Weyl calculus with respect to the Gaussian measure and restricted L^p-L^q boundedness of the Ornstein-Uhlenbeck semigroup in complex time
aa r X i v : . [ m a t h . F A ] J u l WEYL CALCULUS WITH RESPECT TO THE GAUSSIAN MEASUREAND RESTRICTED L p - L q BOUNDEDNESS OF THEORNSTEIN-UHLENBECK SEMIGROUP IN COMPLEX TIME
JAN VAN NEERVEN AND PIERRE PORTAL
Abstract.
In this paper, we introduce a Weyl functional calculus a a ( Q, P ) for the position and mo-mentum operators Q and P associated with the Ornstein-Uhlenbeck operator L = − ∆ + x · ∇ , and givea simple criterion for restricted L p - L q boundedness of operators in this functional calculus. The analysisof this non-commutative functional calculus is simpler than the analysis of the functional calculus of L . Itallows us to recover, unify, and extend, old and new results concerning the boundedness of exp( − zL ) asan operator from L p ( R d , γ α ) to L q ( R d , γ β )f for suitable values of z ∈ C with Re z > p, q ∈ [1 , ∞ ), and α, β >
0. Here, γ τ denotes the centred Gaussian measure on R d with density (2 πτ ) − d/ exp( −| x | / τ ). Introduction
In the standard euclidean situation, pseudo-differential calculus arises as the Weyl joint functional calculusof a non-commuting pair of operators: the position and momentum operators (see, e.g., [9] and [11, ChapterXII]). By transferring this calculus to the Gaussian setting, in this paper we introduce a Gaussian versionof the Weyl pseudo-differential calculus which assigns to suitable functions a : R d × R d → C a boundedoperator a ( Q, P ) acting on L ( R d , γ ). Here, Q = ( Q , . . . , Q d ) and P = ( P , . . . , P d ) are the position andmomentum operators associated with the Ornstein-Uhlenbeck operator L = − ∆ + x · ∇ on L ( R d , γ ), where d γ ( x ) = (2 π ) − d/ exp( − | x | ) d x is the standard Gaussian measure on R d . We showthat the Ornstein-Uhlenbeck semigroup exp( − tL ) can be expressed in terms of this calculus by the formulaexp( − tL ) = (cid:16) − e − t e − t (cid:17) d exp (cid:16) − − e − t e − t ( P + Q ) (cid:17) . (1.1)With s := − e − t e − t , the expression on the right-hand side is defined through the Weyl calculus as (1 + s ) d a s ( Q, P ), where a s ( x, ξ ) = exp( − s ( | x | + | ξ | )). The main ingredient in the proof of (1.1) is the ex-plicit determination of the integral kernel for a s ( Q, P ). By applying a Schur type estimate to this kernel weare able to prove the following sufficient condition for restricted L p - L q -boundedness of a s ( Q, P ): Theorem 1.1.
Let p, q ∈ [1 , ∞ ) and let α, β > . For s ∈ C with Re s > , define r ± ( s ) := Re ( s ± s ) . If s satisfies − αp + r + ( s ) > , βq − r + ( s ) > , and ( r − ( s )) (cid:0) − αp + r + ( s ) (cid:1)(cid:0) βq − r + ( s ) (cid:1) , then the operator exp( − s ( P + Q )) is bounded from L p ( R d , γ α ) to L q ( R d , γ β ) . Date : September 10, 2018.2010
Mathematics Subject Classification.
Primary: 47A60; Secondary: 47D06, 47G30, 60H07, 81S05.
Key words and phrases.
Weyl functional calculus, canonical commutation relation, Schur estimate, Ornstein-Uhlenbeckoperator, Mehler kernel, restricted L p - L q -boundedness, restricted Sobolev embedding.The authors gratefully acknowledge financial support by the ARC discovery Grant DP 160100941. Here, γ τ denotes the centred Gaussian measure on R d with density (2 πτ ) − d/ exp( −| x | / τ ) (so that γ = γ is the standard Gaussian measure). The proof of the theorem provides an explicit estimate for thenorm of this operator that is of the correct order in the variable s , as subsequent corollaries show.Taken together, (1.1) and Theorem 1.1 can then be used to obtain criteria for L p ( R d , γ α )- L q ( R d , γ β )boundedness of exp( − zL ) for suitable values of z ∈ C with Re z >
0. Among other things, in Section5 we show that the operators exp( − zL ) map L ( R d ; γ ) to L ( R d ; γ ) for all Re z >
0. We also prove amore precise boundedness result which, for real values t >
0, implies the boundedness of exp( − tL ) from L ( R d , γ α t ) to L ( R d , γ ), where α t = 1 + e − t . The boundedness of these operators was proved recently byBakry, Bolley, and Gentil [1] as a corollary of their work on hypercontractive bounds on Markov kernels fordiffusion semigroups. As such, our results may be interpreted as giving an extension to complex time of theBakry-Bolley-Gentil result for the Ornstein-Uhlenbeck semigroup.In the final Section 6 we show that Theorem 1.1 also captures the well-known result of Epperson [2](see also Weissler [12] for the first boundedness result of this kind, and part of the contractivity result) for1 < p q < ∞ , the operator exp( − zL ) is bounded from L p ( R d , γ ) to L q ( R d , γ ) if and only if ω := e − z satisfies | ω | < p/q and( q − | ω | + (2 − p − q )(Re ω ) − (2 − p − q + pq )(Im ω ) + p − > . In particular, for p = q the semigroup exp( − tL ) on L p ( R d , γ ) extends analytically to the set (see Figure 1) E p := { z = x + iy ∈ C : | sin( y ) | < tan( θ p ) sinh( x ) } , (1.2)where cos φ p = (cid:12)(cid:12)(cid:12) p − (cid:12)(cid:12)(cid:12) . (1.3) Figure 1.
