On theorems of Chernoff and Ingham on the Heisenberg group
Sayan Bagchi, Pritam Ganguly, Jayanta Sarkar, Sundaram Thangavelu
OON THEOREMS OF CHERNOFF AND INGHAMON THE HEISENBERG GROUP
SAYAN BAGCHI, PRITAM GANGULY, JAYANTA SARKARAND SUNDARAM THANGAVELU
Abstract.
We prove an analogue of Chernoff’s theorem for the sublaplacian on the Heisen-berg group and use it prove a version of Ingham’s theorem for the Fourier transform on thesame group. Introduction
Roughly speaking, the uncertainty principle for the Fourier transform on R n says that afunction f and its Fourier transform (cid:98) f cannot both have rapid decay. Several manifesta-tions of this principle are known: Heisenberg-Pauli-Weyl inequality, Paley-Wiener theorem,Hardy’s uncertainty principle are some of the most well known. But there are lesser knownresults such as theorems of Ingham and Levinson. The best decay a non trivial function canhave is vanishing identically outside a compact set and for such functions it is well knownthat their Fourier transforms extend to C n as entire functions and hence cannot vanish onany open set. For any such function of compact support, its Fourier transform cannot haveany exponential decay for a similar reason: if | (cid:98) f ( ξ ) | ≤ Ce − a | ξ | for some a >
0, then it followsthat f extends to a tube domain in C n as a holomorphic function and hence it cannot havecompact support. So it is natural to ask the question: what is the best possible decay that isallowed of a function of compact support? An interesting answer to this question is providedby the following theorem of Ingham [10] Theorem 1.1 (Ingham) . Let Θ( y ) be a nonnegative even function on R such that Θ( y ) decreases to zero when y → ∞ . There exists a nonzero continuous function f on R , equal tozero outside an interval ( − a, a ) having Fourier transform (cid:98) f satisfying the estimate | (cid:98) f ( y ) | ≤ Ce −| y | Θ( y ) if and only if Θ satisfies (cid:82) ∞ Θ( t ) t − dt < ∞ . This theorem of Ingham and its close relatives Paley -Wiener ([20, 21]) and Levinson ([14])theorems have received considerable attention in recent years. In [2] Bhowmik et al provedanalogues of the above theorem for R n , the n -dimensional torus T n and step two nilpotent Mathematics Subject Classification.
Primary: 43A80. Secondary: 22E25, 33C45, 26E10, 46E35.
Key words and phrases.
Heisenberg group, sublaplacian, quasi-analyticity, sobolev spaces, Chernoff’stheorem, Ingham’s theorem. a r X i v : . [ m a t h . F A ] S e p BAGCHI, GANGULY, SARKAR AND THANGAVELU
Lie groups. See also the recent work of Bowmik-Pusti-Ray [3] for a version of Ingham’stheorem for the Fourier transform on Riemannian symmetric spaces of non-compact type.As we are interested in Ingham’s theorem on the Heisenberg group, let us recall the resultproved in [2]. Let H n = C n × R be the Heisenberg group. For an integrable function f on H n let (cid:98) f ( λ ) be the operator valued Fourier transform of f indexed by non-zero real λ. Measuringthe decay of the Fourier transform in terms of the Hilbert-Schmidt operator norm (cid:107) (cid:98) f ( λ ) (cid:107) HS Bhowmik et al have proved the following result.
