby Theorem 6.9 which proves that no triangle in the plane could be a set E + ( u ) (cf.(6.4.5)) for a generic u in L ( R ) .7. Convex polygons of the plane Convex Cones. We have seen in Proposition 5.30 and Theorem 5.20 that theself-adjoint bounded operator with Weyl symbol H ( x ) H ( ξ ) does satisfy(7.1.1) µ − = m = λ min (cid:0) Op w ( H ( x ) H ( ξ )) (cid:1) ≤ Op w ( H ( x ) H ( ξ )) ≤ λ max (cid:0) Op w ( H ( x ) H ( ξ )) (cid:1) = M = µ +2 , [ µ − , µ +2 ] = spectrum ( Op w ( H ( x ) H ( ξ ))) , (7.1.2)with(7.1.3) µ − ≈ − . , µ +2 ≈ . . This result is true as well for the characteristic function of any convex cone (whichis not a half-plane nor the full plane) in the plane since we can map it to the quarterplane by a transformation in Sl (2 , R ) = Sp (1 , R ) . On the other hand a concave coneis the complement of a convex cone and the diagonalization offered by Theorem 5.19proves that the spectrum of the Weyl quantization of the indicatrix of a concavecone is − Spectrum ( Op w ( H ( x ) H ( ξ ))) . We may sum-up the situation by the following theorem. Theorem 7.1. Let Σ θ be a convex cone in R with aperture θ ∈ [0 , π ] (cf. (5.3.9) )and let A θ be the self-adjoint bounded operator with the indicatrix function of Σ θ asa Weyl symbol. We define a triangle as the intersection of three half-planes, which includes of course the convexenvelope of three points, but also the set with infinite area { ( x, ξ ) ∈ R , x ≥ , ξ ≥ , x + ξ ≥ λ } for some λ > . 00 NICOLAS LERNER (1) If θ = 0 , we have A θ = 0 . (2) If θ ∈ (0 , π ) , the operator A θ is unitarily equivalent to Op w ( H ( x ) H ( ξ )) , thuswith spectrum [ µ − , µ +2 ] with µ − < < < µ +2 as given in Theorem 5.20. (3) If θ = π , Σ π is a half-space and A π is a proper orthogonal projection, thuswith spectrum { , } . (4) If θ ∈ ( π, π ) , Σ θ is a concave cone and the operator A θ is unitarily equivalentto Id − Op w ( H ( x ) H ( ξ )) , thus with spectrum [1 − µ +2 , − µ − ] . (5) If θ = 2 π , we have A π = Id . Remark 7.2. It is only in the trivial cases θ ∈ { , π, π } that A θ is an orthogonalprojection. These cases are also characterized (among cones) by the fact that thespectrum of A θ is included in [0 , . Remark 7.3. It is interesting to remark that all operators A θ for θ ∈ (0 , π ) areunitarily equivalent and thus with constant spectrum [ µ − , µ +2 ] as given in Theorem5.20. Nevertheless the sequence ( A θ ) <θ<π is weakly converging to the orthogonalprojection A π whose spectrum is { , } : indeed for φ ∈ S ( R ) , ψ ∈ S ( R ) , we have (cid:104)A θ φ, ψ (cid:105) L ( R ) = (cid:120) Σ θ W ( φ, ψ ) (cid:124) (cid:123)(cid:122) (cid:125) ∈ S ( R ) ( x, ξ ) dxdξ, and thus the Lebesgue Dominated Convergence Theorem implies that(7.1.4) lim θ → π − (cid:104)A θ φ, ψ (cid:105) L ( R ) = (cid:104)A π φ, ψ (cid:105) L ( R ) . On the other hand for u, v ∈ L ( R ) and sequences ( φ k ) k ≥ , ( ψ k ) k ≥ in S ( R ) withrespective limits u, v in L ( R ) , we have (cid:104)A θ u, v (cid:105) L ( R ) = (cid:104)A θ ( u − φ k ) , v (cid:105) L ( R ) + (cid:104)A θ φ k , v − ψ k (cid:105) L ( R ) + (cid:104)A θ φ k , ψ k (cid:105) L ( R ) , so that (cid:104)A θ u, v (cid:105) L ( R ) − (cid:104)A π u, v (cid:105) L ( R ) = (cid:104)A θ ( u − φ k ) , v (cid:105) L ( R ) + (cid:104)A θ φ k , v − ψ k (cid:105) L ( R ) + (cid:104)A θ φ k , ψ k (cid:105) L ( R ) , − (cid:104)A π ( u − φ k ) , v (cid:105) L ( R ) − (cid:104)A π φ k , v − ψ k (cid:105) L ( R ) − (cid:104)A π φ k , ψ k (cid:105) L ( R ) , implying |(cid:104)A θ u, v (cid:105) L ( R ) − (cid:104)A π u, v (cid:105) L ( R ) |≤ ( µ +2 + 1) (cid:0) (cid:107) u − φ k (cid:107) L ( R ) (cid:107) v (cid:107) L ( R ) + (cid:107) v − ψ k (cid:107) L ( R ) (cid:107) φ k (cid:107) L ( R ) (cid:1) + |(cid:104)A θ φ k , ψ k (cid:105) L ( R ) − (cid:104)A π φ k , ψ k (cid:105) L ( R ) | , and thus, using (7.1.4), we get lim sup θ → + |(cid:104)A θ u, v (cid:105) L ( R ) − (cid:104)A π u, v (cid:105) L ( R ) |≤ ( µ +2 + 1) (cid:0) (cid:107) u − φ k (cid:107) L ( R ) (cid:107) v (cid:107) L ( R ) + (cid:107) v − ψ k (cid:107) L ( R ) (cid:107) φ k (cid:107) L ( R ) (cid:1) . So that we have in particular, from (2), the inequalities − µ +2 < < < − µ − . NTEGRALS OF THE WIGNER DISTRIBUTION 101 Taking now the infimum with respect to k of the right-hand-side in the above in-equality, we obtain indeed the weak convergence(7.1.5) lim θ → + (cid:104)A θ u, v (cid:105) L ( R ) = (cid:104)A π u, v (cid:105) L ( R ) . Of course we cannot have strong convergence of the bounded self-adjoint A θ towards(the bounded self-adjoint) A π because of their respective spectra and the same linescan be written on the weak limit when θ → + of A θ .7.2. Triangles. We may consider general “triangles” in the plane that we define as(7.2.1) T c ,c ,c L ,L ,L = (cid:8) ( x, ξ ) ∈ R , L j ( x, ξ ) ≥ c j , j ∈ { , , } (cid:9) ,c j are real numbers and L j are linear forms. To avoid degenerate situations, we shallassume that(7.2.2) for j (cid:54) = k, dL j ∧ dL k (cid:54) = 0 , |T c ,c ,c L ,L ,L | > and T c ,c ,c L ,L ,L is not a cone . Note that this includes standard triangles (convex envelope of three non-colinearpoints) but also sets with infinite area such as(7.2.3) { ( x, ξ ) ∈ R , x ≥ , ξ ≥ , x + ξ ≥ λ } , where λ is a positive parameter.Without loss of generality, we may assume that L ( x, ξ ) − c = x, L ( x, ξ ) − c = ξ ,so that T c ,c ,c L ,L ,L = { ( x, ξ ) ∈ R , x ≥ , ξ ≥ , ax + bξ ≥ ν } , where a, b, λ are real parameters with a (cid:54) = 0 , b (cid:54) = 0 from the assumption (7.2.2);using the symplectic mapping ( x, ξ ) (cid:55)→ ( µx, ξ/µ ) with µ = (cid:112) | b/a | , we see that thecondition ax + bξ ≥ ν becomes x sign a + ξ sign b ≥ λ = ν/ (cid:112) | ab | i.e x + ξ ≥ ˜ ν,x − ξ ≥ ˜ ν, − x + ξ ≥ ˜ ν, − x − ξ ≥ ˜ ν. The first case requires ˜ ν > and the other cases ˜ ν < . The only case with finitearea is the fourth case(7.2.4) T ,λ = { ( x, ξ ) ∈ R , x ≥ , ξ ≥ , x + ξ ≤ λ } triangle with area λ / , λ > .The second case is(7.2.5) T ,λ = { ( x, ξ ) ∈ R , x ≥ , ξ ≥ , x − ξ ≥ − λ } , λ > , The third case is(7.2.6) T ,λ = { ( x, ξ ) ∈ R , x ≥ , ξ ≥ , ξ − x ≥ − λ } , λ > , and the first case is(7.2.7) T ,λ = { ( x, ξ ) ∈ R , x ≥ , ξ ≥ , ξ + x ≥ λ } , λ > . 02 NICOLAS LERNER Proposition 7.4. Let T ,λ be a triangle with finite non-zero area in the plane givenby (7.2.4) , where λ is a positive parameter. Then the operator Op w ( T ,λ ) is unitarilyequivalent to the operator with kernel (7.2.8) ˜ k ,λ ( x, y ) = [0 ,λ ] (cid:0) x + y (cid:1) sin (cid:0) π ( x − y )( λ − x + y ) (cid:1) π ( x − y ) . The operator Op w ( T ,λ ) is self-adjoint and bounded on L ( R ) so that (7.2.9) (cid:107) Op w ( T ,λ ) (cid:107) B ( L ( R )) ≤ (cid:18) µ +2 + (cid:113) µ +2 ) (cid:19) := ˜ µ , where µ +2 is given in (7.1.1) .Proof. The kernel k ,λ of Op w ( T ,λ ) is such that k ,λ ( x, y ) = [0 ,λ ] (cid:0) x + y (cid:1) (cid:90) λ − x + y e iπ ( x − y ) ξ dξ = [0 ,λ ] (cid:0) x + y (cid:1) (cid:0) e iπ ( x − y )( λ − x + y ) − (cid:1) iπ ( x − y )= e iπ ( λx − x ) [0 ,λ ] (cid:0) x + y (cid:1) sin( π ( x − y )( λ − x + y )) π ( x − y ) e − iπ ( λy − y ) , proving (7.2.8). We note now that the kernel of the operator with Weyl symbol H ( ξ ) H ( λ − ξ − x ) is(7.2.10) (cid:96) λ ( x, y ) = e iπ ( λx − x ) H ( λ − x + y π ( x − y )( λ − x + y )) π ( x − y ) e − iπ ( λy − y ) , and that Op w ( H ( ξ ) H ( λ − ξ − x )) is unitarily equivalent to the operator Op w ( H ( x ) H ( ξ )) as given by Theorem 7.1. Weget then(7.2.11) k ,λ ( x, y ) = H ( x + y ) (cid:96) λ ( x, y ) = H ( x ) (cid:96) λ ( x, y ) H ( y )+ H ( x + y ) (cid:0) H ( x ) ˇ H ( y ) + ˇ H ( x ) H ( y ) (cid:1) H ( λ − x + y × sin( π ( x − y )( λ − x + y )) π ( x − y ) × e iπ ( λx − x ) e − iπ ( λy − y ) , and we have thus(7.2.12) Op w ( T ,λ ) = H Op w (cid:0) H ( ξ ) H ( λ − ξ − x ) (cid:1) H + Ω λ , where the kernel ω λ ( x, y ) of the operator Ω λ verifies | ω λ ( x, y ) | ≤ H ( x + y ) (cid:0) H ( x ) ˇ H ( y ) + ˇ H ( x ) H ( y ) (cid:1) π | x − y | = H ( x + y ) (cid:0) H ( x ) ˇ H ( y ) + ˇ H ( x ) H ( y ) (cid:1) π ( | x | + | y | ) . NTEGRALS OF THE WIGNER DISTRIBUTION 103 We obtain thanks to Proposition 9.13 [2] that(7.2.13) (cid:120) | ω λ ( x, y ) || u ( y ) || u ( x ) | dydx ≤ (cid:107) ˇ Hu (cid:107) L ( R ) (cid:107) Hu (cid:107) L ( R ) . As a result, we find that |(cid:104) Op w ( T ,λ ) u, u (cid:105) L ( R ) | ≤ µ +2 (cid:107) Hu (cid:107) L ( R ) + (cid:107) ˇ Hu (cid:107) L ( R ) (cid:107) Hu (cid:107) L ( R ) , proving (7.2.9). (cid:3) Proposition 7.5. Let T ,λ be a triangle with infinite area in the plane given by (7.2.7) , where λ is a positive parameter. Then the operator Op w ( T ,λ ) is unitarilyequivalent to the operator with kernel (7.2.14) ˜ k ,λ ( x, y ) = [0 ,λ ] (cid:0) x + y (cid:1) sin (cid:0) π ( x − y )( λ − x + y ) (cid:1) π ( x − y ) . The operator Op w ( T ,λ ) is self-adjoint and bounded on L ( R ) so that (7.2.15) (cid:107) Op w ( T ,λ ) (cid:107) B ( L ( R )) ≤ (cid:32) µ +2 + (cid:114) 14 + ( µ +2 ) (cid:33) ≈ . , where µ +2 is given in (7.1.1) .Proof. The kernel k ,λ of Op w ( T ,λ ) is such that k ,λ ( x, y ) = H ( x + y ) e iπ ( x − y ) max(0 ,λ − x + y ) (cid:16) δ ( y − x ) + 1 iπ ( y − x ) (cid:17) = H ( x ) 12 δ ( x − y ) H ( y ) + H ( x ) e iπ ( x − y ) max(0 ,λ − x + y ) iπ ( y − x ) H ( y )+ H ( x + y ) (cid:0) H ( x ) ˇ H ( y ) + ˇ H ( x ) H ( y ) (cid:1) e iπ ( x − y ) max(0 ,λ − x + y ) iπ ( y − x ) . We note that the kernel of the operator Op w ( H ( x + ξ − λ ) H ( ξ )) is (cid:96) ( x, y ) = e iπ ( x − y ) max(0 ,λ − x + y ) (cid:16) δ ( y − x ) + 1 iπ ( y − x ) (cid:17) , so that(7.2.16) Op w ( T ,λ ) = H Op w ( H ( x + ξ − λ ) H ( ξ )) (cid:124) (cid:123)(cid:122) (cid:125) unitarily equivalent toOp w ( H ( x ) H ( ξ )) H + Ω ,λ , where the kernel ω ,λ of the operator Ω ,λ is such that | ω ,λ ( x, y ) | ≤ H ( x + y ) (cid:0) H ( x ) ˇ H ( y ) + ˇ H ( x ) H ( y ) (cid:1) π ( | x | + | y | ) , and, thanks to Proposition 9.13 [2], we get from (7.2.16) As a result, we find that |(cid:104) Op w ( T ,λ ) u, u (cid:105) L ( R ) | ≤ µ +2 (cid:107) Hu (cid:107) L ( R ) + 12 (cid:107) ˇ Hu (cid:107) L ( R ) (cid:107) Hu (cid:107) L ( R ) , which gives (7.2.15). (cid:3) 04 NICOLAS LERNER We leave for the reader to check the two other cases (7.2.5), (7.2.6), which arevery similar as well as the degenerate cases excluded by (7.2.2), which are in facteasier to tackle. Theorem 7.6. Let T = (cid:8) T c ,c ,c L ,L ,L (cid:9) c j ∈ R , L j linear form on R be the set of triangles of R . Forall T ∈ T , the operator Op w ( T ) is bounded on L ( R ) , self-adjoint and we have (7.2.17) . ≈ µ +2 = sup C cone (cid:107) Op w ( C ) (cid:107) B ( L ( R )) ≤ µ +3 = sup T triangle (cid:107) Op w ( T ) (cid:107) B ( L ( R )) ≤ ˜ µ ≈ . . N.B. The L boundedness is easy to prove since it is obvious for triangles with finiteareas and in the case of triangles with infinite area, we may note that in the case(7.2.7) (resp. (7.2.5), (7.2.6)) they are the union of two cones (resp. one cone) witha strip [0 , × R + . What matters most in the above statement is the effective explicitbound. Our result does not give an explicit value for µ +3 and it is quite likely thatthe bound given by ˜ µ is way too large. Proof. The second inequality is proven in Propositions 7.4 & 7.5 and the first in-equality is a consequence of Theorem 5.26. (cid:3) Remark 7.7. This implies that for any u ∈ L ( R ) and any T ∈ T , we have (7.2.18) (cid:12)(cid:12)(cid:12) (cid:120) T W ( u, u )( x, ξ ) dxdξ (cid:12)(cid:12)(cid:12) ≤ ˜ µ (cid:107) u (cid:107) L ( R ) , with ˜ µ ≈ . . Convex Polygons. We want to tackle