The Epperson region E p (red/orange) and the sector with angle θ p for p = 4 / s − e − t e − t , various algebraic simplifications allow one to derive sharp results for the Ornstein-Uhlenbecksemigroup in a unified manner. In fact, as the referee of this paper pointed out to us, Weissler took exactly EYL CALCULUS AND RESTRICTED L p - L q BOUNDEDNESS 3 this approach in [12], and obtained the most important special case of our Theorem 1.1 in 1979. Besidesgeneralising this result to the context of weighted Gaussian measures γ α arising from [1], the point of thenew approach given here is to connect results such as Weissler’s, and other classical hypercontractivitytheorems, to the underlying Weyl calculus. In doing so, one sees the reason why certain crucial algebraicsimplifications occur, and one develops a far more flexible tool to study other spectral multipliers associatedwith the Ornstein-Uhlenbeck operator (and, possibly, perturbations thereof). In such applications, thealgebraic consequences of the fact that the Weyl calculus involves non-commuting operators may not be aseasily unpacked as in (1.1). The L p -analysis of operators in the Weyl calculus of the pair ( Q, P ), however, issimpler than the direct analysis of operators in the functional calculus of L (or perturbations of L ). In futureworks, we plan to develop this theory and include harmonic analysis substantially more advanced than theSchur type estimate employed here, along with applications to non-linear stochastic differential equations. Acknowledgements
We are grateful to the anonymous referee for her/his useful suggestions, and, in particular, for pointingout to us the paper [12].2.
The Weyl calculus with respect to the Gaussian measure
In this section we introduce the Weyl calculus with respect to the Gaussian measure. To emphasise itsFourier analytic content, our point of departure is the fact that Fourier-Plancherel transform is unitarilyequivalent to the second quantisation of multiplication by − i . The unitary operator implementing thisequivalence is used to define the position and momentum operators Q and P associated with the Ornstein-Uhlenbeck operator L . This approach bypasses the use of creation and annihilation operators altogether andleads to the same expressions.2.1. The Wiener-Plancherel transform with respect to the Gaussian measure.
Let d m ( x ) =(2 π ) − d/ d x denote the normalised Lebesgue measure on R d . The mapping E : f ef , where e ( x ) := exp( − | x | ) , is unitary from L ( R d , γ ) onto L ( R d , m ), and the dilation δ : L ( R d , m ) → L ( R d , m ), δf ( x ) := ( √ d f (cid:0) √ x (cid:1) is unitary on L ( R d , m ). Consequently the operator U := δ ◦ E is unitary from L ( R d , γ ) onto L ( R d , m ). It was shown by Segal [8, Theorem 2] that U establishes a unitaryequivalence W = U − ◦ F ◦ U of the Fourier-Plancherel transform F as a unitary operator on L ( R d , m ), F f ( y ) := b f ( y ) := 1(2 π ) d/ Z R d f ( x ) exp( − ix · y ) d x = Z R d f ( x ) exp( − ix · y ) d m ( x ) , with the unitary operator W on L ( R d , γ ), defined for polynomials f by W f ( y ) := Z R d f ( − iy + √ x ) d γ ( x ) . We have the following beautiful representation of this operator, which is sometimes called the
Wiener-Plancherel transform , in terms of the second quantisation functor Γ [8, Corollary 3.2]: W = Γ( − i ) . This identity is not used in the sequel, but it is stated only to demonstrate that both the operator W andthe unitary U are very natural. JAN VAN NEERVEN AND PIERRE PORTAL
Position and momentum with respect to the Gaussian measure.
Consider classical positionand momentum operators X = ( x , . . . , x d ) , D = ( 1 i ∂ , . . . , i ∂ d ) , viewed as densely defined operators mapping from L ( R d ) into L ( R d ; C d ). Explicitly, x j is the denselydefined self-adjoint operator on L ( R d ) defined by pointwise multiplication, i.e., ( x j f )( x ) := x j f ( x ) for x ∈ R d , with maximal domain D ( x j ) = { f ∈ L ( R d ) : x j f ∈ L ( R d ) } , and i ∂ j is the self-adjoint operator f i ∂ h f with maximal domain D ( 1 i ∂ j ) = { f ∈ L ( R d ) : ∂ j f ∈ L ( R d ) } , the partial derivative being interpreted in the sense of distributions.Having motivated our choice of the unitary U , we now use it to introduce the position and momentum op-erators Q = ( q , . . . , q d ) and P = ( p , . . . , p d ) as densely defined closed operators acting from their natural do-mains in L ( R d , γ ) into L ( R d , γ ; C d ) by unitary equivalence with X = ( x , . . . , x d ) and D = ( i ∂ , . . . , i ∂ d ): q j := U − ◦ x j ◦ U,p j := U − ◦ i ∂ j ◦ U. They satisfy the commutation relations(2.1) [ p j , p k ] = [ q j , q k ] = 0 , [ q j , p k ] = 1 i δ jk , as well as the identity(2.2) 12 ( P + Q ) = L + d I. Here, L is the Ornstein-Uhlenbeck operator which acts on test functions f ∈ C ( R d ) by Lf ( x ) := − ∆ f ( x ) + x · ∇ f ( x ) ( x ∈ R d ) . It follows readily from the definition of the Wiener-Plancherel transform W that q j ◦ W = W ◦ p j ,p j ◦ W = − W ◦ q j consistent with the relations x j ◦ F = F ◦ ( i ∂ j ) and ( i ∂ j ) ◦ F = − F ◦ x j for position and momentum inthe Euclidean setting. Remark . Our definitions of P and Q coincide with the physicist’s definitions in the theory of the quantumharmonic oscillator (cf. [4]). Other texts, such as [7], use different normalisations. The present choice makesthe commutation relation between position and momentum as well as the identity relating the Ornstein-Uhlenbeck operator and position and momentum come out right in the sense that (2.1) and (2.2) hold.The former says that position and momentum satisfy the ‘canonical commutation relations’ and the lattersays that the Hamiltonian ( P + Q ) of the quantum harmonic oscillator equals the number operator L (physicists would write N ) plus the ground state energy d . EYL CALCULUS AND RESTRICTED L p - L q BOUNDEDNESS 5
The Weyl calculus with respect to the Gaussian measure.
The Weyl calculus for the pair (
X, D )is defined, for Schwartz functions a : R d → C , by a ( X, D ) f ( y ) = Z R d b a ( u, v ) exp( i ( uX + vD )) f ( y ) d m ( u ) d m ( v ) . Here m (d x ) = (2 π ) − d/ d x as before, b a := F a is the Fourier-Plancherel transform of a , and the unitaryoperators exp( i ( uX + vD )) on L ( R d , γ ) are defined through the actionexp( i ( uX + vD )) f ( y ) := exp( iuy + iuv ) f ( v + y )(2.3)(cf. [11, Formula 51, page 550]). This definition can be motivated by a formal application of the Baker-Campbell-Hausdorff formula to the (unbounded) operators X and D ; alternatively, one may look upon it asdefining a unitary representation of the Heisenberg group encoding the commutation relations of X and D ,the so-called Schr¨odinger representation.Motivated by the constructions in the preceding subsection, we make the following definition. Definition 2.2.
For u, v ∈ R d , on L ( R d , γ ) we define the unitary operators exp( i ( uQ + vP )) on L ( R d ; γ ) by exp( i ( uQ + vP )) := U − ◦ exp( i ( uX + vD )) ◦ U. This allows us to define, for Schwartz functions a : R d → C , the bounded operator a ( Q, P ) on L ( R d , γ )by a ( Q, P ) = U − ◦ a ( X, D ) ◦ U = Z R d b a ( u, v ) exp( i ( uQ + vP )) d m ( u ) d m ( v ) , (2.4)the integral being understood in the strong sense. An explicit expression for a ( Q, P ) can be obtained asfollows. By (2.3) and a change of variables one has (cf. [11, Formula (52), page 551]) a ( X, D ) f ( y ) = Z R d a ( ( v + y ) , ξ ) exp( − iξ ( v − y )) f ( v ) d m ( v ) d m ( ξ ) . By (2.4) and the definition of U , this gives the following explicit formula for the Gaussian setting:(2.5) a ( Q, P ) f ( y ) = 1(2 π ) d Z R d a ( ( x + y √ , ξ ) exp( − iξ ( x − y √ − | x | + | y | ) f ( x √
2) d ξ d x = 1(2 √ π ) d Z R d a ( x + y √ , ξ ) exp( − iξ ( x − y √ − | x | + | y | ) f ( x ) d ξ d x = Z R d K a ( y, x ) f ( x ) d x, where K a ( y, x ) := 1(2 √ π ) d exp( − | x | + | y | ) Z R d a ( x + y √ , ξ ) exp( − iξ ( x − y √ ξ. (2.6) 3. Expressing the Ornstein-Uhlenbeck semigroup in the Weyl calculus
In order to translate results about the Weyl functional calculus of (
Q, P ) into results regarding thefunctional calculus of L , we first need to relate these two calculi. This is done in the next theorem. It is theonly place where we rely on the concrete expression of the Mehler kernel. Theorem 3.1.
For all t > we have, with s := − e − t e − t , (3.1) exp( − tL ) = (1 + s ) d a s ( Q, P ) , where a s ( x, ξ ) := exp( − s ( | x | + | ξ | )) . JAN VAN NEERVEN AND PIERRE PORTAL
In the next section we provide restricted L p - L q estimates for a s ( Q, P ) for complex values of s purely basedon the Weyl calculus.We need an elementary calculus lemma which is proved by writing out the inner product and square normin terms of coordinates, thus writing the integral as a product of d integrals with respect to a single variable. Lemma 3.2.