now the general case of a convex poly-gon in the plane. We consider L , . . . , L N , to be N linear forms of x, ξ ( L j ( x, ξ ) = a j ξ − α j x = [( x, ξ ); ( a j , α j )] ) and c , . . . , c N some real constants. We consider the convex polygon(7.3.1) P = { ( x, ξ ) ∈ R , ∀ j ∈ { , . . . , N } , L j ( x, ξ ) − c j ≥ } , so that P ( x, ξ ) = (cid:89) ≤ j ≤ N H (cid:0) L j ( x, ξ ) − c j (cid:1) . Definition 7.8. Let N ∈ N ∗ , let L , . . . L N be linear forms on R and let c , . . . , c N be real numbers. The polygon with N sides P c ,...,c N L ,...,L N is defined by (7.3.1) . We shalldenote by P N the set of all polygons with N sides. N.B. Since we may take some L j = 0 in (7.3.1) , we see that P N ⊂ P N +1 . Note as above that it includes some convex subsets of the plane with infinite areasuch as (7.2.3). NTEGRALS OF THE WIGNER DISTRIBUTION 105 Theorem 7.9. Let P N be the set of convex polygons with N sides of the plane R .We define (7.3.2) µ + N = sup P∈ P N (cid:107) Op w ( P ) (cid:107) B ( L ( R )) Then µ +2 is given by Theorem 5.20 and (7.3.3) ∀ N ≥ , µ + N ≤ (cid:112) N/ . Proof. Using an affine symplectic transformation, we may assume that L N ( x, ξ ) − c N = x , so that P ( x, ξ ) = H ( x ) (cid:89) ≤ j ≤ N − H (cid:0) a j ξ − α j x − c j (cid:1) . and the kernel of the operator Op w ( P ) is k N ( x, y ) = H ( x + y ) (cid:90) e iπ ( x − y ) ξ (cid:89) ≤ j ≤ N − H (cid:0) a j ξ − α j ( x + y − c j (cid:1) dξ. As a result, we have k N ( x, y ) = H ( x + y ) k N − ( x, y ) , where k N − is the kernel of Op w ( P N − ) , where P N − = { ( x, ξ ) ∈ R , ∀ j ∈ { , . . . , N − } , L j ( x, ξ ) − c j ≥ } . We may assume inductively that for any convex polygon P k with k ≤ N − sides,there exist µ + k such that(7.3.4) Op w ( P k ) ≤ µ + k , where µ + k depends only on k and not on the area of the polygon, a fact alreadyproven for k = 1 , , . We note that with A N = Op w ( P N ) , we have with H standingfor the operator of multiplication by H ( x ) , HA N H = HA N − H, A N − = Op w ( P N − ) , since the kernel of HA N H is H ( x ) H ( y ) k N ( x, y ) = H ( x + y ) H ( x ) H ( y ) k N − ( x, y ) = H ( x ) H ( y ) k N − ( x, y ) . Also we have, with ˇ H ( x ) = H ( − x ) , that ˇ HA N ˇ H = 0 , since the kernel of that operator is ˇ H ( x ) ˇ H ( y ) H ( x + y ) k N − ( x, y ) = 0 . We have thus(7.3.5) A N = HA N − H + 2 Re ˇ HA N H, and the kernel of HA N H is(7.3.6) ω N ( x, y ) = H ( x + y ) (cid:0) ˇ H ( x ) H ( y ) + ˇ H ( y ) H ( x ) (cid:1) k N − ( x, y ) . We calculate now(7.3.7) k N − ( x, y ) = (cid:90) e iπ ( x − y ) ξ (cid:89) ≤ j ≤ N − H (cid:0) a j ξ − α j ( x + y − c j (cid:1) dξ. 06 NICOLAS LERNER We check first the j such that a j = 0 (and thus α j (cid:54) = 0 ) . Without loss of generality,we may assume that this happens for ≤ j < N so that with some interval J ofthe real line, ˜ α j = α j /a j , ˜ c j = c j /a j , k N − ( x, y ) = J ( x + y (cid:90) e iπ ( x − y ) ξ (cid:89) N ≤ j ≤ N − a j > H (cid:0) ξ − ˜ α j ( x + y − ˜ c j (cid:1) × (cid:89) N ≤ j ≤ N − a j < ˇ H (cid:0) ξ − ˜ α j ( x + y − ˜ c j (cid:1) dξ. We note that the integration domain is ψ ( x + y N ≤ j ≤ N − a j > (cid:0) ˜ α j ( x + y c j (cid:1) ≤ ξ ≤ min N ≤ j ≤ N − a j < ˜ α j ( x + y c j = − φ ( x + y , with φ, ψ convex piecewise affine functions; since φ + ψ is also a convex function, weget the – convex – constraint ( φ + ψ )(( x + y ) / ≤ , so that ( x + y ) / must belongto a subinterval ˜ J of the interval J . As a result we get that k N − ( x, y ) = ˜ J ( x + y e − iπ ( x − y ) φ ( x + y ) − e iπ ( x − y ) ψ ( x + y ) iπ ( x − y )= ˜ J ( x + y e − iπ ( x − y )( φ − ψ )( x + y ) e − iπ ( x − y )( φ + ψ )( x + y ) − e iπ ( x − y )( φ + ψ )( x + y ) iπ ( x − y )= ˜ J ( x + y e − iπ ( x − y )( φ − ψ )( x + y ) sin (cid:0) π ( x − y )( φ + ψ )( x + y ) (cid:1) π ( y − x ) , and thus the kernel of HA N H is ω N ( x, y ) = H ( x + y ) (cid:0) ˇ H ( x ) H ( y ) + ˇ H ( y ) H ( x ) (cid:1) ˜ J ( x + y × e − iπ ( x − y )( φ − ψ )( x + y ) sin (cid:0) π ( x − y )( φ + ψ )( x + y ) (cid:1) π ( y − x ) , so that, thanks to Proposition 9.13 [2],(7.3.8) (cid:104) ˇ HA N Hu, u (cid:105) ≤ (cid:107) Hu (cid:107)(cid:107) ˇ Hu (cid:107) , and with (7.3.5), (cid:104) A N u, u (cid:105) ≤ µ + N − (cid:107) Hu (cid:107) + (cid:107) Hu (cid:107)(cid:107) ˇ Hu (cid:107) , we get(7.3.9) µ + N ≤ µ + N − + (cid:113) ( µ + N − ) + 12 , which implies that(7.3.10) ∀ N ≥ , µ + N ≤ (cid:112) N/ , In this induction proof, we may assume that all the linear forms L j , ≤ j ≤ N are differentfrom 0, otherwise we may use the induction hypothesis. NTEGRALS OF THE WIGNER DISTRIBUTION 107 since it is true for N = 3 and if we assume that it is true for some N ≥ , we get µ + N +1 ≤ µ + N + (cid:112) ( µ + N ) + 12 ≤ (cid:0)(cid:114) N (cid:114) N + 22 (cid:1) ≤ (cid:114) N + 12 , where the latter inequality follows from the concavity of the square-root functionsince we have for a concave function F , N N + 22 = N + 12 and thus F (cid:0) N (cid:1) + 12 F (cid:0) N + 22 (cid:1) ≤ F (cid:0) N + 12 (cid:1) . The proof of Theorem 7.9 is complete. (cid:3) Remark 7.10. The above result is weak by its dependence on the number of sides,but it should be pointed out that it is independent of the area of the polygon (whichcould be infinite). Another general comment is concerned with convexity: althoughFlandrin’s conjecture is not true, there is still something special about convex subsetsof the phase space and it is in particular interesting that an essentially explicitcalculation of the kernel of the operator Op w ( P ) is tractable when P is a polygonwith N sides of R .7.4. Symbols supported in a half-space.Theorem 7.11. [ ] Let A be a bounded self-adjoint operator on L ( R n ) such that its Weyl symbol a ( x, ξ ) is supported in R + × R n − . Then with ˇ H standing for the orthogonal pro-jection onto (7.4.1) { u ∈ L ( R n ) , supp u ⊂ R − × R n − } , we have ˇ HA ˇ H = 0 . [ ] Let A be as above; if A is a non-negative operator, then with H = I − ˇ H , wehave ˇ HA = A ˇ H = 0 , A = HAH, N.B. We have seen explicit examples of bounded self-adjoint operators such that theWeyl symbol is supported in x ≥ but for which ˇ HAH (cid:54) = 0 : the quarter-planeoperator (see Section 5.1) has the Weyl symbol H ( x ) H ( ξ ) , the kernel of ˇ H Op w ( H ( x ) H ( ξ )) H is ˇ H ( x ) H ( y ) H ( x + y ) 12 iπ pv y − x , which is not the zero distribution and, according to the above result, this alone impliesthat Op w ( H ( x ) H ( ξ )) cannot be non-negative.Proof. Let us prove first that ˇ HA ˇ H = 0 ; let φ, ψ ∈ C ∞ c ( R n ) such that supp φ ∪ supp ψ ⊂ ( −∞ , × R n − . Since the Wigner distribution W ( φ, ψ ) belongs to S ( R n ) and is given by the integral W ( φ, ψ )( x, ξ ) = (cid:90) R n φ ( x + z ψ ( x − z e − iπz · ξ dz, Indeed we have µ +3 ≤ ˜ µ < . < . ≈ (cid:112) / . 08 NICOLAS LERNER we infer right away that supp W ( φ, ψ ) ⊂ ( −∞ , × R n − . We know also that (cid:104) Aφ, ψ (cid:105) L ( R n ) = (cid:104) Aφ, ψ (cid:105) S (cid:48) ( R n ) , S ( R n ) = (cid:104) a, W ( φ, ψ ) (cid:105) S (cid:48) ( R n ) , S ( R n ) = 0 . As a result, the L ( R n ) bounded operator ˇ HA ˇ H is such that, for u, v ∈ L ( R n ) , φ, ψ as above, (cid:104) ˇ HA ˇ Hu, v (cid:105) L ( R n ) = (cid:104) ˇ HA ˇ H ˇ Hu, ˇ Hv (cid:105) L ( R n ) = (cid:104) ˇ HA ˇ H ( ˇ Hu − φ ) , ˇ Hv (cid:105) L ( R n ) + (cid:104) ˇ HA ˇ Hφ, ˇ Hv − ψ (cid:105) L ( R n ) + (cid:104) ˇ HA ˇ Hφ, ψ (cid:105) L ( R n ) (cid:124) (cid:123)(cid:122) (cid:125) (cid:104) Aφ,ψ (cid:105) L R n ) =0 , so that |(cid:104) ˇ HA ˇ Hu, v (cid:105) L ( R n ) | ≤ (cid:107) A (cid:107) B ( L ( R n )) (cid:16) (cid:107) ˇ Hu − φ (cid:107) L ( R n ) (cid:107) v (cid:107) L ( R n ) + (cid:107) ˇ Hv − ψ (cid:107) L ( R n ) (cid:107) φ (cid:107) L ( R n ) (cid:17) . Using now that the set { φ ∈ C ∞ c ( R n ) , supp φ ⊂ ( −∞ , × R n − } is dense in(7.4.2) { w ∈ L ( R n ) , supp w ⊂ ( −∞ , × R n − } , we obtain that (cid:104) ˇ HA ˇ Hu, v (cid:105) L ( R n ) = 0 and the first result. Let us assume that theoperator A is non-negative. We have A = B , B = B ∗ bounded self-adjoint.It implies with L ( R n ) norms and dot-products, (cid:104) Au, u (cid:105) = (cid:104) HAHu, u (cid:105) + 2 Re (cid:104) ˇ HAHu, ˇ Hu (cid:105) = (cid:104) HBBHu, u (cid:105) + 2 Re (cid:104) ˇ HBBHu, ˇ Hu (cid:105) = (cid:107) BHu (cid:107) + 2 Re (cid:104) BHu, B ˇ Hu (cid:105) = (cid:107) BHu + B ˇ Hu (cid:107) − (cid:107) B ˇ Hu (cid:107) = (cid:107) Bu (cid:107) − (cid:107) B ˇ Hu (cid:107) = (cid:104) Au, u (cid:105) − (cid:107) B ˇ Hu (cid:107) , and thus B ˇ H = 0 , so that ˇ HB = 0 and thus ˇ HB = ˇ HA = 0 = A ˇ H , so that ˇ HAH = 0 = HA ˇ H , and A = HAH, concluding the proof of [2]. (cid:3) In the integrand, we must have, x + z ≤ − (cid:15) < , x − z ≤ − (cid:15) < and thus x ≤− ( (cid:15) + (cid:15) ) / Let χ be a function satisfying (5.2.1) and let w be in the set (7.4.2). Let ( φ k ) k ≥ be asequence in C ∞ c ( R n ) converging in L ( R n ) towards w ; the function defined by ˜ φ k ( x ) = χ ( − kx ) φ k ( x ) , belongs to C ∞ c ( R n ) , is supported in ( −∞ , − /k ] × R n − , and that sequence converges in L ( R n ) towards w since (cid:107) ˜ φ k − w (cid:107) L ( R n ) ≤ (cid:107) χ ( − kx ) (cid:0) φ k ( x ) − w ( x ) (cid:1) (cid:107) L ( R n ) (cid:124) (cid:123)(cid:122) (cid:125) ≤(cid:107) φ k − w (cid:107) L R n ) → when k → + ∞ . + (cid:107) ( χ ( − kx ) − w ( x ) (cid:107) L ( R n ) and (cid:107) ( χ ( − kx ) − w ( x ) (cid:107) L ( R n ) ≤ (cid:82) (cid:8) − k ≤ x ≤ (cid:9) | w ( x ) | dx which has also limit 0 when k goes to + ∞ by the Lebesgue Dominated Convergence Theorem. NTEGRALS OF THE WIGNER DISTRIBUTION 109 Corollary 7.12. Let A be a bounded self-adjoint operator on L ( R n ) such that itsWeyl symbol is supported in R + × R n − and such that Re( ˇ HAH ) (cid:54) = 0 , then thespectrum of A intersects ( −∞ , .Proof. We have from [1] in the previous theorem, A = ( H + ˇ H ) A ( H + ˇ H ) = HAH + 2 Re HA ˇ H, and from [2] , if A were non-negative, we would have A ˇ H = 0 and Re HA ˇ H = 0 ,contradicting the assumption. (cid:3) Remark 7.13. If C is a compact convex body of R n , we may use the fact (see e.g.[31]) that C = (cid:92) H j closed half-spacescontaining K H j . Then of course Op w ( C ) is a bounded self-adjoint operator on L ( R n ) , and if H j isdefined by H j = { ( x, ξ ) ∈ R , L j ( x, ξ ) ≥ c j } , where L j is a linear form on R and c j a real constant, we obtain with the symplecticcovariance of the Weyl calculus, setting H j ( x, ξ ) = H ( L j ( x, ξ ) − c j ) , that for all H j closed half-spaces containing K , we have(7.4.3) Op w ( K ) = Op w ( H j ) Op w ( K ) Op w ( H j ) + 2 Re Op w ( ˇ H j ) Op w ( K ) Op w ( H j ) , where ˇ H ( x, ξ ) = H ( − L j ( x, ξ ) + c j ) .8. Open questions & Conjectures In this section we review the rather long list of conjectures formulated in the textand we try to classify their statements by rating their respective interest, relevanceand difficulty. We should keep in mind that the study of Op w ( E ) for a subset E ofthe phase space is highly correlated to some particular set of special functions relatedto E : Hermite functions and Laguerre polynomials for ellipses, Airy functions forparabolas, homogeneous distributions for hyperbolas and so on. It is quite likelythat the “shape” of E will determine the type of special functions to be studied togetting a diagonalization of the operator Op w ( E ) .8.1. Anisotropic Ellipsoids & Paraboloids.Conjecture 8.1. Let E be an ellipsoid in R n equipped with its canonical symplecticstructure. Then the operator Op w ( E ) is bounded on L ( R n ) (which is obvious from (1.2.5) ) and we have (8.1.1) Op w ( E ) ≤ Id . 