For all
A > , B ∈ R , and y ∈ R d , Z R d exp( − A | y | + Bxy ) d x = (cid:0) πA (cid:1) d/ exp (cid:0) B A | y | (cid:1) . Proof of Theorem 3.1.
By (2.6) we have(3.2) K a s ( y, x ) = 1(2 √ π ) d exp( − | x | + | y | ) Z R d exp( − s ( | ξ | + | x + y | )) exp( − iξ ( x − y √ ξ = 1(2 √ π ) d exp( − s | x + y | ) exp( − | x | + | y | ) Z R d exp( − s ( | ξ | + is ξ ( x − y √ ξ = 1(2 √ π ) d exp( − s | x − y | ) exp( − s | x + y | ) exp( − | x | + | y | ) Z R d exp( − s | η | ) d η = 12 d (2 πs ) d/ exp( − s | x − y | ) exp( − s | x + y | ) exp( − | x | + | y | )= 12 d (2 πs ) d/ exp( − s (1 − s ) ( | x | + | y | ) + ( s − s ) xy ) exp( − | x | )and therefore(3.3)exp( − s ( P + Q )) f ( y ) = Z R d K a s ( y, x ) f ( x ) d x = 12 d (2 πs ) d/ Z R d exp( − s (1 − s ) ( | x | + | y | ) + ( s − s ) xy ) f ( x ) e − / | x | d x. Taking s := − e − t e − t in this identity we obtain(3.4) (cid:0) − e − t e − t (cid:17) d exp (cid:16) − − e − t e − t ( P + Q ) (cid:17) f ( y )= 1(2 π ) d/ (cid:16)
21 + e − t (cid:17) d d (cid:16) e − t − e − t (cid:17) d/ × Z R d exp (cid:16) − e − t − e − t ( | x | + | y | ) + e − t − e − t xy (cid:17) f ( x ) exp( − | x | ) d x = 1(2 π ) d/ (cid:16) − e − t (cid:17) d/ Z R d exp (cid:16) − | e − t y − x | − e − t (cid:17) f ( x ) d x = Z R d M t ( y, x ) f ( x ) d x = exp( − tL ) f ( y ) , where M t ( y, x ) = 1(2 π ) d/ (cid:16) − e − t (cid:17) d/ exp (cid:16) − | e − t y − x | − e − t (cid:17) denotes the Mehler kernel; the last step of (3.4) uses the classical Mehler formula for exp( − tL ). (cid:3) For any z ∈ C with Re z >
0, the operator exp( − zL ) is well defined and bounded as a linear operator on L ( R d , γ ), and the same is true for the expression on the right-hand side in (3.1) by analytically extendingthe kernel defining it. By uniqueness of analytic extensions, the identity (3.1) persists for complex time. EYL CALCULUS AND RESTRICTED L p - L q BOUNDEDNESS 7
The identity (3.1), extended analytically into the complex plane, admits the following deeper interpreta-tion. The transformation(3.5) s = 1 − e − z e − z , which is implicit in Theorem 3.1, is bi-holomorphic from { z ∈ C : Re z > , | Im ( z ) | < π } onto { s ∈ C : Re s > , s [1 , ∞ ) } . For 1 < p < ∞ it maps E p ∩ { z ∈ C : | Im ( z ) | < π } , where E p is the Epperson region defined by (1.2), ontoΣ θ p \ [1 , ∞ ), where Σ θ p = { s ∈ C : s = 0 , | arg( s ) | < θ p } is the open sector with angle θ p given by (1.3) (seeFigure 1). Using the periodicity modulo 2 πi of the exponential function, the mapping (3.5) maps E p ontoΣ θ p \ { } .Using this information, the analytic extendibility of the semigroup exp( − tL ) on L p ( R d , γ ) to E p can nowbe proved by showing that that exp( − s ( P + Q )) extends analytically to Σ θ p ; the details are presented inTheorem 6.4. This shows that exp( − s ( P + Q )) is a much simpler object than exp( − zL ). Remark . By (2.5) and (3.4), the theorem can be interpreted as giving a representation of the Mehlerkernel in terms of the variable − e − t e − t . This representation could be taken as the starting point for theresults in the next section without any reference to the Weyl calculus. As we already pointed out inthe Introduction, this would obscure the point that the Weyl calculus explains why the ensuing algebraicsimplifications occur. What is more, the calculus can be applied to other functions a ( x, ξ ) beyond the specialchoice a s ( ξ, x ) = exp( − s ( | x | + | ξ | )) and may serve as a tool to study spectral multipliers associated withthe Ornstein-Uhlenbeck operator.4. Restricted L p - L q estimates for exp( − s ( P + Q ))Restricting the operators exp( − s ( P + Q )) to C ∞ c ( R d ), we now take up the problem of determining whenthese restrictions extend to bounded operators from L p ( R d , γ α ) into L q ( R d , γ β ). Here, for τ >
0, we setd γ τ ( x ) = (2 πτ ) − d/ exp( −| x | / τ ) d x (so that γ = γ is the standard Gaussian measure). Boundedness (or rather, contractivity) from L p ( R d , γ ) to L q ( R d , γ ) corresponds to classical hypercontractivity of the Ornstein-Uhlenbeck semigroup. For other valuesof α, β > L p ( R d , γ α )- L q ( R d , γ β ) boundedness (Theorem 4.2 below). Re-calling that exp( − s ( P + Q )) equals the integral operator with kernel K a s given by (3.2), an immediatesufficient condition for boundedness derives from H¨older’s inequality: if p, q ∈ [1 , ∞ ) and p + p ′ = 1, and(4.1) Z R d (cid:16)Z R d | K a ( y, x ) | p ′ exp (cid:0) p ′ αp | x | (cid:1) d x (cid:17) q/p ′ exp( −| y | / β ) d y =: C < ∞ (with the obvious change if p = 1) then a ( Q, P ) extends to a bounded operator from L p ( R d , γ α ) to L q ( R d , γ β )with norm at most C . A much sharper criterion can be given by using the following Schur type estimate(which is a straightforward refinement of [10, Theorem 0.3.1]). Lemma 4.1.
Let p, q, r ∈ [1 , ∞ ) be such that r = 1 − ( p − q ) . If K ∈ L ( R ) and φ, ψ : R → (0 , ∞ ) areintegrable functions such that sup y ∈ R d (cid:16)Z R d | K ( y, x ) | r ψ r/q ( y ) φ r/p ( x ) d x (cid:17) /r =: C < ∞ , and sup x ∈ R d (cid:16)Z R d | K ( y, x ) | r ψ r/q ( y ) φ r/p ( x ) d y (cid:17) /r =: C < ∞ JAN VAN NEERVEN AND PIERRE PORTAL then T K f ( y ) := Z R K ( y, x ) f ( x ) d x ( f ∈ C c ( R )) defines a bounded operator T K from L p ( R d , φ ( x ) d x ) to L q ( R d , ψ ( x ) d x ) with norm k T K k L p ( R d ,φ ( x ) d x ) ,L q ( R d ,ψ ( x ) d x ) C − rq C rq . Proof.
For strictly positive functions η ∈ L ( R ) denote by L sη ( R ) the Banach space of measurable functions g such that ηg ∈ L s ( R ), identifying two such functions g if they are equal almost everywhere. From | T K f ( y ) | Z R d | K ( y, x ) || f ( x ) | φ /p ( x ) φ /p ( x ) d x (cid:16)Z R d | K ( y, x ) | r φ r/p ( x ) d x (cid:17) /r k f k L r ′ φ /p ( R ) = 1 ψ /q ( y ) (cid:16)Z R d | K ( y, x ) | r ψ r/q ( y ) φ r/p ( x ) d x (cid:17) /r k f k L r ′ φ /p ( R ) we find that k T K f k L ∞ ψ /q ( R ) C k f k L r ′ φ /p ( R ) . This means that T K : L r ′ φ /p ( R ) → L ∞ ψ /q ( R )is bounded with norm at most C . With K ′ ( y, x ) := K ( x, y ), the same argument gives that T ∗ K = T K ′ extends to a bounded operator from L r ′ (1 /ψ ) /q ( R ) to L ∞ (1 /φ ) /p ( R ) with norm at most C . Dualising, thisimplies that T K : L φ /p ( R ) → L rψ /q ( R )is bounded with norm at most C .This puts us into a position to apply the Riesz-Thorin theorem. Choose 0 < θ < p = − θr ′ + θ , that is, θr = p − (1 − r ) = q , so θ = rq . In view of q = − θ ∞ + θr it follows that T K : L pφ /p ( R ) → L qψ /q ( R )is bounded with norm at most C = C − θ C θ = C − rq C rq . But this means that T K : L p ( R , φ ( x ) d x ) → L q ( R , ψ ( x ) d x )is bounded with norm at most C . (cid:3) Motivated by (3.3), for s ∈ C with Re s > b s := s (1 − s ) , c s := ( s − s ) . (4.2)Setting r ± ( s ) := 12 Re ( 1 s ± s )we have the identities + Re b s = r + ( s ) and Re c s = r − ( s ) . Theorem 4.2 (Restricted L p - L q boundedness) . Let p, q ∈ [1 , ∞ ) , let r = 1 − ( p − q ) , and let α, β > . If s ∈ C with Re s > satisfies − αp + r + ( s ) > , βq − r + ( s ) > , and ( r − ( s )) (cid:0) − αp + r + ( s ) (cid:1)(cid:0) βq − r + ( s ) (cid:1) , (4.3) then the operator exp( − s ( P + Q )) is bounded from L p ( R d , γ α ) to L q ( R d , γ β ) with norm k exp( − s ( P + Q )) k L ( L p ( R d ,γ α ) ,L q ( R d ,γ β )) rs ) d/ ( αr ) d/ p ( βr ) − d/ q (1 − αp + r + ( s )) d (1 − p ) ( βq − r + ( s )) d q . EYL CALCULUS AND RESTRICTED L p - L q BOUNDEDNESS 9
Remark . We have no reason to believe that the numerical constant ( r ) d/ ( αr ) d/ p ( βr ) − d/ q is sharp,but the examples that we are about to work out indicate that the dependence on s is of the correct order. Remark . For s = x + iy ∈ C with x > r + ( s ) = ( xx + y + x ) > . It follows that the positivityassumptions 1 − αp + r + ( s ) > βq − r + ( s ) > s > αp > βq Proof.