10 NICOLAS LERNER A sharp version of this result was proven for n = 1 in the 1988 P. Flandrin’s article[9], and was improved to an isotropic higher dimensional setting in the paper [27] byE. Lieb and Y. Ostrover. Without isotropy, it remains a conjecture. As describedin more details in Section 3.4, it can be reformulated as a problem on Laguerrepolynomials. That conjecture is a very natural one and it would be quite surprisingthat a counterexample to (8.1.1) could occur from an anisotropic ellipsoid . Weintroduced in Section 4.4 a conjecture on anisotropic paraboloids directly related toConjecture 8.1. Conjecture 8.2. Let E be an anisotropic paraboloid in R n equipped with its canon-ical symplectic structure. Then the operator Op w ( E ) is bounded on L ( R n ) and wehave (8.1.2) Op w ( E ) ≤ Id . In terms of special functions, it is related to a property of Airy-type functions. Asa contrast with ellipses, we do not expect (8.1.2) to leave any room for improvementwhereas (8.1.1) can certainly be improved with its right-hand-side replaced by asmaller operator as in (3.2.5).8.2. Balls for the (cid:96) p norm. We have seen in Section 5.3.2 that the quantization ofthe indicatrix of a (cid:96) p ball could have a spectrum intersecting (1 , + ∞ ) when p (cid:54) = 2 .More generally one could raise the following question. Question 8.3. Let p ∈ [1 , + ∞ ] , p (cid:54) = 2 and let B np be the unit (cid:96) p ball in R n . For λ > , we define the operator (8.2.1) P n,p,λ = Op w ( λ B np ) . Is it possible to say something on the spectrum of the operator P n,p,λ , even in a two-dimensional phase space ( n = 1 )? Is there an asymptotic behaviour for the upperbound of the spectrum of P n,p,λ when λ goes to + ∞ ? On generic pulses in L ( R n ) . We have seen that the set G defined in (6.3.4) isgeneric in the Baire category sense, but our explicit examples were quite simplistic. Question 8.4. Let G be defined in (6.3.4) . Does there exist u ∈ G such that the set E + ( u ) (defined in (6.4.5) ) is connected? On convex bodies.Conjecture 8.5. For N ≥ , we define (8.4.1) µ + N = sup P convex boundedpolygon with N sides Spectrum ( Op w ( P )) . Then the sequence ( µ + N ) N ≥ is increasing and there exists α > such that (8.4.2) ∀ N ≥ , µ + N ≤ α ln N. We mean by anisotropic ellipsoid a set of type (3.3.2) where < a < a < · · · < a n . According to our Definition 7.8 of the set P N of polygons with N sides is increasing withrespect to N . NTEGRALS OF THE WIGNER DISTRIBUTION 111 N.B. Theorem 7.9 is a small step in this direction. A stronger version of Conjecture 8.5 would be Conjecture 8.6. We define (8.4.3) µ + = sup C convexbounded Spectrum ( Op w ( C )) . Then we have µ + < + ∞ . The invalid Flandrin’s conjecture was µ + = 1 and we know now that µ + ≥ µ +2 > as given by (7.1.3). Question 8.7. There is a diagonalization of the quantization of the indicatrix func-tion of Ellipsoids, Paraboloids and Hyperbolic regions. Is there a non-quadraticexample of diagonalization? Question 8.8. The value of µ +2 is known explicitly, but for µ +3 , we have only theupperbound ˜ µ as given by Theorem 7.6. Is it possible to determine explicitly thevalue of µ +3 , either by answering Question 8.7, or via another argument? Conjecture 8.9. Let C be a proper closed convex subset of R with positive Lebesguemeasure such that Op w ( C ) is bounded self-adjoint on L ( R ) (that assumption isuseless if Conjecture 8.6 is proven) with a spectrum included in [0 , . Then C is thestrip [0 , × R , up to an affine symplectic map. All the explicitly avalaible examples are compatible with that conjecture (see alsoRemark 7.2) and the second part of Theorem 7.11 is also an indication in thatdirection. It would be nice in that instance to reach a spectral characterization of asubset modulo the affine symplectic group.9. Appendix Fourier transform, Weyl quantization, Harmonic Oscillator. Fourier transform. We use in this paper the following normalization for theFourier transform and inversion formula: for u ∈ S ( R n ) ,(9.1.1) ˆ u ( ξ ) = (cid:90) R n e − iπx · ξ u ( x ) dx, u ( x ) = (cid:90) R n e iπx · ξ ˆ u ( ξ ) dξ, a formula that can be extended to u ∈ S (cid:48) ( R n ) , with defining the distribution ˆ u bythe duality bracket(9.1.2) (cid:104) ˆ u, φ (cid:105) S (cid:48) ( R n ) , S ( R n ) = (cid:104) u, ˆ φ (cid:105) S (cid:48) ( R n ) , S ( R n ) . Checking (9.1.1) for u ∈ S (cid:48) ( R n ) is then easy, that is(9.1.3) ˇˆˆ u = u, where the distribution ˇ u is defined by(9.1.4) (cid:104) ˇ u, φ (cid:105) S (cid:48) ( R n ) , S ( R n ) = (cid:104) u, ˇ φ (cid:105) S (cid:48) ( R n ) , S ( R n ) , with ˇ φ ( x ) = φ ( − x ) . 12 NICOLAS LERNER It is useful to notice that for u ∈ S (cid:48) ( R n ) ,(9.1.5) ˇˆ u = ˆˇ u. This normalization yields simple formulas for the Fourier transform of Gaussianfunctions: for A a real-valued symmetric positive definite n × n matrix, we definethe function v A in the Schwartz space by(9.1.6) v A ( x ) = e − π (cid:104) Ax,x (cid:105) , and we have (cid:99) v A ( ξ ) = (det A ) − / e − π (cid:104) A − ξ,ξ (cid:105) . Similarly when B is a real-valued symmetric non-singular n × n matrix, the function w B defined by w B ( x ) = e iπ (cid:104) Bx,x (cid:105) is in L ∞ ( R n ) and thus a tempered distribution and we have(9.1.7) (cid:99) w B ( ξ ) = | det B | − / e iπ sign B e − iπ (cid:104) B − ξ,ξ (cid:105) , where sign B stands for the signature of B that is, with E the set of eigenvalues of B (which are real and non-zero),(9.1.8) sign B = Card( E ∩ R + ) − Card( E ∩ R − ) . With H standing for the characteristic function of R + , we have H + ˇ H, δ = ˆ H + ˆˇ H,D sign = δ iπ , (cid:92) D sign = 1 iπ , ξ (cid:100) sign = 1 iπ , (cid:100) sign = 1 iπ pv ξ , (principal value) the latter formula following from the fact that ξ (cid:0) (cid:100) sign − pv iπξ (cid:1) = 0 , which implies (cid:100) sign − pv iπξ = cδ = 0 , since (cid:100) sign − iπξ is odd. We infer from that ˆ H − (cid:98) ˇ H = (cid:100) sign = pv iπξ , and(9.1.9) ˆ H = δ pv iπξ . Lemma 9.1. Let T be a compactly supported distribution on R n such that (9.1.10) ∀ N ∈ N , (cid:104) ξ (cid:105) N ˆ T ( ξ ) is bounded, with (cid:104) ξ (cid:105) = (cid:112) | ξ | .Then T is a C ∞ function.Proof. Note that ˆ T is an entire function, as the Fourier transform of a compactlysupported distribution. Moreover, from (9.1.10) with N = n + 1 , we get that ˆ T belongs to L ( R n ) and thus T is a continuous function. Moreover, we have for any α ∈ N n , ( D αx T )( x ) = (cid:90) e iπx · ξ ξ α ˆ T ( ξ ) (cid:124) (cid:123)(cid:122) (cid:125) ∈ L ( R n ) dξ, so that T is a C ∞ function. (cid:3) NTEGRALS OF THE WIGNER DISTRIBUTION 113 Proposition 9.2. Let ρ > and let f be an holomorphic function on a neighborhoodof { z ∈ C , | Im z | ≤ ρ } such that ∀ y ∈ [ − ρ, ρ ] , (cid:90) | f ( x + iy ) | dx < + ∞ , (9.1.11) lim R → + ∞ (cid:90) | y |≤ ρ | f ( ± R + iy ) | dy = 0 . (9.1.12) Then we have (9.1.13) ∀ ξ ∈ R , | ˆ f ( ξ ) | ≤ Ce − πρ | ξ | , with C = max( C + , C − ) , C ± = (cid:82) R | f ( x ± iρ ) | dx. Conversely, if f is a bounded mea-surable function such that ˆ f ( ξ ) is O ( e − πr | ξ | ) for some r > , then f is holomorphicon { z ∈ C , | Im z | < r } .Proof. If f is holomorphic near { z ∈ C , | Im z | ≤ ρ } , satisfies (9.1.11) and (9.1.12),then Cauchy’s formula shows that for | y | ≤ ρ , (cid:90) R e − iπ ( x + iy ) ξ f ( x + iy ) dx = e πyξ lim R → + ∞ (cid:90) R − R e − iπxξ f ( x + iy ) dx = lim R → + ∞ (cid:90) [ − R + iy,R + iy ] e − iπzξ f ( z ) dz = lim R → + ∞ (cid:90) [ − R + iy, − R ] ∪ [ − R,R ] ∪ [ R,R + iy ] e − iπzξ f ( z ) dz = ˆ f ( ξ ) + lim R → + ∞ (cid:18)(cid:90) y e − iπ ( R + it ) ξ f ( R + it ) idt − (cid:90) y e − iπ ( − R + it ) ξ f ( − R + it ) idt (cid:19) . We have for | y | ≤ ρ , (cid:12)(cid:12)(cid:12)(cid:90) y e − iπ ( ± R + it ) ξ f ( ± R + it ) idt (cid:12)(cid:12)(cid:12) ≤ (cid:90) | t |≤ ρ | f ( ± R + it ) | dt e πρ | ξ | , which goes to 0 when R goes to + ∞ , thanks to (9.1.12), so that for all y ∈ [ − ρ, ρ ] ,we have (cid:90) R e − iπ ( x + iy ) ξ f ( x + iy ) dx = ˆ f ( ξ ) , which implies for y = − ρ sign ξ (taken as 0, if ξ = 0 ) | ˆ f ( ξ ) | ≤ (cid:90) R | f ( x ∓ iρ ) | dx e − πρ | ξ | ≤ (cid:124)(cid:123)(cid:122)(cid:125) from (9.1.11) Ce − πρ | ξ | , proving the first part of the proposition. Let us consider now a function f in L ∞ ( R ) such that ˆ f ( ξ ) is O ( e − πr | ξ | ) for some r > , and let ρ ∈ (0 , r ) . We have f ( x ) = (cid:82) e iπxξ ˆ f ( ξ ) dξ and for | y | ≤ ρ , we have (cid:82) R e π | y || ξ | | ˆ f ( ξ ) | dξ < + ∞ , so that f is holomorphic on { z ∈ C , | Im z | < r } with f ( x + iy ) = (cid:90) R e iπ ( x + iy ) ξ ˆ f ( ξ ) dξ, concluding the proof. (cid:3) 14 NICOLAS LERNER Weyl quantization. Let a ∈ S (cid:48) ( R n ) . We define the operator Op w ( a ) , con-tinuous from S ( R n ) into S (cid:48) ( R n ) , given by the formula(9.1.14) ( Op w ( a ) u )( x ) = (cid:120) e iπ ( x − y ) · ξ a ( x + y , ξ ) u ( y ) dydξ, to be understood weakly as(9.1.15) (cid:104) Op w ( a ) u, ¯ v (cid:105) S (cid:48) ( R n ) , S ( R n ) = (cid:104) a, W ( u, v ) (cid:105) S (cid:48) ( R n ) , S ( R n ) , where the so-called Wigner function W ( u, v ) is defined for u, v ∈ S ( R n ) by(9.1.16) W ( u, v )( x, ξ ) = (cid:90) e − iπz · ξ u ( x + z v ( x − z dz. We note that the sesquilinear mapping S ( R n ) × S ( R n ) (cid:51) ( u, v ) (cid:55)→ W ( u, v ) ∈ S ( R n ) is continuous so that the above bracket of duality (cid:104) a, W ( u, v ) (cid:105) S (cid:48) ( R n ) , S ( R n ) , makes sense. We note as well that a temperate distribution a ∈ S (cid:48) ( R n ) getsquantized by a continuous operator a w from S ( R n ) into S (cid:48) ( R n ) . Moreover, for a ∈ S (cid:48) ( R n ) and b a polynomial in C [ x, ξ ] , we have the composition formula, a w b w = ( a(cid:93)b ) w , (9.1.17) ( a(cid:93)b )( x, ξ ) = (cid:88) k ≥ iπ ) k (cid:88) | α | + | β | = k ( − | β | α ! β ! ( ∂ αξ ∂ βx a )( x, ξ )( ∂ αx ∂ βξ b )( x, ξ ) , (9.1.18)which involves here a finite sum. This follows from (2.1.26) in [21] where severalgeneralizations can be found.Also, we find that W ( u, u ) is real-valued since W ( u, u )( x, ξ ) = (cid:90) e iπz · ξ ¯ u ( x + z u ( x − z dz = (cid:90) e − iπz · ξ ¯ u ( x − z u ( x + z dz = W ( u, u )( x, ξ ) . A particular case of Segal’s formula (see e.g. Theorem 2.1.2 in [21]) is with F standing for the Fourier transformation,(9.1.19) F ∗ a w F = a ( ξ, − x ) w . Lemma 9.3. Let a be a tempered distribution on R n and let b be a polynomial ofdegree d on R n . Then we have a(cid:93)b = (cid:88) ≤ k ≤ d ω k ( a, b ) , with (9.1.20) ω k ( a, b ) = 1(4 iπ ) k (cid:88) | α | + | β | = k ( − | β | α ! β ! ( ∂ αξ ∂ βx a )( x, ξ )( ∂ αx ∂ βξ b )( x, ξ ) , (9.1.21) ω k ( b, a ) = ( − k ω k ( a, b ) . (9.1.22) NTEGRALS OF THE WIGNER DISTRIBUTION 115 The Weyl symbol of the commutator [ Op w ( a ) , Op w ( b )] is (9.1.23) c ( a, b ) = 2 (cid:88) ≤ k ≤ dk odd ω k ( a, b ) . If the degree of b is smaller than 2, we have (9.1.24) c ( a, b ) = 2 ω ( a, b ) = 12 πi { a, b } , and if a is a function of b , the commutator [ Op w ( a ) , Op w ( b )] = 0 . Remark 9.4. In particular if q ( x, ξ ) is a quadratic polynomial and a ( x, ξ ) = H (cid:0) − q ( x, ξ ) (cid:1) , is the characteristic function of the set { ( x, ξ ) , q ( x, ξ ) ≤ } , then we have (9.1.25) (cid:2) Op w ( a ) , Op w ( q ) (cid:3) = 0 . Proof. Applying (9.1.17), (9.1.18), we obtain that this lemma follows from (9.1.22),that we check now: (4 iπ ) k ω k ( a, b ) = (cid:88) | α | + | β | = k ( − | β | α ! β ! ( ∂ αξ ∂ βx a )( x, ξ )( ∂ αx ∂ βξ b )( x, ξ )= (cid:88) | α | + | β | = k ( − | α | α ! β ! ( ∂ βξ ∂ αx a )( x, ξ )( ∂ βx ∂ αξ b )( x, ξ )= (cid:88) | α | + | β | = k ( − k −| β | α ! β ! ( ∂ βξ ∂ αx a )( x, ξ )( ∂ βx ∂ αξ b )( x, ξ ) = ( − k (4 iπ ) k ω k ( b, a ) , which is the sought result. (cid:3) Remark 9.5. We can note that Formula (1.2.54) is non-local in the sense that for a, b ∈ S ( R n ) with disjoint supports, although all ω k ( a, b ) (given by (9.1.21)) areidentically 0, the function a(cid:93)b (which belongs to S ( R n ) ) is different from 0; let usgive an example. Let χ ∈ C ∞ c ( R ; [0 , with support [ − (cid:15) , − (cid:15) ] with (cid:15) ∈ (0 , and let us consider in R , a ( x, ξ ) = χ ( x ) e − πξ , b ( x, ξ ) = χ ( x − e − πξ , so that a, b both belong to S ( R ) and supp a = [ − (cid:15) , − (cid:15) ] × R , supp b = [1 + (cid:15) , − (cid:15) ] × R , so that the supports are disjoint and all ω k ( a, b ) are identically vanishing. We checknow ( a(cid:93)b )( x, ξ ) = 4 (cid:120)(cid:120) χ ( y ) e − πη χ ( z − e − πζ e − iπ ( ξ − η )( x − z ) e iπ ( x − y )( ξ − ζ ) dydηdzdζ = 4 (cid:120) χ ( y ) χ ( z − e − π ( x − z ) e − π ( x − y ) e iπξ ( z − x + x − y ) dydz = 4 (cid:16)(cid:90) χ ( y ) e − iπξy e − π ( x − y ) dy (cid:17)(cid:16)(cid:90) χ ( z ) e iπξz e − π ( x − − z ) dz (cid:17) , 16 NICOLAS LERNER so that ( a(cid:93)b )(0 , 0) = 4 (cid:16)(cid:90) χ ( y ) e − πy dy (cid:17) > (cid:16)(cid:90) χ ( z ) e − π (2+ z ) dz (cid:17) > > . Some explicit computations. We may also calculate with(9.1.26) u a ( x ) = (2 a ) / e − πax , a > , (9.1.27) W ( u a , u a )( x, ξ ) = (2 a ) / (cid:90) e − iπz · ξ e − πa | x − z | e − πa | x + z | dz = (2 a ) / (cid:90) e − iπz · ξ e − πax e − πaz / dz = (2 a ) / e − πax / a − / e − π a ξ = 2 e − π ( ax + a − ξ ) , which is also a Gaussian function on the phase space (and positive function). Thecalculation of W ( u (cid:48) a , u (cid:48) a )( x, ξ ) is interesting since we have π (cid:104) D x b w D x u a , ¯ u a (cid:105) S (cid:48) ( R n ) , S ( R n ) = (cid:104) b w u (cid:48) a , ¯ u (cid:48) a (cid:105) S (cid:48) ( R n ) , S ( R n ) = (cid:104) b, W ( u (cid:48) a , u (cid:48) a ) (cid:105) S (cid:48) ( R n ) , S ( R n ) , and for b ( x, ξ ) real-valued we have ξ(cid:93)b(cid:93)ξ = (cid:0) ξb + b (cid:48) x iπ (cid:1) (cid:93)ξ = ξ b + b (cid:48) x ξ iπ − ∂ x iπ (cid:0) ξb + b (cid:48) x iπ (cid:1) = ξ b + b (cid:48)(cid:48) xx π , so that π (cid:120) e − π ( ax + a − ξ ) (cid:0) ξ b + b (cid:48)(cid:48) xx π (cid:1) dxdξ = (cid:104) b, W ( u (cid:48) a , u (cid:48) a ) (cid:105) , proving that W ( u (cid:48) a , u (cid:48) a )( x, ξ ) = 2 e − π ( ax + a − ξ ) π ξ + 14 2 ∂ x (cid:0) e − π ( ax + a − ξ ) (cid:1) = 2 e − π ( ax + a − ξ ) (cid:0) π ξ + 14 (( − πax ) − πa ) (cid:1) = 8 π e − π ( ax + a − ξ ) a (cid:0) a − ξ + ax − π (cid:1) . We obtain that the function W ( u (cid:48) a , u (cid:48) a ) is negative on a − ξ + ax < π , which has area / . We may note as well for consistency that for u a given by (9.1.26),we have u (cid:48) a = (2 a ) / ( − πax ) e − πax , (cid:107) u (cid:48) a (cid:107) L = πa, and (cid:120) W ( u (cid:48) a , u (cid:48) a )( x, ξ ) dxdξ = 8 π a (cid:120) e − π ( y + η ) ( y + η − π ) dydη = 8 π a π = πa = (cid:107) u (cid:48) a (cid:107) L . For λ > and a ∈ S (cid:48) ( R n ) , we define(9.1.28) a λ ( x, ξ ) = a ( λ − x, λξ ) , NTEGRALS OF THE WIGNER DISTRIBUTION 117 and we find that ( a λ ) w = U ∗ λ a w U λ , (9.1.29) for f ∈ S ( R n ) , ( U λ f )( x ) = f ( λx ) λ n/ , U ∗ λ = U λ − = ( U λ ) − . (9.1.30)We note that the above formula is a particular case of Segal’s Formula (see e.g.Theorem 2.1.2 in [21]).9.1.4. The Harmonic Oscillator. The Harmonic oscillator H n in n dimensions isdefined as the operator with Weyl symbol π ( | x | + | ξ | ) and thus from (9.1.29), wefind that H = U √ π (cid:0) | x | + 4 π | ξ | (cid:1) w U ∗√ π = U √ π (cid:0) − ∆ + | x | (cid:1) U ∗√ π . We shall define in one dimension the Hermite function of level k ∈ N , by(9.1.31) ψ k ( x ) = ( − k k √ k ! 2 / e πx (cid:18) d √ πdx (cid:19) k ( e − πx ) , and we find that ( ψ k ) k ∈ N is a Hilbertian orthonormal basis on L ( R ) . The one-dimensional harmonic oscillator can be written as(9.1.32) H = (cid:88) k ≥ ( 12 + k ) P k , where P k is the orthogonal projection onto ψ k .In n dimensions, we consider a multi-index ( α , . . . , α n ) = α ∈ N n and we defineon R n , using the one-dimensional (9.1.31),(9.1.33) Ψ α ( x ) = (cid:89) ≤ j ≤ n ψ α j ( x j ) , E k = Vect (cid:8) Ψ α (cid:9) α ∈ N n , | α | = k , | α | = (cid:88) ≤ j ≤ n α j . We note that(9.1.34) the dimension of E k,n is (cid:18) k + n − n − (cid:19) and that (9.1.32) holds with P k ; n standing for the orthogonal projection onto E k,n ; thelowest eigenvalue of H n is n/ and the corresponding eigenspace is one-dimensional inall dimensions, although in two and more dimensions, the eigenspaces correspondingto the eigenvalue n + k, k ≥ are multi-dimensional with dimension (cid:0) k + n − n − (cid:1) . The n -dimensional harmonic oscillator can be written as(9.1.35) H n = (cid:88) k ≥ ( n k ) P k ; n , where P k ; n stands for the orthogonal projection onto E k,n defined above. We have inparticular(9.1.36) P k ; n = (cid:88) α ∈ N n , | α | = k P α , where P α is the orthogonal projection onto Ψ α . 18 NICOLAS LERNER On the spectrum of the anisotropic harmonic oscillator. The standard n -dimensional harmonic oscillator is the operator H n = π (cid:88) ≤ j ≤ n ( D j + x j ) , D j = 12 πi ∂ x j , and its spectral decomposition is H = (cid:88) k ≥ ( n k ) P k ; n , P k ; n = (cid:88) α ∈ N n ,α + ··· + α n = k P α ⊗ · · · ⊗ P α n , where P α j stands for the orthogonal projection onto the one-dimensional Hermitefunction with level α j . Now let us consider for µ = ( µ , . . . , µ n ) with µ j > , theoperator(9.1.37) H ( µ ) = π (cid:88) ≤ j ≤ n µ j ( D j + x j ) = π Op w ( q µ ( x, ξ )) , with(9.1.38) q µ ( x, ξ ) = (cid:88) ≤ j ≤ n µ j ( x j + ξ j ) . With the notation | µ | = (cid:80) ≤ j ≤ n µ j and µ · α = (cid:80) ≤ j ≤ n µ j α j , we have(9.1.39) H ( µ ) = (cid:88) α ∈ N n ( | µ | µ · α ) ( P α ⊗ · · · ⊗ P α n ) (cid:124) (cid:123)(cid:122) (cid:125) P α , so that the eigenspaces are the same as for H n but the arithmetic properties of µ make possible that all eigenvalues ( | µ | + µ · α ) are simple. For instance for n = 2 , < µ < µ , µ µ / ∈ Q , if β ∈ Z is such that µ β + µ β = 0 , this implies that β = 0 and thus that all theeigenvalues of H ( µ ) are simple. Remark 9.6. If < µ ≤ · · · ≤ µ n and if for all j ∈ [2 , n ] we have µ j /µ ∈ N , wethen have for α ∈ N n , α · µ = µ (cid:16) α + (cid:88) ≤ j ≤ n α j µ j µ (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) β = β · µ, β = ( β , , . . . , ∈ N n . Sinus cardinal . It is a classical result of Distribution Theory that the weak limitwhen λ → + ∞ of the Sinus Cardinal sin( λx ) x is πδ , where δ is the Dirac mass at 0,but we wish to extend that result to more general test functions. Lemma 9.7. Let f be a function in L loc ( R ) such that (cid:90) | τ |≥ | f ( τ ) || τ | dτ < + ∞ and ∃ a ∈ C so that (cid:90) | τ |≤ | f ( τ ) − a || τ | dτ < + ∞ . Then we have (9.1.40) lim λ → + ∞ (cid:90) R sin( λτ ) πτ f ( τ ) dτ = a. NTEGRALS OF THE WIGNER DISTRIBUTION 119 N.B. In particular if f is an Hölderian function such that f ( τ ) /τ ∈ L ( {| τ | ≥ } ) we get that the left-hand-side of (9.1.40) equals f (0) .Proof. Let χ be a function in C ∞ c ( R ) equal to 1 near the origin and let us define χ = 1 − χ . We have (cid:90) R sin( λτ ) πτ f ( τ ) dτ = (cid:90) R sin( λτ ) π (cid:0) f ( τ ) − a (cid:1) τ χ ( τ ) (cid:124) (cid:123)(cid:122) (cid:125) ∈ L ( R ) dτ + a (cid:90) R sin( λτ ) πτ χ ( τ ) dτ + (cid:90) R sin( λτ ) π f ( τ ) τ − χ ( τ ) (cid:124) (cid:123)(cid:122) (cid:125) ∈ L ( R ) dτ, so that the limit when λ → + ∞ of the first and the third integral is zero, thanks tothe Riemann-Lebesgue Lemma. We note also that sin( λτ ) πτ = (cid:92) [ − λ π , λ π ] ( τ ) , and applying Plancherel’s Formula to the second integral yields (cid:90) R sin( λτ ) πτ χ ( τ ) dτ = (cid:90) | t |≤ λ/ (2 π ) (cid:99) χ ( t ) dt, whose limit when λ → + ∞ is (cid:82) R (cid:99) χ ( t ) dt = χ (0) = 1 , thanks to the LebesgueDominated Convergence Theorem, completing the proof of the lemma. (cid:3) Mehler’s formula. We provide here a couple of statements related to the so-called Mehler’s formula, appearing as particular cases of L. Hörmander’s study in[15] (see also the recent K. Pravda-Starov’ article [30]). In the general framework,we consider a complex-valued quadratic form Q on the phase space R n such that Re Q ≤ : we want to quantize the Gaussian function (here X stands for ( x, ξ ) ) a ( X ) = e (cid:104) QX,X (cid:105) , and to relate the operator with Weyl symbol a to the operator exp { Op w ( (cid:104) QX, X (cid:105) ) } . Lemma 9.8. For Re t ≥ , t / ∈ iπ (2 Z + 1) , we have in n dimensions, (9.2.1) (cid:0) cosh( t/ (cid:1) n exp − tπ Op w ( | x | + | ξ | ) = Op w (cid:0) e − t ) π ( x + ξ ) (cid:1) . In particular, for t = − is, s ∈ R , s / ∈ π (1 + 2 Z ) , we have in n dimensions(9.2.2) (cos s ) n exp (cid:0) iπs Op w ( | x | + | ξ | ) (cid:1) = Op w (cid:0) e iπ tan s ( | x | + | ξ | ) (cid:1) . Lemma 9.9. For any z ∈ C , Re z ≥ , we have in n dimensions (9.2.3) Op w (cid:16) exp − (cid:0) zπ (cid:0) | ξ | + | x | (cid:1)(cid:1)(cid:17) = 1(1 + z ) n (cid:88) k ≥ (cid:16) − z z (cid:17) k P k ; n , where P k ; n is defined in Section 9.1.4 and the equality holds between L ( R n ) -boundedoperators. 20 NICOLAS LERNER We provide first a proof of a particular case of the results of [15]. Lemma 9.10. For Re t ≥ , t / ∈ iπ (2 Z + 1) , we have in n dimensions, (9.2.4) (cid:0) cosh( t/ (cid:1) n exp − tπ Op w ( | x | + | ξ | ) = Op w (cid:0) e − t ) π ( x + ξ ) (cid:1) . Proof. By tensorisation, it is enough to prove that formula for n = 1 , which weassume from now on. We define L = ξ + ix, ¯ L = ξ − ix, M ( t ) = β ( t ) Op w ( e − α ( t ) πL ¯ L ) , where α, β are smooth functions of t to be chosen below. Assuming β (0) = 1 , α (0) =0 , we find that M (0) = Id and ˙ M + π Op w ( | L | ) M = Op w (cid:16) ˙ βe − απ | L | − β ˙ απ | L | e − απ | L | + π ( | L | ) (cid:93)βe − απ | L | (cid:17) . We have from (9.1.18), since ∂ x ∂ ξ | L | = 0 , | L | (cid:93)e − απ | L | = | L | e − απ | L | + 14 iπ =0 (cid:122) (cid:125)(cid:124) (cid:123)(cid:110) | L | , e − απ | L | (cid:111) + 1(4 iπ ) (cid:16) ∂ ξ ( | L | ) ∂ x e − απ | L | + ∂ x ( | L | ) ∂ ξ e − απ | L | (cid:17) = | L | e − απ | L | + 1(4 iπ ) e − απ | L | (cid:16) (cid:0) ( − απx ) − απ (cid:1) + 2 (cid:0) ( − απξ ) − απ (cid:1)(cid:17) = | L | e − απ | L | (cid:16) − α π π (cid:17) + απ π e − απ | L | , so that ˙ M + π Op w ( | L | ) M = Op w (cid:16) ˙ βe − απ | L | − β ˙ απ | L | e − απ | L | + πβ | L | e − απ | L | (cid:16) − α π π (cid:17) + απβ π e − απ | L | (cid:17) = Op w (cid:16) e − απ | L | (cid:110) | L | (cid:0) − π ˙ αβ + πβ (1 − α (cid:1) + ˙ β + αβ (cid:111)(cid:17) . We solve now ˙ α = 1 − α , α (0) = 0 ⇐⇒ α ( t ) = 2 tanh( t/ , and β + αβ = 0 , β (0) = 1 ⇐⇒ β ( t ) = 1cosh( t/ . We obtain that ˙ M + π Op w ( | L | ) M = 0 , M (0) = Id , and this implies β ( t ) Op w ( e − α ( t ) πL ¯ L ) = M ( t ) = exp − tπ ( | L | ) w , which proves (9.2.4). (cid:3) In particular, for t = − is, s ∈ R , s / ∈ π (1 + 2 Z ) , we have in n dimensions(9.2.5) (cos s ) n exp (cid:0) iπs Op w ( | x | + | ξ | ) (cid:1) = Op w (cid:0) e iπ tan s ( | x | + | ξ | ) (cid:1) . NTEGRALS OF THE WIGNER DISTRIBUTION 121 Lemma 9.11. For any z ∈ C , Re z ≥ , we have in n dimensions (9.2.6) Op w (cid:0) exp − (cid:0) zπ ( | ξ | + | x | ) (cid:1)(cid:1) = 1(1 + z ) n (cid:88) k ≥ (cid:16) − z z (cid:17) k P k ; n , where P k ; n is defined in Section 9.1.4 and the equality holds between L ( R n ) -boundedoperators.Proof. Starting from (9.2.5), we get for τ ∈ R , in n dimensions, (cos(arctan τ )) n exp (cid:0) iπ arctan τ Op w ( | x | + | ξ | ) (cid:1) = Op w (cid:0) e iπτ ( | x | + | ξ | ) (cid:1) , so that using the spectral decomposition of the ( n -dimensional) Harmonic Oscillatorand (9.7.2), we get (1 + τ ) − n/ (cid:88) k ≥ e i (arctan τ )( k + n ) P k ; n = Op w (cid:0) e iπτ ( | x | + | ξ | ) (cid:1) , which implies (1 + τ ) − n/ (cid:88) k ≥ (1 + iτ ) k + n (1 + τ ) k + n P k ; n = Op w (cid:0) e iπτ ( | x | + | ξ | ) (cid:1) , entailing (cid:88) k ≥ (1 + iτ ) k (1 − iτ ) k + n P k ; n = Op w (cid:0) e iπτ ( | x | + | ξ | ) (cid:1) , proving the lemma by analytic continuation (we may refer the reader as well to [34](pp. 204-205) and note that for any z ∈ C , Re z ≥ , we have | − z z | ≤ ). (cid:3) Laguerre polynomials. Classical Laguerre polynomials. The Laguerre polynomials { L k } k ∈ N are de-fined by(9.3.1) L k ( x ) = (cid:88) ≤ l ≤ k ( − l l ! (cid:18) kl (cid:19) x l = e x k ! (cid:18) ddx (cid:19) k (cid:8) x k e − x (cid:9) = (cid:18) ddx − (cid:19) k (cid:8) x k k ! (cid:9) , and we have L = 1 ,L = − X + 1 ,L = 12 ( X − X + 2) ,L = 16 ( − X + 9 X − X + 6) ,L = 124 ( X − X + 72 X − X + 24) ,L = 1120 ( − X + 25 X − X + 600 X − X + 120) ,L = 1720 (cid:0) X − X + 450 X − X + 5400 X − X + 720 (cid:1) ,L = − X + 49 X − X + 7350 X − X + 52920 X − X + 50405040 . 