Using the notation of (4.2), the condition (4.3) is equivalent to(Re c s ) (cid:0) − αp + Re b s (cid:1)(cid:0) βq + Re b s (cid:1) . We prove the theorem by checking the criterion of Lemma 4.1 for K = K a s with a s ( x, ξ ) = exp( − s ( | x | + | ξ | )), and φ ( x ) = (2 πα ) − d/ exp( −| x | / α ), ψ ( x ) = (2 πβ ) − d/ exp( −| x | / β ).By (3.2), for almost all x, y ∈ R d we have K a s ( y, x ) = 12 d (2 πs ) d/ exp( − b s ( | x | + | y | ) + c s xy ) exp( − | x | ) . Let r ∈ [1 , ∞ ) be such that r = 1 − ( p − q ). Using Lemma 3.2, applied with A = r ( − αp + Re b s ) and B = r Re c s , we may estimatesup y ∈ R d (cid:16)Z R d | K a s ( y, x ) | r (2 πα ) rd/ p exp( p r | x | / α )(2 πα ) − rd/ p exp( − q r | y | / β ) d x (cid:17) /r = (2 πα ) d/ p (2 πβ ) − d/ q d (2 πs ) d/ × sup y ∈ R d (cid:16)Z R d exp( − r Re b s ( | x | + | y | ) + r Re c s xy ) exp( − r ( − αp ) | x | ) exp( − βq r | y | ) d x (cid:17) /r = (2 πα ) d/ p (2 πβ ) − d/ q d (2 πs ) d/ × sup y ∈ R d h exp( − ( βq + Re b s ) | y | ) (cid:16)Z R d exp( − r ( − αp + Re b s ) | x | + r Re c s xy ) d x (cid:17) /r i = (2 πα ) d/ p (2 πβ ) − d/ q d (2 πs ) d/ (cid:16) πr ( − αp + Re b s ) (cid:17) d/ r × sup y ∈ R d h exp( − ( βq + Re b s ) | y | ) exp (cid:16) (Re c s ) − αp + Re b s ) | y | (cid:17)i = (2 πα ) d/ p (2 πβ ) − d/ q d (2 π ) d/ (cid:16) πr (cid:17) d/ r s d/ (cid:16) − αp + Re b s (cid:17) d/ r = (2 α ) d/ p (2 β ) − d/ q d/ r d/ r s d/ (cid:16) − αp + Re b s (cid:17) d/ r . In the same way, using Lemma 3.2 applied with A = r ( βq + Re b s ) and B = r Re c s ,sup x ∈ R d (cid:16)Z R d | K a s ( y, x ) | r exp( αp r | x | ) exp( − βq r | y | ) d y (cid:17) /r = (2 α ) d/ p (2 β ) − d/ q d/ r d/ r s d/ (cid:16) βq + Re b s (cid:17) d/ r . Denoting these two bounds by C and C , Lemma 4.1 bounds the norm of the operator by C − rq C rq = C r (1 − p )1 C rq . After rearranging the various constants a bit, this gives the estimate in the statement of thetheorem. (cid:3) Remark . In the above proof one could replace the Schur test (Lemma 4.1) by the weaker condition (4.1)based on H¨older’s inequality. This would have the effect of replacing the suprema by integrals throughoutthe proof. This leads not only to sub-optimal estimates, but more importantly it would not allow to handlethe critical case when (4.3) holds with equality.Combining Theorems 3.1 and 4.2, we obtain the following boundedness result for the operators exp( − zL ). Corollary 4.6.
Let s ∈ C with Re s > satisfy the conditions of the theorem and define z ∈ C by s = − e − z e − z . Then, k exp( − zL ) k L ( L p ( R d ,γ α ) ,L q ( R d ,γ β )) d C | − e − z | d − αp + Re e − z − e − z ) d (1 − p ) ( βq − e − z − e − z ) d q , where C is the numerical constant in Theorem 4.2 (cf. Remark 4.3).Proof. Noting that 2 / (1 + e − z ) = 1 + s , we have k exp( − zL ) k | s | d k exp( − s ( P + Q )) k C | s | d | s | d/ | − αp + r + ( s ) | d (1 − p ) | βq − r + ( s ) | d q . The result follows from this by substituting r + ( s ) = Re ( s + s ) = Re e − z − e − z . (cid:3) Restricted L p - L boundedness and Sobolev embedding As a first application of Theorem 4.2 we have the following ‘hyperboundedness’ result for real times t > Corollary 5.1.