22 NICOLAS LERNER We get also easily from the above definition that(9.3.2) L (cid:48) k +1 = L (cid:48) k − L k , since with T = d/dX − L (cid:48) k − L k = T L k = T k +1 ( X k k ! ) = T k +1 ( ddX X k +1 ( k + 1)! ) = ddX L k +1 . Formula (6.8) and Theorem 12 in the R. Askey & G. Gasper’s article [2] provide theinequalities(9.3.3) ∀ k ∈ N , ∀ x ≥ , (cid:88) ≤ l ≤ k ( − l L l ( x ) ≥ . This result follows as well from Formula (73) in the 1940 paper [8] by E. Feldheim.Let us calculate the Fourier transform of the Laguerre polynomials: we have L k ( x ) = (cid:18) ddx − (cid:19) k (cid:8) x k k ! (cid:9) , so that (cid:99) L k ( ξ ) = (2 iπξ − k (cid:18) − iπ (cid:19) k δ ( k )0 k ! = ( − k k ! ( ξ − iπ ) k δ ( k )0 ( ξ ) . As a result, defining for k ∈ N , t ∈ R ,(9.3.4) M k ( t ) = ( − k H ( t ) e − t L k (2 t ) , H = R + , we find, using the homogeneity of degree − k − of δ ( k )0 , (cid:99) M k ( τ ) = 12 ( − k k ! (cid:0) τ − iπ (cid:1) k δ ( k )0 ( τ ∗ ( − k iπτ = ( − k ( ddσ ) k (cid:26) ( σ − iπ ) k /k !1 + 2 iπ ( τ − σ ) (cid:27) | σ =0 (cid:99) M k ( τ ) = (cid:88) l ( − k (cid:18) kl (cid:19) ( σ − iπ ) k − l ( k − l )! ( k − l )!(2 iπ ) k − l (cid:0) iπ ( τ − σ ) (cid:1) k − l | σ =0 = (cid:88) l ( − k (cid:18) kl (cid:19) ( − k − l (cid:0) iπτ (cid:1) k − l = ( − k (1 + 2 iπτ ) (cid:88) l (cid:18) kl (cid:19) ( − k − l (cid:0) iπτ (cid:1) k − l = ( − k (1 + 2 iπτ ) (cid:32) − (cid:0) iπτ (cid:1) (cid:33) k = ( − k (1 + 2 iπτ ) (cid:18) − iπτ iπτ (cid:19) k = 1(1 + 2 iπτ ) (cid:18) − iπτ iπτ (cid:19) k NTEGRALS OF THE WIGNER DISTRIBUTION 123 so that(9.3.5) (cid:99) M k ( τ ) = (1 − iπτ ) k (1 + 2 iπτ ) k +1 = (1 − iπτ ) k +1 (1 + 4 π τ ) k +1 . Generalized Laguerre polynomials. Let α be a complex number and let k bea non-negative integer such that α + k / ∈ ( − N ∗ ) . We define the generalized Laguerrepolynomial L αk by(9.3.6) L αk ( x ) = x − α e x (cid:18) ddx (cid:19) k (cid:8) e − x x k + α k ! (cid:9) = x − α (cid:18) ddx − (cid:19) k (cid:8) x k + α k ! (cid:9) . We note that L αk is indeed a polynomial with degree k with the formula L αk ( x ) = (cid:88) k + k = k k ! (cid:18) kk (cid:19) ( − k Γ( k + α + 1) x k − k Γ( k + α + 1 − k )= (cid:88) ≤ k ≤ k ( − k k !( k − k )! Γ( k + α + 1) x k − k Γ( k + α + 1 − k )= (cid:88) ≤ l ≤ k (cid:18) k + αk − l (cid:19) ( − l x l l ! . (9.3.7) N.B. We recall that the function / Γ is an entire function with simple zeroes at − N .As a result to make sense for the binomial coefficient (cid:18) k + αk − l (cid:19) = Γ( k + α + 1)( k − l )!Γ( l + α + 1) , we need to make sure that k + α + 1 / ∈ − N , i.e. α / ∈ − N ∗ − k . Lemma 9.12. Let α ∈ C \ ( − N ∗ ) and let k be a non-negative integer. For α = 0 ,we have L αk = L k , where L k is the classical Laguerre polynomial defined in (9.3.1) .Moreover we have for l ≤ k , (9.3.8) (cid:18) ddX (cid:19) l L αk = ( − l L α + lk − l . Proof. Indeed, we have from (9.3.7) (cid:18) ddX (cid:19) l L αk = ( − l (cid:88) l ≤ m ≤ k (cid:18) k + αk − m (cid:19) ( − m − l X m − l ( m − l )!= ( − l (cid:88) ≤ r ≤ k − l (cid:18) k − l + α + lk − r − l (cid:19) ( − r X r r ! = ( − l L α + lk − l , proving the sought formula. (cid:3) Singular integrals.Proposition 9.13. [ ] The (Hardy) operator with distribution kernel H ( x ) H ( y ) π ( x + y ) 24 NICOLAS LERNER is self-adjoint bounded on L ( R ) with spectrum [0 , and thus norm 1. [ ] The (modified Hardy) operators with respective distribution kernels H ( x − y ) H ( x ) H ( y ) π ( x + y ) , H ( y − x ) H ( x ) H ( y ) π ( x + y ) , are bounded on L ( R ) with norm / .Proof. Let us prove [ ] : for φ ∈ L ( R + ) , we define for t ∈ R , ˜ φ ( t ) = φ ( e t ) e t/ , andwe have to check the kernel e t/ e s/ π ( e t + e s ) = 1 π ( e ( t − s ) / + e − ( t − s ) / ) = 12 π sech (cid:0) t − s (cid:1) , which is a convolution kernel. Using now the classical formula(9.4.1) (cid:90) e − iπxξ sech xdx = π sech( π ξ ) , we get that π (cid:82) sech( t ) e − iπtτ dt = sech( π τ ) , a smooth function whose range is (0 , , proving the first part of the proposition. To obtain [ ] , we observe with thenotations φ ( t ) = u ( e t ) e t/ , ψ ( s ) = v ( e s ) e s/ that we have to check (cid:120) H ( s − t ) e t/ e s/ π ( e t + e s ) φ ( t ) ¯ ψ ( s ) dtds = (cid:120) H ( s − t ) π ( e ( t − s ) / + e − ( t − s ) / ) φ ( t ) ¯ ψ ( s ) dtds = (cid:104) R ∗ φ, ψ (cid:105) L ( R ) , with(9.4.2) R ( t ) = H ( t )2 π cosh( t/ , ˆ R ( τ ) = 12 π (cid:90) + ∞ sech( t/ e − iπtτ dt, so that (9.4.3) | ˆ R ( τ ) | ≤ ˆ R (0) = 12 π (cid:90) + ∞ sech( t/ dt = 12 , yielding the sought result. (cid:3) On some auxiliary functions. A preliminary quadrature. Lemma 9.14. We have (9.5.1) (cid:90) π/ (csc s − csch s ) ds = (cid:90) + ∞ π/ csch sds = Log(coth π , with csc s = 1 / sin s, csch s = 1 / sinh s .Proof. Note that the function [0 , π/ (cid:51) s (cid:55)→ sinh s − sin s sinh s sin s , We recall that dds arctan(sinh s ) = sech s . NTEGRALS OF THE WIGNER DISTRIBUTION 125 is continuous. Moreover, we have (cid:90) ds sin s = 12 Log (cid:0) − cos s s (cid:1) and (cid:90) ds sinh s = 12 Log (cid:0) cosh s − s + 1 (cid:1) , so that (cid:90) π/ (cid:15) (csc s − csch s ) ds = 12 (cid:20) Log (cid:0) − cos s s (cid:1)(cid:21) π/ (cid:15) − (cid:20) 12 Log (cid:0) cosh s − s + 1 (cid:1)(cid:21) π/ (cid:15) = 12 Log (cid:16) (cid:0) (cid:15) − cos (cid:15) (cid:1)(cid:0) cosh (cid:15) − (cid:15) + 1 (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) = (2+ O ( (cid:15) (cid:15) 22 + O ( (cid:15) (cid:15) 22 + O ( (cid:15) O ( (cid:15) → for (cid:15) → (cid:17) + 12 Log (cid:0) cosh π + 1cosh π − (cid:1) , so that we obtain (cid:90) π/ (csc s − csch s ) ds = 12 Log (cid:0) e π/ + e − π/ + 2 e π/ + e − π/ − (cid:1) = Log cosh( π/ π/ , which is the first result. Also we have (cid:82) + ∞ π/ csch sds = Log (cid:0) cosh( π/ π/ − (cid:1) , yieldingthe second result. (cid:3) Study of the function ρ σ . We study in this section the real-valued Schwartzfunction ρ σ given in (5.2.14). Using the notations(9.5.2) ω = 2 πτ, κ = 2 πσ, ν = (cid:112) κ/ω, we have(9.5.3) ρ σ ( τ ) = (cid:90) R s sinh s e iω ( s − ν tanh s ) ds = (cid:90) R s sinh s cos (cid:0) ω ( s − ν tanh s ) (cid:1) ds. Defining the holomorphic function F by(9.5.4) F ( z ) = z sinh z e iω ( z − ν tanh z ) , we see that F has simple poles at iπ Z ∗ and essential singularities at iπ ( + Z ) . Wealready know that the function ρ σ belongs to the Schwartz space, but we want toprove a more precise exponential decay. We start with the calculation of(9.5.5) (cid:90) R + i π F ( z ) dz = (cid:90) R t + i π sinh( t + i π ) e iω ( t + i π − ν tanh( t + i π )) dt = e − πω/ √ (cid:90) R t + i π (1 + i ) e t − (1 − i ) e − t e iωt e − iων et (1+ i ) − e − t (1 − i ) et (1+ i )+ e − t (1 − i ) dt = e − πω/ √ (cid:90) R t + i π sinh t + i cosh t e iωt e − iων et (1+ i ) − e − t (1 − i ) et (1+ i )+ e − t (1 − i ) dt. We have Im (cid:18) e t (1 + i ) − e − t (1 − i ) e t (1 + i ) + e − t (1 − i ) (cid:19) = Im (cid:18) sinh t + i cosh t cosh t + i sinh t (cid:19) = 1cosh t + sinh t , 26 NICOLAS LERNER so that(9.5.6) (cid:12)(cid:12)(cid:12)(cid:90) R + i π F ( z ) dz (cid:12)(cid:12)(cid:12) ≤ e − πω √ (cid:90) R (cid:112) t + ( π ) (cid:112) sinh t + cosh t e ων t +cosh2 t dt = e − πω √ e κ (cid:90) R (cid:112) t + ( π ) (cid:112) sinh t + cosh t dt ≤ e − πω e κ . Claim 9.15. We have lim R → + ∞ (cid:73) [ R,R + iπ/ F ( z ) dz = lim R → + ∞ (cid:73) [ − R, − R + iπ/ F ( z ) dz = 0 . Proof of the Claim. We note first that (cid:73) [ − R, − R + iπ/ F ( z ) dz = − (cid:73) [ R,R + iπ/ F ( z ) dz, so that it is enough to prove one equality. Indeed for R > , we have (cid:73) [ R,R + iπ/ F ( z ) dz = (cid:90) π/ R + it sinh( R + it ) e iω ( R + it − ν tanh( R + it )) idt, so that (cid:12)(cid:12)(cid:12)(cid:73) [ R,R + iπ/ F ( z ) dz (cid:12)(cid:12)(cid:12) ≤ (cid:90) π/ √ R + t | e R + it || − e − R − it | e − ωt e κ Im(tanh( R + it )) dt ≤ e − R (cid:112) R + π / − e − R (cid:90) π/ e κ | − e − R − it e − R − it | dt ≤ e − R (cid:112) R + π / − e − R π e κ (1 − e − R ) , proving the claim. (cid:3) Lemma 9.16. We have for τ > , σ ≥ , ρ σ given in (5.2.14) , (9.5.7) | ρ σ ( τ ) | ≤ e − π τ e πσ . Proof. We have, with the notations (9.5.2), F given in (9.5.4) and γ R = [ − R, − R + i π ] ∪ [ − R + i π , R + i π ] ∪ [ R + i π , R ] , ρ σ ( τ ) = lim R → + ∞ (cid:90) [ − R,R ] F ( s ) ds = lim R → + ∞ (cid:18)(cid:73) γ R F ( z ) dz (cid:19) = (cid:124)(cid:123)(cid:122)(cid:125) Claim (9.15) (cid:73) R + iπ F ( z ) dz, so that (9.5.6) implies the lemma. (cid:3) On the function ψ ν . Let ν ∈ (0 , be given. We study first the function φ ν defined on [0 , π/ by(9.5.8) φ ν ( s ) = s − ν tan s, so that φ (cid:48) ν ( s ) = 1 − ν (1 + tan s ) = cos s − ν cos s , NTEGRALS OF THE WIGNER DISTRIBUTION 127 so that(9.5.9) s s ν t ν π φ (cid:48) ν ( s ) − ν + 0 − − φ ν ( s ) 0 (cid:37) φ ν ( s ν ) (cid:38) (cid:38) −∞ We have(9.5.10) (cid:40) s ν = arccos ν = π − ν + O ( ν ) ,φ ν ( s ν ) = arccos ν − ν √ − ν = π − ν + O ( ν ) , for ν → . The function φ ν is concave on (0 , π/ since we have there φ (cid:48)(cid:48) ν ( s ) = − ν ( − s ) − ( − sin s ) = − ν s ) − sin s ≤ . We have defined in (5.2.58)(9.5.11) ψ ν ( ω ) = e − πω π (cid:90) π/ e ωφ ν ( s ) − s ds. Let us start with an elementary lemma. Lemma 9.17. Let λ > be given. Defining (9.5.12) J ( λ ) = e − λ (cid:90) λ e σ − σ dσ, we have J ( λ ) = λ − + O ( λ − ) , λ → + ∞ , (9.5.13) ∀ λ > , J ( λ ) ≥ λ − − λ − . (9.5.14) Proof. Indeed we have for λ > ,(9.5.15) λJ ( λ ) = λe − λ (cid:88) k ≥ (cid:90) λ σ k − k ! dσ = λe − λ (cid:88) k ≥ λ k k ! k = e − λ (cid:88) k ≥ λ k +1 ( k + 1)! k + 1 k = e − λ (cid:88) k ≥ λ k +1 ( k + 1)! + e − λ (cid:88) k ≥ λ k +1 ( k + 1)! 1 k = e − λ ( e λ − − λ ) + λ − (cid:32) e − λ (cid:88) k ≥ λ k +2 ( k + 1)! 1 k (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) R ( λ ) , with ≤ R ( λ ) ≤ e − λ (cid:88) k ≥ λ k +2 ( k + 2)! k + 2 k ≤ e − λ ( e λ − − λ − λ × O (1) , (9.5.16)so that λJ ( λ ) = e − λ ( e λ − − λ ) + λ − O (1) = 1 + λ − O (1) − (1 + λ ) e − λ = 1 + λ − O (1) , proving (9.5.13). Note also that (9.5.15), (9.5.16) imply, since R ( λ ) ≥ , λJ ( λ ) ≥ − e − λ (1 + λ ) , 28 NICOLAS LERNER so that J ( λ ) ≥ λ − − e − λ (1 + λ − ) , and thus the sought result (9.5.14). (cid:3) Remark 9.18. Considering now the function ϕ defined by(9.5.17) ϕ ( ω ) = e − πω π (cid:90) π/ e ωs − s ds, we find that, for ω ≥ , using Lemma 9.17, ϕ ( ω ) ≥ e − πω π (cid:90) π/ e ωs − s ds = e − πω π (cid:90) πω e σ − σ dσ = 12 π J ( πω ) , so that(9.5.18) ϕ ( ω ) ≥ π ω − π ω . It is our goal now to prove a minoration of the same flavour for the function (9.5.11)defined above.Assuming ν ∈ (0 , / , we have π < s ν < t ν < π ( s ν , t ν are defined in (9.5.9), ψ ν in (9.5.11)), πe πω ψ ν ( ω ) = (cid:90) t ν e ωφ ν ( s ) − s ds + (cid:90) π/ t ν e ωφ ν ( s ) − s ds (9.5.19) ≥ (cid:90) t ν e ωφ ν ( s ) − s ds (cid:124) (cid:123)(cid:122) (cid:125) on (0 , t ν ) , φ ν ( s ) ≥ − (cid:90) π/ t ν ds sin s ≥ (cid:90) s ν e ωφ ν ( s ) − s ds − (cid:90) π/ π/ ds sin s = (cid:90) s ν e ωφ ν ( s ) − s ds (cid:124) (cid:123)(cid:122) (cid:125) on (0 , s ν ) φ ν ( s ) > and φ (cid:48) ν ( s ) > − ln 32 . Claim 9.19. For s ∈ (0 , π/ , we have φ ν ( s ) ≥ φ (cid:48) ν ( s ) sin s . Moreover, for s ∈ (0 , s ν ) , we have s ≥ φ (cid:48) ν ( s ) φ ν ( s ) .Proof of the Claim. Indeed, we have φ ν ( s ) − φ (cid:48) ν ( s ) sin s = s − ν tan s − sin s + ν (1 + tan s ) sin s = ν (cid:0) sin s + sin s tan s − tan s (cid:1) + s − sin s = ν (cid:0) sin s cos s − sin s cos s (cid:1) + s − sin s = ν sin s cos s (cid:0) − cos s (cid:1) + s − sin s ≥ , for s ∈ (0 , π/ . (9.5.20)The last part of the claim follows from the first part and the fact that sin s and φ ν ( s ) are both positive on (0 , s ν ) . (cid:3) We leave for the reader to check that for λ > , e − λ (1 + λ − ) ≤ λ − , which boils dow to study q ( λ ) = e − λ ( λ + λ ) reaching its maximum for λ ∈ R + , at λ = (1 + √ / with q ( λ ) ≈ . < . NTEGRALS OF THE WIGNER DISTRIBUTION 129 Going back now to (9.5.19), we obtain that for ν ∈ (0 , / and ω > , we have(9.5.21) πe πω ψ ν ( ω ) ≥ (cid:90) s ν e ωφ ν ( s ) − φ ν ( s ) φ (cid:48) ν ( s ) ds − ln 32= (cid:90) ωφ ν ( s ν )0 e σ − σ dσ − ln 32 = e ωφ ν ( s ν ) J (2 ωφ ν ( s ν )) − ln 32 , so that, using (9.5.14), we get ψ ν ( ω ) ≥ π e − πω e ωφ ν ( s ν ) (cid:16) ωφ ν ( s ν ) − ωφ ν ( s ν )) (cid:17) − ln 32 12 π e − πω , and since φ ν ( s ν ) = π − (cid:15) ν , with (cid:15) ν ∈ (0 , π/ , we find also that (cid:15) ν is a concavefunction of ν ∈ (0 , and πν ≤ (cid:15) ν ≤ ν so that φ ν ( s ν ) = π − (cid:15) ν ∈ [ π − ν, π − πν ] , so that for ν ∈ (0 , / , we have (assuming ω > ), ψ ν ( ω ) ≥ π e − πω e ω ( π − (cid:15) ν ) (cid:16) ω ( π − (cid:15) ν ) − ω ( π − (cid:15) ν )) (cid:17) − ln 32 12 π e − πω , ≥ π e − νω (cid:16) ωπ − ω ( π − (cid:17) − ln 32 12 π e − πω , We recall the notations (9.5.2), so that ν = (cid:112) κ/ω i.e. νω = √ κω and we get(9.5.22) ∀ ω > , ψ ν ( ω ) ≥ π e − √ κω (cid:16) πω − ω (cid:17) − ln 32 12 π e − πω , ν = (cid:112) κ/ω. An explicit expresssion for a . According to (5.2.31), we have(9.5.23) a ( τ, σ ) = 12 + 12 π (cid:90) + ∞ sin(2 πtτ − πσ tanh( t/ t/ dt. We have used in Section 5.2 the equivalent expression a ( τ, σ ) = + ˆ T σ ( τ ) , where T σ is defined in (5.2.12) and we were able to prove the estimate in Lemma 5.18.It turns out that (9.5.7) is not optimal, and it is interesting to give an “explicit”expression for a as displayed in [37].Using the notations (9.5.2), we can write (9.5.23) as(9.5.24) a ( τ, σ ) = 12 + 14 π (cid:90) R Im exp i ( ωt − κ tanh( t/ t/ dt = 12 + Im lim R → + ∞ π (cid:90) [ − R,R ] exp 2 i ( ωs − κ tanh s )sinh s ds. Defining the holomorphic function G by(9.5.25) G ( z ) = exp 2 i ( ωz − κ tanh z )2 π sinh z , We have from (9.5.10) (cid:15) ν = π − arccos ν + ν √ − ν , d(cid:15) ν dν = 2 √ − ν , d (cid:15) ν dν = − ν/ √ − ν < , so that the concavity gives π ν ≤ (cid:15) ν ≤ ν. We know that ω ( π − (cid:15) ν ) ≥ ω ( π − ν ) ≥ ω ( π − so that to ensure ω ( π − (cid:15) ν ) ≥ , it sufficesto assume ω ≥ / ( π − . 30 NICOLAS LERNER we see that G has simple poles at iπ Z and essential singularities at iπ ( + Z ) . For R ∈ R + \ π Z , (cid:15) ∈ (0 , π/ , we have(9.5.26) (cid:73) [ − R, − (cid:15) ] ∪ [ (cid:15),R ] G ( z ) dz + (cid:73) γ − (cid:15) γ − (cid:15) ( θ )= (cid:15)e iθ − π ≤ t ≤ G ( z ) dz + (cid:73) γ + R γ + R ( θ )= Re iθ ≤ t ≤ π G ( z ) dz = 2 iπ (cid:88) k ∈ N kπ< R Res ( G, ikπ/ . Claim 9.20. We have lim (cid:15) → (cid:72) γ − (cid:15) G ( z ) dz = i .Proof. Indeed we have (cid:90) − π exp 2 i ( ω(cid:15)e iθ − κ tanh( (cid:15)e iθ ))2 π sinh( (cid:15)e iθ ) i(cid:15)e iθ dθ = i π (cid:90) − π e iω(cid:15)e iθ (cid:15)e iθ sinh( (cid:15)e iθ ) exp ( − iκ tanh( (cid:15)e iθ )) dθ, and since the function z (cid:55)→ ze iωz sinh z e − iκ tanh z is holomorphic near 0 with value 1 at 0,we get the result of the claim. (cid:3) Lemma 9.21. We have lim N (cid:51) m → + ∞ Im (cid:18)(cid:72) γ + π m π G ( z ) dz (cid:19) = 0 . Proof. Indeed we have with R = π + m π , Im (cid:90) π exp 2 i ( ωRe iθ − κ tanh( Re iθ ))2 π sinh( Re iθ ) iRe iθ dθ = Rπ Re (cid:90) π e iωR cos θ e − Rω sin θ e iθ − e − Re iθ e − Re iθ exp ( − iκ tanh( Re iθ )) dθ = 2 Rπ (cid:90) π/ Re (cid:26) e iωR cos θ e − Rω sin θ e iθ − e − Re iθ e − Re iθ exp ( − iκ tanh( Re iθ )) (cid:27) dθ, so that(9.5.27) Im (cid:16)(cid:73) γ + π m π G ( z ) dz (cid:17) = 2 Rπ (cid:90) π/ e − R cos θ e − Rω sin θ Re (cid:26) e iωR cos θ e iθ − e − Re iθ e − iR sin θ exp ( − iκ tanh( Re iθ )) (cid:27) dθ. We have also tanh( Re iθ ) = 1 − e − Re iθ e − Re iθ . (9.5.28) Claim 9.22. Defining for m ∈ N , θ ∈ [0 , π ] , g m ( θ ) = 1 − e − ( π + mπ ) e iθ , we find that (9.5.29) inf θ ∈ [0 ,π ] m ∈ N | g m ( θ ) | = β > , inf θ ∈ [0 ,π ] m ∈ N | − g m ( θ ) | = β > . NTEGRALS OF THE WIGNER DISTRIBUTION 131 Proof of the claim. If it were not the case, we could find sequences θ l ∈ [0 , π ] , m l ∈ N such that(9.5.30) lim l → + ∞ e − ( π + m l π ) e iθl = 1 . Taking the logarithm of the modulus of both sides, we would get lim l → + ∞ ( π m l π ) cos θ l = 0 , i.e. cos θ l = (cid:15) lπ + m l π , lim l → + ∞ (cid:15) l = 0 . Going back to (9.5.30), we find then lim l → + ∞ e − i ( π + m l π ) sin θ l = 1 , i.e. since sin θ l ≥ , lim l → + ∞ exp − i (cid:110) ( π m l π ) (cid:16) − (cid:15) l ( π + m l π ) (cid:17) / (cid:111) = 1 , implying lim l → + ∞ e − i ( π + m l π ) = 1 , which is not possible since e − i ( π + m l π ) = − i ( − m l ∈ {± i } , proving the first inequality of the claim. The second inequality follows from thesame reductio ad absurdum , starting with(9.5.31) lim l → + ∞ e − ( π + m l π ) e iθl = − , ending-up with an impossibility since − / ∈ {± i } . (cid:3) As a consequence of Claim 9.22 and (9.5.28), we obtain for R = π + m π , θ ∈ (0 , π ) ,(9.5.32) | tanh( Re iθ ) | ≤ β . Formula (9.5.27) gives then(9.5.33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Im (cid:73) γ + π m π G ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Rπ (cid:90) π/ e − R cos θ e − Rω sin θ β exp (4 κ/β ) dθ, where, for ω > , the right-hand-side goes to zero when R goes to + ∞ , completingthe proof of Lemma 9.21. (cid:3) Lemma 9.23. With G defined in (9.5.25) , we have (9.5.34) π (cid:88) k ∈ N Res ( G, ikπ/ 2) = 11 + e − πω + e − πω i (1 + e − πω ) Res (cid:18) e iωz − iκ coth z cosh z , (cid:19) . Proof. We have Res ( G, ikπ/ 2) = Res ( G k , and with k = 2 l , G k ( z ) = exp 2 i ( ω ( z + ikπ ) − κ tanh( z + ikπ ))2 π sinh( z + ikπ ) = e − lπω e iωz e − iκ tanh z π ( − l sinh z , so that(9.5.35) Res ( G l , 0) = ( − l e − lπω π , 32 NICOLAS LERNER whereas for k = 2 l + 1 , we have G l +1 ( z ) = exp 2 i ( ω ( z + ilπ + iπ ) − κ tanh( z + ilπ + iπ ))2 π sinh( z + ilπ + iπ ) = e − (2 l +1) πω e iωz e − iκ coth z π ( − l i cosh z , so that(9.5.36) Res ( G l +1 , 0) = ( − l e − (2 l +1) πω πi Res (cid:16) e iωz − iκ coth z cosh z , (cid:17) , yielding π (cid:88) k ∈ N Res ( G, ikπ/ (cid:88) l ∈ N ( − l e − lπω + (cid:88) l ∈ N ( − l e − (2 l +1) πω i Res (cid:16) e iωz − iκ coth z cosh z , (cid:17) , = 11 + e − πω + e − πω i (1 + e − πω ) Res (cid:16) e iωz − iκ coth z cosh z , (cid:17) , concluding the proof of the lemma. (cid:3) Proposition 9.24. Using the notations (9.5.2) , with a defined in (9.5.23) (seealso (9.5.24) ), we have for τ > , σ ≥ , (9.5.37) a ( τ, σ ) = 11 + e − πω + e − πω e − πω Im (cid:26) Res (cid:18) e i ( ωz − κ coth z ) cosh z , (cid:19)(cid:27) . Proof. Taking the imaginary part of both sides in (9.5.26), and letting R → + ∞ , (cid:15) → + , we get, using (9.5.34), (9.5.24), Claim 9.20, a − 12 + Im i i (cid:16) 11 + e − πω + e − πω i (1 + e − πω ) Res (cid:0) e iωz − iκ coth z cosh z , (cid:1)(cid:17) , which is (9.5.37). (cid:3) Remark 9.25. In particular, when σ = 0 , we find for τ > (9.5.38) − a ( τ, 0) = e − π τ e − π τ , and since (5.2.33) implies that π Re a ( τ, 0) = (cid:90) + ∞ sin(4 πtτ )cosh t dt = Im (cid:104) e i πτt H ( t ) , sech t (cid:105) S (cid:48) ( R t ) , S ( R t ) = Im 14 iπτ (cid:104) ddt (cid:8) e i πτt (cid:9) H ( t ) , sech t (cid:105) = Im 14 iπτ (cid:18) (cid:104) ddt (cid:8) e i πτt H ( t ) (cid:9) , sech t (cid:105) − (cid:104) δ , sech (cid:105) (cid:19) = 14 πτ − Im 14 iπτ (cid:104) e i πτt H ( t ) , sech (cid:48) ( t ) (cid:105) = 14 πτ + O ( τ − ) , τ → + ∞ , NTEGRALS OF THE WIGNER DISTRIBUTION 133 we readily find that Re a ( τ, (cid:29) − a ( τ, , τ → + ∞ , providing another proof of Theorem 5.20 in the case σ = 0 . Remark 9.26. The equation (5.2.53) gives also Im a ( τ, σ ) = e − π τ a ( τ, σ ) , where(5.2.31) gives, using the notations (9.5.2), Im a ( τ, σ ) = 14 π (cid:90) + ∞ cos( tω − κ coth( t/ t/ dt (9.5.39) = 12 π (cid:90) + ∞ cos (cid:0) tω − κ coth t ) (cid:1) cosh t dt = 14 π (cid:90) R cos (cid:0) tω − κ coth t ) (cid:1) cosh t dt. With G given by (9.5.25), we note that(9.5.40) ˜ G ( z ) = ie πω G ( z + iπ i ( ωz − κ coth z )4 π cosh z , an holomorphic function with simple poles at iπ ( + Z ) and essential singularitiesat iπ Z . Following now for ˜ G the track of G in Claim 9.20, Lemmas 9.21, 9.23 andProposition 9.24, we get(9.5.41) Im a ( τ, σ ) = lim m → + ∞ (cid:15) → + Re (cid:73) [ − R m , − (cid:15) ] ∪ [ (cid:15),R m ] ˜ G ( z ) dz, R m = π m π , and we have also(9.5.42) (cid:73) [ − R m , − (cid:15) ] ∪ [ (cid:15),R m ] ˜ G ( z ) dz − (cid:73) γ + (cid:15) γ + (cid:15) ( θ )= (cid:15)e iθ ≤ t ≤ π ˜ G ( z ) dz + (cid:73) γ + Rm γ + Rm ( θ )= R m e iθ ≤ t ≤ π ˜ G ( z ) dz = 2 iπ (cid:88) ≤ k ≤ m Res ( ˜ G, ikπ/ 2) = − πe πω (cid:88) ≤ k ≤ m Res (cid:18) G ( ζ + ikπ iπ , (cid:19) = − πe πω (cid:88) ≤ l ≤ m +1 Res (cid:18) G ( ζ + ilπ , (cid:19) . Claim 9.27. We have lim (cid:15) → (cid:72) γ + (cid:15) ˜ G ( z ) dz = 0 . Proof. Indeed, we have − iκ coth (cid:15)e iθ = − iκ e − (cid:15)eiθ − e − (cid:15)eiθ and for θ ∈ (0 , π ) , Im (cid:0) e − (cid:15)e iθ − e − (cid:15)e iθ (cid:1) = Im (1 + e − (cid:15)e iθ )(1 − e − (cid:15)e − iθ ) | − e − (cid:15)e iθ | = Im e − (cid:15)e iθ − e − (cid:15)e − iθ | − e − (cid:15)e iθ | = e − (cid:15) cos θ Im e − (cid:15)i sin θ − e (cid:15)i sin θ | − e − (cid:15)e iθ | = e − (cid:15) cos θ Im − i sin(2 (cid:15) sin θ ) | − e − (cid:15)e iθ | = − e − (cid:15) cos θ sin(2 (cid:15) sin θ ) | − e − (cid:15)e iθ | ≤ , if (cid:15) ≤ π/ , 34 NICOLAS LERNER so that | e − iκ coth (cid:15)e iθ | ≤ , implying π (cid:12)(cid:12)(cid:12)(cid:12)(cid:73) γ + (cid:15) ˜ G ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) π | e iω(cid:15)e iθ || cosh (cid:15)e iθ | (cid:15) | ie iθ | dθ = (cid:15) (cid:90) π e − ω(cid:15) sin θ | cosh (cid:15)e iθ | dθ, which goes to zero when (cid:15) → + , concluding the proof of Claim 9.27. (cid:3) Claim 9.28. We have lim N (cid:51) m → + ∞ (cid:72) γ + π m π ˜ G ( z ) dz = 0 . Proof. Indeed, we have, using Claim 9.22, | coth( R m e iθ ) | = (cid:12)(cid:12)(cid:12) e − R m e iθ − e − R m e iθ (cid:12)(cid:12)(cid:12) ≤ e − Rm cos θ β ≤ β , for θ ∈ [0 , π/ , (cid:12)(cid:12)(cid:12) e Rmeiθ − e Rmeiθ (cid:12)(cid:12)(cid:12) ≤ β , for θ ∈ [ π , π ] , so that | ˜ G ( R m e iθ ) iR m e iθ | ≤ R m e κ/β e − ωR m sin θ (cid:12)(cid:12)(cid:12) e − Rmeiθ e − Rmeiθ (cid:12)(cid:12)(cid:12) ≤ e − Rm cos θ β for θ ∈ [0 , π ] , (cid:12)(cid:12)(cid:12) e Rmeiθ e Rmeiθ (cid:12)(cid:12)(cid:12) ≤ e Rm cos θ β for θ ∈ [ π , π ] , ≤ R m β e κ/β e − ωR m sin θ − R m | cos θ | , which goes to 0 when m goes to + ∞ , proving the claim. (cid:3) Using (9.5.34), we calculate now π (cid:88) l ≥ Res (cid:18) G ( ζ + ilπ , (cid:19) = 11 + e − πω + e − πω i (1 + e − πω ) Res (cid:18) e iωz − iκ coth z cosh z , (cid:19) − π (cid:16) Res ( G, iπ/ 2) + Res ( G, (cid:17) = 11 + e − πω + e − πω i (1 + e − πω ) Res (cid:18) e iωz − iκ coth z cosh z , (cid:19) + ie − πω Res (cid:18) e iωz − iκ coth z cosh z , (cid:19) − − e − πω e − πω − i (cid:18) e − πω e − πω − e − πω (cid:19) Res (cid:18) e iωz − iκ coth z cosh z , (cid:19) = − e − πω e − πω + ie − πω (cid:18) e − πω e − πω (cid:19) Res (cid:18) e iωz − iκ coth z cosh z , (cid:19) , so that from (9.5.41), (9.5.42), Claims 9.27 & 9.28, we obtain Im a ( τ, σ )= − πe πω π (cid:18) − e − πω e − πω − e − πω (cid:18) e − πω e − πω (cid:19) Im (cid:110) Res (cid:16) e iωz − iκ coth z cosh z , (cid:17)(cid:111)(cid:19) = e πω (cid:18) e − πω e − πω + e − πω (cid:18) e − πω e − πω (cid:19) Im (cid:110) Res (cid:16) e iωz − iκ coth z cosh z , (cid:17)(cid:111)(cid:19) , NTEGRALS OF THE WIGNER DISTRIBUTION 135 so that(9.5.43) Im a ( τ, σ )= e − πω e − πω ) + e − πω e − πω ) Im (cid:26) Res (cid:18) e iωz − iκ coth z cosh z , (cid:19)(cid:27) , recovering (9.5.37) from (5.2.53). N.B. We note that Res (cid:18) e iωz − iκ coth z cosh z , (cid:19) = 12 Res (cid:18) e i ( ωz − κ coth( z/ cosh( z/ , (cid:19) , (9.5.44)so that (9.5.43) corroborates ( A in [37]; however, we were not able to understandformulas ( A , ( A and (20) in [37].9.6. Airy function. Standard results on the Airy function. We collect in this section a couple ofclassical results on the Airy function (see e.g. Definition 7.6.8 in Section 7.6 of [16]or the references [35], [33], [20]). For all the statements of this section whose proofsare not included, we refer the reader to Chapter 9 of [23]. Definition 9.29. The Airy function Ai is defined as the inverse Fourier transformof ξ (cid:55)→ e i (2 πξ ) / . Proposition 9.30. For any h > and all x ∈ C , we have (9.6.1) Ai ( x ) = 12 π (cid:90) e i ( ξ + ih ) e ix ( ξ + ih ) dξ = e − xh e h π (cid:90) e − hξ e i ( ξ − ξh ) e ixξ dξ. We note that the function R (cid:51) ξ (cid:55)→ e i ( ξ + ih ) belongs to the Schwartz space for any h > since i ξ + ih ) = − hξ + h i (cid:0) ξ − ξh (cid:1) , so that e i ( ξ + ih ) = e − hξ e i ( ξ − ξh ) e h / . Theorem 9.31. The Airy function Ai is an entire function on C , real-valued onthe real line, which is the unique solution of the initial value problem for the Airyequation (9.6.2) Ai (cid:48)(cid:48) ( x ) − x Ai ( x ) = 0 , Ai (0) = 3 − / Γ(1 / π , Ai (cid:48) (0) = − / Γ(2 / π . We have also, for any x ∈ C , (9.6.3) Ai ( x ) = 1 π (cid:90) + ∞ e − ξ / e − xξ/ cos (cid:0) xξ √ 32 + π (cid:1) dξ, and the power series expansion of the Airy function is (9.6.4) Ai ( x ) = 1 π / (cid:88) k ≥ (3 / x ) k k ! Γ (cid:0) k + 13 (cid:1) sin (cid:0) k + 1) π (cid:1) . 36 NICOLAS LERNER Lemma 9.32. For x ∈ C \ R − , we have (9.6.5) Ai ( x ) = 12 π e − x / (cid:90) R e − x / ξ e iξ / dξ. Proof. Using Proposition 9.30, we get (9.6.5) for x > (choosing h = x / ), andthen we may use an analytic continuation argument. (cid:3) Theorem 9.33. For all M ∈ N , for all x ∈ C \ R − , we have (9.6.6) Ai ( x ) = 12 π e − x / x − / (cid:40) (cid:88) ≤ l ≤ M ( − l l (2 l )! Γ (cid:0) l + 12 (cid:1) x − l/ + R M ( x ) (cid:41) , with | R M ( x ) | ≤ Γ (cid:0) M + 3 + (cid:1) M +2 (2 M + 2)! | x | − M +1)2 ) (cid:16) cos( arg x (cid:17) − M +1) − . For x < , we have Ai ( x ) = 1 | x | / √ π (cid:16) sin (cid:0) π | x | / (cid:1) + O ( | x | − / ) (cid:17) , (9.6.7) Ai (cid:48) ( x ) = − | x | / √ π (cid:16) cos (cid:0) π | x | / (cid:1) + O ( | x | − / ) (cid:17) . (9.6.8) Lemma 9.34. With j = e iπ/ we have for all x ∈ C , (9.6.9) Ai ( x ) + j Ai ( jx ) + j Ai ( j x ) = 0 . In particular for r ≥ , we have (9.6.10) Ai ( − r ) = 2 Re (cid:0) e iπ Ai ( re iπ ) (cid:1) . Lemma 9.35. The zeroes of the Airy function are simple and located on ( −∞ , .We shall use the notation (9.6.11) Ai − ( { } ) = { η k } k ≥ , η k +1 < η k < , lim k → + ∞ η k = −∞ . The largest zero of Ai is η ≈ − . and Ai ( η ) is positive for η > η . Wehave also for all k ≥ , Ai ( η k +1 ) = 0 , Ai (cid:48) ( η k +1 ) < , Ai ( η k ) = 0 , Ai (cid:48) ( η k ) > , (9.6.12) Ai ( η ) < for η ∈ ( η k +1 , η k ) , Ai ( η ) > for η ∈ ( η k +2 , η k +1 ) , (9.6.13) Ai (cid:48)(cid:48) ( η ) > for η ∈ ( η k +1 , η k ) , Ai (cid:48)(cid:48) ( η ) < for η ∈ ( η k +2 , η k +1 ) . (9.6.14) N.B. The simplicity of the zeroes of the Airy function holds true for any non-zerosolution of the Airy differential equation y (cid:48)(cid:48) = xy. The solutions of this ODE areanalytic functions and if a is a double zero, we have y ( a ) = y (cid:48) ( a ) = 0 and thus fromthe Airy equation, we get y (cid:48)(cid:48) ( a ) = 0 ; we may then prove by induction on k ≥ that y ( l ) ( a ) = 0 for ≤ l ≤ k + 1 : it is proven for k = 1 , and if true for some k ≥ , weget y ( k +2) ( x ) = (cid:0) xy ( x ) (cid:1) ( k ) = ⇒ y ( k +2) ( a ) = 0 , NTEGRALS OF THE WIGNER DISTRIBUTION 137 proving the final step in the induction; as a consequence, the function has a zero ofinfinite order, which is impossible for a non-zero analytic function. Assertion (9.6.14)follows from the Airy differential equation (9.6.2), from (9.6.13) and η k < . Remark 9.36. For M = 0 , | arg x | ≤ π/ , we have | R ( x ) | ≤ Γ (cid:0) (cid:1) (2)! | x | − (cid:16) √ (cid:17) − = | x | − √ π / √ ≤ | x | − × . , so that | R ( x ) | ≤ . | x | − / if | arg x | ≤ π/ ,(9.6.15) and for | x | ≥ , | arg x | ≤ π/ we have | R ( x ) | ≤ . . (9.6.16)We get then for λ > , using (9.6.10) Ai ( − λ ) = 1 π Re (cid:16) e iπ/ λ − / e − i λ / (cid:0) √ πe − iπ/ + R ( λe iπ/ ) (cid:1)(cid:17) = 1 √ π λ − / cos( π − λ / ) + 1 π Re (cid:110) λ − / R ( re iπ/ ) e iπ/ e − i λ / (cid:111) = 1 √ π λ − / (cid:16) sin( π λ / ) + 1 √ π Re (cid:8) R ( λe iπ/ ) e iπ/ e − i λ / (cid:9)(cid:17) , so that for λ > , Ai ( − λ ) = 1 √ π λ − / (cid:16) sin( π λ / ) + ˜ R ( λ ) (cid:17) , (9.6.17) with | ˜ R ( λ ) | ≤ λ − / × . , (9.6.18) and for λ ≥ , | ˜ R ( λ ) | ≤ . . (9.6.19) Remark 9.37. For M = 1 , | arg x | ≤ π/ , we have(9.6.20) | R ( x ) | ≤ Γ (cid:0) (cid:1) (4)! | x | − (cid:16) √ (cid:17) − − = | x | − √ π / × / × ≤ | x | − × . , and(9.6.21) for | x | ≥ , | R ( x ) | ≤ . , so that Ai ( − r ) = 1 √ π r − / (cid:16) sin( π r / ) + Γ(7 / √ π sin( 23 r / − π r − / + 1 √ π Re (cid:8) R ( re iπ/ ) e iπ/ e − i r / (cid:9)(cid:17) = 1 √ π r − / (cid:16) sin( π r / ) + Γ(7 / √ π sin( 23 r / − π r − / + 1 √ π ˜ R ( r ) (cid:17) , with for r > , | ˜ R ( r ) | ≤ r − × . , (9.6.22) for r ≥ , | ˜ R ( r ) | ≤ . . (9.6.23) 38 NICOLAS LERNER We find for λ > ,(9.6.24) G ( − λ )= (cid:90) + ∞ λ r / √ π (cid:16) sin (cid:0) π r / (cid:1) + Γ(7 / √ π r − / sin (cid:0) r / − π (cid:1) + 1 √ π ˜ R ( r ) (cid:17) dr, and we have (cid:90) + ∞ λ r / √ π r / sin (cid:0) π r / (cid:1) dr = cos (cid:0) π λ / (cid:1) λ / √ π − (cid:90) + ∞ λ r / √ π cos (cid:0) π r / (cid:1) dr, as well as − (cid:90) + ∞ λ r / √ π cos (cid:0) π r / (cid:1) dr = − (cid:90) + ∞ λ r / √ π r / cos (cid:0) π r / (cid:1) dr = 34 √ π sin (cid:0) π λ / (cid:1) λ − / − √ π (cid:90) + ∞ λ r − / sin (cid:0) π r / (cid:1) dr, so that(9.6.25) (cid:90) + ∞ λ r / √ π sin (cid:0) π r / (cid:1) dr = cos (cid:0) π λ / (cid:1) λ / √ π + 34 √ π sin (cid:0) π λ / (cid:1) λ − / − √ π (cid:90) + ∞ λ r − / sin (cid:0) π r / (cid:1) dr. We have also(9.6.26) (cid:90) + ∞ λ r / Γ(7 / π r − / sin (cid:0) r / − π (cid:1) dr = Γ(7 / π (cid:90) + ∞ λ r − / sin (cid:0) r / − π (cid:1) dr = − Γ(7 / π cos (cid:0) λ / − π (cid:1) λ − / + Γ(7 / π (cid:90) + ∞ λ cos (cid:0) r / − π (cid:1) r − / dr, so that (9.6.25), (9.6.26) and (9.6.24) entail G ( − λ ) = cos (cid:0) π λ / (cid:1) λ / √ π + 34 √ π sin (cid:0) π λ / (cid:1) λ − / − √ π (cid:90) + ∞ λ r − / sin (cid:0) π r / (cid:1) dr − Γ(7 / π cos (cid:0) λ / − π (cid:1) λ − / + Γ(7 / π (cid:90) + ∞ λ cos (cid:0) r / − π (cid:1) r − / dr + 1 π (cid:90) + ∞ λ r − / ˜ R ( r ) . NTEGRALS OF THE WIGNER DISTRIBUTION 139 We get then G ( − λ ) = λ − / √ π (cid:32) cos (cid:0) π λ / (cid:1) + 34 sin (cid:0) π λ / (cid:1) λ − / − × λ / (cid:90) + ∞ λ r − / sin (cid:0) π r / (cid:1) dr − Γ(7 / √ π cos (cid:0) λ / − π (cid:1) λ − / + Γ(7 / √ π λ / (cid:90) + ∞ λ cos (cid:0) r / − π (cid:1) r − / dr + λ / √ π (cid:90) + ∞ λ r − / ˜ R ( r ) (cid:33) , so that(9.6.27) G ( − λ ) = λ − / √ π (cid:16) cos (cid:0) π λ / (cid:1) + λ − / S ( λ ) (cid:17) , with(9.6.28) | S ( λ ) | ≤ 34 + 34 + Γ(7 / √ π + Γ(7 / √ π + 49 √ π × . ≤ . where we have used (9.6.22) for the bound of the last term above. As a consequence,if λ ≥ , we get that(9.6.29) | λ − / S ( λ ) | ≤ . . This is allowing us to extend the proof of Lemma 9.43 to all values. Note that thefirst 10 values (and more) are accessible numerically.Since we have η = − . < − , Formulas (9.6.17), (9.6.19), (9.6.27),(9.6.29) imply the following result. Lemma 9.38. With Ai and G defined above, we have for − λ ≤ η Ai ( − λ ) = 1 √ π λ − / (cid:16) sin( π λ / ) + ˜ R ( λ ) (cid:17) , (9.6.30) | ˜ R ( λ ) | ≤ λ − / × . ≤ . , (9.6.31) G ( − λ ) = λ − / √ π (cid:16) cos (cid:0) π λ / (cid:1) + ˜ S ( λ ) (cid:17) , (9.6.32) | ˜ S ( λ ) | ≤ λ − / × . ≤ . . (9.6.33)9.6.2. More on the Airy function. Proposition 9.39. We have (9.6.34) (cid:90) + ∞ Ai ( x ) dx = 13 . 