For p ∈ [1 , and t > set α p,t := (1 + e − t ) /p. (1) For all t > the operator exp( − tL ) is bounded from L ( R d , γ α ,t ) to L ( R d , γ ) , with norm k exp( − tL ) k L ( L ( R d ,γ α ,t ) ,L ( R d ,γ ) . d (1 − e − t ) − d/ . (2) For all p ∈ [1 , and t > the operator exp( − tL ) is bounded from L p ( R d , γ α p,t ) to L ( R d , γ ) , withnorm k exp( − tL ) k L ( L p ( R d ,γ αp,t ) ,L ( R d ,γ ) . d,p t − d ( p − ) as t ↓ . Proof.
Elementary algebra shows that with α p,t = p (1 + s s ) and s = − e − t e − t , the criterion of Theorem 4.2holds for all t > p = 1, the second by noting that Re ( e − t − e − t ) ∼ t for small values of t . (cid:3) A sharp version of this corollary is due to Bakry, Bolley and Gentil [1, Section 4.2, Eq. (28)], who showed(for p = 1) the hypercontractivity bound k exp( − tL ) k L ( L ( R d ,γ α ,t ) ,L ( R d ,γ )) (1 − e − t ) − d/ . Their proof relies on entirely different techniques which seem not to generalise to complex time so easily.The next corollary gives ‘ultraboundedness’ of the operators exp( − zL ) for arbitrary Re z > L p ( R d , γ /p ) into L ( R d , γ ): Corollary 5.2.
Let p ∈ [1 , . For all z ∈ C with Re z > the operator exp( − zL ) maps L p ( R d , γ /p ) into L ( R d , γ ) . As a consequence, the semigroup generated by − L extends to a strongly continuous holomorphicsemigroup of angle π on L p ( R d , γ /p ) . For each θ ∈ (0 , π ) this semigroup is uniformly bounded on thesector { z ∈ C : z = 0 , | arg( z ) | < θ } .Proof. This follows from Corollary 4.6 upon realising that the assumptions of Theorem 4.2 are satisfied when q = 2, β = 1 and α = p , or q = p and α = β = p . (cid:3) EYL CALCULUS AND RESTRICTED L p - L q BOUNDEDNESS 11
A notable consequence of Corollary 4.6 is the following (restricted) Sobolev embedding result. It isinteresting because ( I + L ) − maps L p ( R d , γ ) into L ( R d , γ ) only when p = 2 (i.e. no full Sobolev embeddingtheorem holds in the Ornstein-Uhlenbeck context). Corollary 5.3 (Restricted Sobolev embedding) . Let p ∈ ( dd +2 , . The resolvent ( I + L ) − maps L p ( R d , γ /p ) into L ( R d , γ ) .Proof. Let p ∈ ( dd +2 ,
2] and fix u ∈ L p ( R d , γ /p ) ∩ L ( R d , γ ). Then k exp( − t ( I + L )) u k L ( R d ,γ ) . d,p k u k L p ( R d ,γ /p ) exp( − t ) t − d ( p − ) ∀ t > , and thus k ( I + L ) − u k L ( R d ,γ ) . d,p k u k L p ( R d ,γ /p ) Z ∞ exp( − t ) t − d ( p − ) d t . k u k L p ( R d ,γ /p ) , since p ∈ ( dd +2 ,
2] implies d ( p − ) < (cid:3) L p - L q Boundedness
We now turn to the classical setting of the spaces L p ( R d , γ ), where γ is the standard Gaussian measure.For α = β = 1 and s = x + iy the first positivity condition of Theorem 4.2 takes the form1 − p + r + ( s ) > ⇐⇒ − p + 12 (cid:16) xx + y + x (cid:17) > p − q )( x + xx + y ) + pq ( x x + y −
1) + 2 p + 2 q − > . (6.2)Let us also observe that if these two conditions hold, together they enforce the second positivity condition q − r + ( s ) >
0; this is apparent from the representation in (4.3).As a warm up for the general case, let us first consider real times t ∈ (0 ,
1) in the z -plane, which correspondto the values s = x ∈ (0 ,
1) in the s -plane. The conditions (6.1) and (6.2) then reduce to(1 − p ) x + 12 ( x + 1) > p − q )( x + 1 x ) + 2 p + 2 q − > , respectively. The first condition is automatic. Substituting x = − e − t e − t in the second and solving for e − t ,assuming p q we find that it is equivalent to the condition e − t p − q − . Thus we recover the boundedness part of Nelson’s celebrated hypercontractivity result [6].Turning to complex time, with some additional effort we also recover the following result due to Weissler[12] (see also Epperson [2] for further refinements), essentially as a Corollary of Theorem 4.2.
Theorem 6.1 ( L p - L q -boundedness of exp( − zL )) . Let < p q < ∞ . If z ∈ C satisfies Re z > , | e − z | < p/q (6.3) and (6.4) ( q − | e − z | + (2 − p − q )(Re e − z ) − (2 − p − q + pq )(Im e − z ) + p − > , then the operator exp( − zL ) maps L p ( R d , γ ) into L q ( R d , γ ) . Before turning to the proof we make a couple of preliminary observations. By a simple argument involvingquadratic forms (see [2, page 3]), the conditions (6.3) and (6.4) taken together are equivalent to the singlecondition (Im ( we − z )) + ( q − we − z )) < (Im w ) + ( p − w ) ∀ w ∈ C . (6.5)Let us denote the set of all z ∈ C , Re z >
0, for which (6.5) holds by E p,q . The following two facts hold: Facts . • E p,p = E p . • E p,q ⊆ E p and E p,q ⊆ E q .The first is implicit in [2, 3], can be proved by elementary means, and is taken for granted. The secondis an immediate consequence of the assumption p q .Let us now start with the proof of Theorem 6.1. It is useful to dispose of the positivity condition (6.1) inthe form of a lemma; see also Figure 2. Figure 2.
The region R p := { s ∈ C : 1 − p + r + ( s ) > } (red/orange) and the sector Σ p (orange), both for p = 4 /
3. Lemma 6.3 implies that Σ p is indeed contained in R p .Let z ∈ C satisfy Re z >
0. By the remarks at the end of Section 3, z belongs to E p if and only if s = − e − z e − z belongs to Σ φ p \ { } . Lemma 6.3.
Every s ∈ Σ φ p satisfies the positivity condition (6.1) .Proof. Writing s = x + iy , we then have x x + y = cos θ p > (1 − p ) , where the angle θ p is given by (1.3). To see that this implies (6.1), note that1 − p + 12 (cid:16) xx + y + x (cid:17) > − p ) x + (1 − p ) + x = (cid:16) x + (1 − p ) (cid:17) and the latter is trivially true. (cid:3) EYL CALCULUS AND RESTRICTED L p - L q BOUNDEDNESS 13
Proof of Theorem 6.1.
Fix Re z > s := − e − z e − z . We show that the assumptions of the theorem implythe conditions of Theorem 4.2, so that exp( − s ( P + Q )) maps L p ( R d , γ ) into L q ( R d , γ ). In combinationwith Theorem 3.1, this gives the result.We begin by checking the condition (6.1). For this, the second fact tells us that there is no loss of generalityin assuming that q = p . In that situation, the first fact tells us that z belongs to E p . But then Lemma 6.3gives us the desired result.It remains to check (6.2). Multiplying both sides with x + y , this can be rewritten as( p − q ) x (1 + x + y ) + pqx − ( pq − p − q + 4)( x + y ) > . (6.6)The proof of the theorem is completed by showing that (6.4) implies (6.6).Towards this end, we rewrite (6.4) in a similar way. Setting e − z = − s s with s = x + iy , and using thatRe 1 − x − iy x + iy = 1 − ( x + y )(1 + x ) + y , Im 1 − x − iy x + iy = − y (1 + x ) + y , (6.4) takes the form( q − − ( x + y )) + 4 y ) + (2 − p − q )(1 − ( x + y )) ((1 + x ) + y ) − (2 − p − q + pq )4 y ((1 + x ) + y ) + ( p − x ) + y ) > . This factors as(6.7) (cid:2) x ) + y ) (cid:3) × (cid:2) ( p − q ) x (1 + x + y ) + (2 p + 2 q − x − ( pq − p − q + 4) y (cid:3) . Quite miraculously, the second term in straight brackets precisely equals the term in (6.6). Since 4((1 + x ) + y ) > (cid:3) It is shown in [2] (see also [5]) that the operator exp( − zL ) is bounded from L p ( R d , γ ) to L q ( R d , γ ) if andonly if z ∈ E p , and then the operators exp( − zL ) are in fact contractions. Our proof does not recover thecontractivity of exp( − zL ). Nevertheless it is remarkable that the boundedness part does follow from ourmethod, which just uses (2.3), elementary calculus, the Schur test, and some algebraic manipulations.For p = q , Theorem 6.1 combined with the fact that E p,p = E p contains as a special case that, for a given z ∈ C with Re z >
0, the operator exp( − zL ) is bounded on L p ( R d , γ ) if z belongs to E p . A more direct -and more transparent - proof of this fact may be obtained as a consequence of the following theorem. Theorem 6.4.
For all < p < ∞ and s ∈ Σ θ p the operator exp( − s ( P + Q )) is bounded on L p ( R d , γ ) . As we explained in Section 3, this result translates into Epperson’s result that the semigroup exp( − tL )on L p ( R d , γ ) can be analytically extended to to E p . Proof.
Lemma 6.3 shows that (6.1) holds. Since q = p , (6.2) reduces to the condition p ( x x + y −
1) + 4 p − > , which is equivalent to saying that s ∈ Σ θ p . (cid:3) Remark . More generally, for an arbitrary pair ( α, p ) ∈ [1 , ∞ ) × [1 , ∞ ) satisfying αp >
2, by the samemethod we obtain that exp( − zL ) is bounded on L p ( R d , γ α ) if s = − e − z e − z satisfiesRe s | s | > − αp . This corresponds to the sector of angle θ α,p = arccos(1 − αp ) in the s -plane. In the z -plane, this correspondsto the Epperson region E αp . Acknowledgment –
We thank Emiel Lorist for generating the figures.
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Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, TheNetherlands
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