40 NICOLAS LERNER Proof. According to Theorem 9.33, the Airy function Ai is rapidily decreasing on thepositive half-line and thus belongs to L ( R + ) , so that the integral in (9.6.34) makessense. Also we have from Theorem 9.33 and the Lebesgue Dominated ConvergenceTheorem that,(9.6.35) (cid:90) + ∞ Ai ( x ) dx = lim h → + (cid:90) + ∞ Ai ( x ) e xh dxe − h / , and we shall now calculate the right-hand-side of (9.6.35). We have for h > , (cid:90) + ∞ Ai ( x ) e xh dxe − h / = (cid:90) + ∞ π (cid:90) e − hξ e i ( ξ − ξh ) e ixξ dξdx = (cid:90) + ∞ (cid:99) ψ h ( − x ) dx, with(9.6.36) ψ h ( ξ ) = e − h (2 πξ ) e i ( (2 πξ )33 − (2 πξ ) h ) , so that (cid:90) + ∞ Ai ( x ) e xh dxe − h / = (cid:104) δ − πi pv ξ , ψ h (cid:105) S (cid:48) , S = 12 − πi (cid:104) pv ξ , e − h (2 πξ ) e i ( (2 πξ )33 − (2 πξ ) h ) (cid:105) = 12 − π (cid:104) pv ξ , e − hξ sin( ξ − ξh ) (cid:105) . We note at this point that, according to (4.2.9), the right-hand-side of the aboveequality is for h = 0 equal to − π π , so that, with (9.6.35), we are left to proving that(9.6.37) lim h → + (cid:104) pv ξ , e − hξ sin( ξ − ξh ) (cid:105) = π . We have (cid:90) sin( ξ − ξh ) ξ e − hξ dξ = π (cid:90) sin( ξ − ξh ) e − hξ − sin( ξ ) ξ dξ = π (cid:90) sin( ξ ) ξ (cid:0) cos( ξh ) e − hξ − (cid:1) dξ (cid:124) (cid:123)(cid:122) (cid:125) I ( h ) − (cid:90) sin( ξh ) ξ cos( ξ e − hξ dξ (cid:124) (cid:123)(cid:122) (cid:125) I ( h ) . We have I , ( h ) = (cid:90) + ∞ ξ sin( ξ ) ξ (cid:0) cos( ξh ) e − hξ − (cid:1) dξ = (cid:90) + ∞ ddξ (cos( ξ )) ξ (cid:0) cos( ξh ) e − hξ − (cid:1) dξ, NTEGRALS OF THE WIGNER DISTRIBUTION 141 and a simple integration by parts shows that lim h → I , ( h ) = 0 ; we have alsotrivially that h → (cid:90) ξ sin( ξ ) ξ (cid:0) cos( ξh ) e − hξ − (cid:1) dξ. On the other hand, we have | I ( h ) | ≤ (cid:90) h e − hξ dξ = O ( h / ) , which completes the proof of (9.6.37) as well as the proof of Proposition 9.39. (cid:3) Lemma 9.40. We have (9.6.38) lim R → + ∞ (cid:90) − R Ai ( x ) dx = 23 . Proof. Using (9.6.7), we find for R ≥ , (cid:90) − R Ai ( x ) dx = (cid:90) R Ai ( − r ) dr = (cid:90) Ai ( − r ) dr + (cid:90) R (cid:16) r / √ π sin (cid:0) π r / (cid:1) + O ( r − / ) (cid:17) dr, proving that the limit in the left-hand-side of (9.6.38) is existing. Claim 9.41. lim h → + (cid:82) −∞ Ai ( x ) e xh dx = (cid:82) −∞ Ai ( x ) dx. Proof of the Claim. We have (cid:90) −∞ Ai ( x ) e xh dx = (cid:90) − −∞ Ai ( x ) e xh dx + (cid:90) − Ai ( x ) e xh dx (cid:124) (cid:123)(cid:122) (cid:125) with limit (cid:82) − Ai ( x ) dx and using (9.6.7), we have only to check (cid:90) − −∞ | x | − / e xh + i | x | / dx = (cid:90) + ∞ t − / e − th + i t / dt = − (cid:90) + ∞ ddt (cid:110) e − th + i t / (cid:111) ( h − it / ) − t − / dt = e − h + i ( h − i ) − + (cid:90) + ∞ e − th + i t / (cid:0) ( h − it / ) − i t − / − ( h − it / ) − t − / (cid:1) dt, and since the absolute value of the integrand in the last integral is bounded aboveby t − / , we get the result of the Claim. (cid:3) The boundary term is easy to handle and for the derivative falling on ξ − , we use that | cos( ξh ) e − hξ − | ≤ ; if the derivative falls on the other term we get (cid:90) + ∞ cos( ξ ) ξ (cid:0) hξ cos( ξh ) e − hξ + e − hξ sin( ξh ) h (cid:1) dξ, which goes trivially to 0 with h . 42 NICOLAS LERNER With (9.6.35), (9.6.36), this gives (cid:90) + ∞−∞ Ai ( x ) dx = lim h → + (cid:90) + ∞−∞ Ai ( x ) e xh dxe − h / = lim h → + (cid:18)(cid:90) R (cid:99) ψ h ( − ξ ) dξ = ψ h (0) (cid:19) = 1 , and Proposition 9.39 provides the result of the lemma. (cid:3) Asymptotic expansion for the function G defined in (4.2.8) . Lemma 9.42. With G defined in (4.2.8) , we have (9.6.39) G ( − λ ) = λ − / π − / sin( 3 π λ / ) + O ( λ − / ) , λ → + ∞ . Proof. Property (9.6.38) and (9.6.7) give for η = − λ < , G ( η ) = 23 + (cid:90) η Ai ( ξ ) dξ = (cid:90) η −∞ Ai ( ξ ) dξ = (cid:90) + ∞ λ Ai ( − r ) dr = (cid:90) + ∞ λ (cid:0) e iπ Ai ( e iπ r ) (cid:1) dr (we have used (9.6.10)); we use now (9.6.6) for M = 1 , x ∈ e iπ/ R + ) = (cid:90) + ∞ λ (cid:16) r / √ π sin (cid:0) π r / (cid:1) + Γ(7 / π r − / sin (cid:0) r / − π (cid:1) + O ( r − / ) (cid:17) dr = (2 / / π − / (cid:90) + ∞ λ / s − / sin (cid:0) π s (cid:1) ds + (2 / / Γ(7 / π (cid:90) + ∞ λ / s − / sin (cid:0) s − π (cid:1) ds + O ( λ − / ) . We integrate by parts in the first integral with (cid:90) + ∞ λ / s − / sin (cid:0) π s (cid:1) ds = − (cid:90) + ∞ λ / s − / dds (cid:110) cos (cid:0) π s (cid:1)(cid:111) ds = ( 23 λ / ) − / cos( π λ / ) + (cid:90) + ∞ λ / ( − / s − / cos( π/ s ) ds. We have to deal with two integrals of type (cid:90) + ∞ λ / s − / dids e is ds = i ( λ / ) ( − / e iλ / − i (cid:90) + ∞ λ / ( − / s − / e is ds = O ( λ − / ) . Eventually we find G ( − λ ) = λ − / π − / cos( π + λ / ) + O ( λ − / ) . (cid:3) With ( η k ) k ≥ standing for the decreasing sequence of the zeroes of the Airy func-tion (cf. Lemma 9.35), we have the following table of variation for the function G . η −∞ . . . η k +2 η k +1 η k . . . η η + ∞ G (cid:48)(cid:48) ( η ) = Ai (cid:48) ( η ) + − + . . . − + G (cid:48) ( η ) = Ai ( η ) + 0 − . . . − + G ( η ) G ( η k +2 ) (cid:37) G( η k +1 ) (cid:38) G( η k ) . . . G ( η ) (cid:38) G ( η ) (cid:37) NTEGRALS OF THE WIGNER DISTRIBUTION 143 η η = − . η = − . η = − . η = − . η = − . G ( η ) − . − . − . η η = − . η = − . η = − . η = − . , η = − . G ( η ) − . − . Lemma 9.43. The zeroes of the function G on the real line are simple and make adecreasing sequence of negative numbers ( ξ l ) l ≤ such that (9.6.40) . . . η k +2 < ξ k +2 < η k +1 < ξ k +1 < η k < ξ k . . . , ξ ≈ − . . The largest ten zeroes of G are given by the following table ξ ξ ξ ξ ξ − . − . − . − . − . ξ ξ ξ ξ ξ − . − . − . − . − . For all k ∈ N , we have (9.6.41) G ( η k ) < < G ( η k +1 ) , and G ( η k ) (resp. G ( η k +1 ) ) is a local minimum (resp. maximum) of G near η k (resp. η k +1 ). Moreover, G ( η ) is an absolute minimum of the function G on thereal line. N.B. We claim also that (9.6.42) | G ( η k ) | > G ( η k +1 ) > | G ( η k +2 ) | , but shall not provide a complete proof for that statement, which is anyway not neededis our Section 4.3.Proof. In the first place, we know that G ( η ) < and G strictly increases on [ η , + ∞ ) so that ξ ≈ − . is defined as the unique zero of G on ( η , since G (0) = 2 / .We may note that we found in particular that(9.6.43) ∀ η > η , > G ( η ) > G ( η ) . 44 NICOLAS LERNER Also, the first ten zeroes of G are simple and satisfy (9.6.40), (9.6.41) and (9.6.42).Moreover, using Lemma 9.38, we obtain that for λ ≥ , G ( − λ ) = 0 = ⇒ (cid:12)(cid:12) cos (cid:0) π λ / (cid:1)(cid:12)(cid:12) ≤ . , Ai ( − λ ) = 0 = ⇒ (cid:12)(cid:12) sin (cid:0) π λ / (cid:1)(cid:12)(cid:12) ≤ . , As a result, if − λ is a double zero of G we must have both inequalities above, whichis impossible. As a result all zeroes of G are simple and located on ( −∞ , . Letus consider the interval [ η k +1 , η k ] : we have Ai ( η k +1 ) = Ai ( η k ) = 0 , Ai (cid:48) ( η k +1 ) < < Ai (cid:48) ( η k ) , Ai (cid:48)(cid:48) > on ( η k +1 , η k ) .As a result, we obtain that G has a local minimum at η k and a local maximum at η k +1 . Moreover we find from (9.6.31) in Lemma 9.38 and k ≥ that max (cid:16) | sin( π | η k | / ) | , | sin( π | η k +1 | / ) | (cid:17) ≤ . which implies that min (cid:16) | cos( π | η k | / ) | , | cos( π | η k +1 | / ) | (cid:17) ≥ . . We know that Ai (cid:48) ( η k ) > , which implies, thanks to (9.6.8) cos (cid:0) π | η k | / (cid:1) ≤ − . , cos (cid:0) π | η k +1 | / (cid:1) ≥ . , and Lemma 9.38 implies that G ( η k ) < < G ( η k +1 ) , which is (9.6.41). Since thefunction G is strictly monotone decreasing on the interval [ η k +1 , η k ] , it has a uniquesimple zero ξ k +1 on the interior of this interval. Analogously, we can prove that onthe interval [ η k +2 , η k +1 ] , it has a unique simple zero ξ k +2 on the interior of thisinterval, proving that the sequence of zeroes of the function G is decreasing strictlywith η k +2 < ξ k +2 < η k +1 < ξ k +1 < η k < ξ k , k ≥ . We shall prove a weaker statement than (9.6.42): we know that | G ( η l ) | < | G ( η ) | ) for ≤ l ≤ from the numerical values obtained above. Moreover if λ ≥ we find | G ( − λ ) | ≤ λ − / π − / (1 + 0 . ≤ . < | G ( η ) | = 0 . , It is not hard to obtain an asymptotic version of this, namely the same result for λ largeenough. However, asymptotic methods provide asymptotic results and to get a result at a finitedistance, we had to use the numerical results of Lemma 9.38, grounded on a numerical estimate ofthe constants appearing in Theorem 9.33. Here this is proven if k is large enough from (9.6.8), and we leave to the reader the proof ofa numerical estimate analogous to Lemma 9.38 for the derivative of the Airy function. A directestimate is possible, using (9.6.5) and the identity (to be differentiated) for λ > , Ai ( − λ ) = λ − / √ π (cid:110) sin (cid:0) π λ / (cid:1) + a ( λ ) λ − / (cid:111) , (9.6.44) a ( λ ) = λ / π e i ( π − λ / ) (cid:90) R e − ξ λ / e iπ/ (cid:0) cos( ξ / − (cid:1) dξ. (9.6.45) NTEGRALS OF THE WIGNER DISTRIBUTION 145 proving indeed that G ( η ) is the absolute minimum of the function G on the realline, since the desired estimate is proven for η > η and for η < η , either G ( η ) ≥ ,or − . ≤ G ( η ) < if η ≤ − . As said above, the values less than 12 aretreated directly by a numerical calculation. The proof of the lemma is complete. (cid:3) - - - - - - - - . - . . . . Figure 9. The function G and its derivative the Airy function, on R − . 46 NICOLAS LERNER Miscellaneous formulas. Some elementary formulas. We define for τ ∈ R ,(9.7.1) arctan τ = (cid:90) τ dt t , and we note that arctan τ ∈ ( − π/ , π/ , ∀ τ ∈ R , tan(arctan τ ) = τ, ∀ θ ∈ ( − π/ , π/ , arctan(tan θ ) = θ. Moreover we have for τ ∈ R ,(9.7.2) e i arctan τ = 1 √ τ (1 + iτ ) , since for θ ∈ ( − π/ , π/ , τ = tan θ , we have τ = θ and thus cos θ > ⇒ cos θ = 1 √ τ = ⇒ − sin θ = − 12 (1 + τ ) − / τ (1 + τ ) , so that e iθ = √ τ (1 + iτ ) . Let a ∈ R + be given. The Fourier transform of [ − a,a ] is(9.7.3) (cid:90) a − a e − iπxξ dx = 2 (cid:90) a cos(2 πxξ ) dx = 22 πξ [sin(2 πxξ )] x = ax =0 = sin(2 πaξ ) πξ . Taking the derivative of F k on R + . We have, using a parity argument, F k ( a ) = (cid:90) R sin aτπτ (1 + iτ ) k +1 (1 + τ ) k +1 dτ = (cid:88) ≤ l ≤ k (cid:90) R sin aτπτ (cid:0) k +12 l (cid:1) ( − l τ l (1 + τ ) k +1 dτ. We see also that k + 2 − l = 2 k + 3 − l ≥ so that we can take the derivativeof F k and get F (cid:48) k ( a ) = (cid:88) ≤ l ≤ k (cid:90) R cos aτπ (cid:0) k +12 l (cid:1) ( − l τ l (1 + τ ) k +1 dτ = 1 π (cid:90) R (cos aτ ) Re (cid:18) (1 + iτ ) k (1 − iτ ) k +1 (cid:19) dτ, with absolutely converging integrals. For a > , we have(9.7.4) F (cid:48) k ( a ) = 1 π (cid:90) R (cos aτ ) (1 + iτ ) k (1 − iτ ) k +1 dτ, since(9.7.5) lim λ → + ∞ (cid:90) λ − λ τ j cos( aτ )(1 + τ ) k +1 dτ makes sense for j ≤ k + 1 (and vanishes for j odd). NTEGRALS OF THE WIGNER DISTRIBUTION 147 A proof of the weak limit. We have for u ∈ S ( R n ) , according to (1.2.2), (cid:104) (cid:0) { π ( x + ξ ) ≤ a } (cid:1) w u, u (cid:105) = (cid:120) π ( x + ξ ) ≤ a W ( u, u )( x, ξ ) dxdξ, so that implies (cid:88) k ≥ F k ( a ) (cid:104) P k u, u (cid:105) L ( R n ) = (cid:120) π ( x + ξ ) ≤ a W ( u, u )( x, ξ ) dxdξ. Choosing now u = u k as a normalized eigenfunction of the Harmonic Oscillator witheigenvalue k + 1 / , we obtain F k ( a ) = (cid:120) π ( x + ξ ) ≤ a W ( u k , u k )( x, ξ ) dxdξ. Since the function ( x, ξ ) (cid:55)→ W ( u k , u k )( x, ξ ) belongs to the Schwartz class of R n , wefind that lim a → + ∞ F k ( a ) = (cid:120) R n H ( u k , u k )( x, ξ ) dxdξ = (cid:107) u k (cid:107) L ( R n ) = 1 , qed. A different normalization for the Wigner function. The paper [27] is using adifferent normalization for the Wigner distribution in n dimensions with(9.7.6) (cid:102) W ( u, v )( x, ξ ) = (2 π ) − n (cid:90) R n u ( x + z v ( x − z e − iz · ξ dz. The relationship with our definition (1.1.6) is(9.7.7) (cid:102) W ( u, v )( x, ξ ) = W ( u, v )( x, ξ π )(2 π ) − n . As a result, we find that E lo (cid:0) B n ( R ) (cid:1) = sup (cid:107) u (cid:107) L R n ) =1 (cid:120) | x | + | ξ | ≤ R (cid:102) W ( u, u )( x, ξ ) dxdξ, is equal to sup (cid:107) u (cid:107) L R n ) =1 (cid:120) | x | +4 π | ξ | ≤ R W ( u, u )( x, ξ ) dxdξ = sup (cid:107) u (cid:107) L R n ) =1 (cid:120) π ( | x | + | ξ | ) ≤ R W ( u, u )( x, ξ ) dxdξ, and we have proven here that for u ∈ L ( R n ) with norm 1 (cid:120) | x | + | ξ | ≤ a π = R π W ( u, u )( x, ξ ) dxdξ ≤ − n − (cid:90) + ∞ a e − t t n − dt = 1 − Γ( n, R )Γ( n ) , where the upper incomplete Gamma function Γ( z, x ) is given by(9.7.8) Γ( z, x ) = (cid:90) + ∞ x t z − e − t dt. This is indeed the result of Theorem 1 in [27]. 48 NICOLAS LERNER N.B. Let x > be given and let z ∈ C with Re z > . 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(2005), no. 4, 042103, 14. MR 2131219 N. Lerner, Institut de Mathématiques de Jussieu, Sorbonne Université (formerlyParis VI), Campus Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris cedex 05,France Email address ::