Theorem 1.2 (Bhowmik-Ray-Sen) . Let Θ( λ ) be a nonnegative even function on R such that Θ( λ ) decreases to zero when λ → ∞ . There exists a nonzero, compactly supported continuousfunction f on H n , whose Fourier transform satisfies the estimate (cid:107) (cid:98) f ( λ ) (cid:107) HS ≤ C | λ | n/ e −| λ | Θ( λ ) if the integral (cid:82) ∞ Θ( t ) t − dt < ∞ . On the other hand, if the above estimate is valid for afunction f and the integral (cid:82) ∞ Θ( t ) t − dt diverges, then the vanishing of f on any set of theform { z ∈ C n : | z | < δ } × R forces f to be identically zero. As the Fourier transform on the Heisenberg group is operator valued, it is natural tomeasure the decay of (cid:98) f ( λ ) by comparing it with the Hermite semigroup e − aH ( λ ) generated by H ( λ ) = − ∆ R n + λ | x | . In this connection, let us recall the following two versions of Hardy’suncertainty principle. Let p a ( z, t ) stand for the heat kernel associated to the sublaplacian L on the Heisenberg group whose Fourier transform turns out to be the Hermite semigroup e − aH ( λ ) . The version in which one measures the decay of (cid:98) f ( λ ) in terms of its Hilbert-Schmidtoperator norm reads as follows. If | f ( z, t ) | ≤ Ce − a ( | z | + t ) , (cid:107) (cid:98) f ( λ ) (cid:107) HS ≤ Ce − bλ (1.1)then f = 0 whenever ab > / . This is essentially a theorem in the t -vairable and can beeasily deduced from Hardy’s theorem on R , see Theorem 2.9.1 in [31]. Compare this withthe following version, Theorem 2.9.2 in [31]. If | f ( z, t ) | ≤ Cp a ( z, t ) , (cid:98) f ( λ ) ∗ (cid:98) f ( λ ) ≤ Ce − bH ( λ ) (1.2)then f = 0 whenever a < b. This latter version is the exact analogue of Hardy’s theorem forthe Heisenberg group, which we can view not merely as an uncertainty principle but also as acharacterisation of the heat kernel. Hardy’s theorem in the context of semi-simple Lie groupsand non-compact Riemannian symmetric spaces are also to be viewed in this perspective.We remark that the Hermite semigroup has been used to measure the decay of the Fouriertransform in connection with the heat kernel transform [12], Pfannschmidt’s theorem [33]and the extension problem for the sublaplacian [23] on the Heisenberg group. In connectionwith the study of Poisson integrals, it has been noted in [32] that when the Fourier transformof f satisfies an estimate of the form (cid:98) f ( λ ) ∗ (cid:98) f ( λ ) ≤ Ce − a √ H ( λ ) , then the function extends to atube domain in the complexification of H n as a holomorphic function and hence the vanishing HERNOFF AND INGHAM ON THE HEISENBERG GROUP 3 of f on an open set forces it to vanish identically. It is therefore natural to ask if the sameconclusion can be arrived at by replacing the constant a in the above estimate by an operatorΘ( (cid:112) H ( λ )) for a function Θ decreasing to zero at infinity. Our investigations have led us tothe following analogue of Ingham’s theorem for the Fourier transform on H n . Theorem 1.3.
Let Θ( λ ) be a nonnegative even function on R such that Θ( λ ) decreases tozero when λ → ∞ . There exists a nonzero compactly supported continuous function f on H n whose Fourier transform (cid:98) f satisfies the estimate (cid:98) f ( λ ) ∗ (cid:98) f ( λ ) ≤ Ce − √ H ( λ )) √ H ( λ ) (1.3) if and only if the function Θ satisfies (cid:82) ∞ Θ( t ) t − dt < ∞ . Theorem 1.1 was proved in [10] by Ingham by making use of Denjoy-Carleman theorem onquasi-analytic functions. In [2] the authors have used Radon transform and a several variableextension of Denjoy-Carleman theorem due to Bochner and Taylor [5] in order to prove the n -dimensional version of Theorem 1.1. An L variant of the result of Bochner-Taylor whichwas proved by Chernoff in [8] has turned out to be very useful in establishing Ingham typetheorems. Theorem 1.4. [8, Chernoff]
Let f be a smooth function on R n . Assume that ∆ m f ∈ L ( R n ) for all m ∈ N and (cid:80) ∞ m =1 (cid:107) ∆ m R n f (cid:107) − m = ∞ . If f and all its partial derivatives vanish at ,then f is identically zero. This theorem shows how partial differential operators generate the class of quasi-analyticfunctions. Recently, Bhowmik-Pusti-Ray [3] have established an analogue of Chernoff’s the-orem for the Laplace-Beltrami operator on non-compact Riemannian symmetric spaces anduse the same in proving a version of Ingham’s theorem for the Helgason Fourier transform. Itis therefore natural to look for an analogue of this result for sublaplacian on the Heisenberggroup. In this paper, we prove the following result.
Theorem 1.5.
Let L be the sublaplacian on the Heisenberg group and let f be a smoothfunction on H n such that L m f ∈ L ( H n ) for all m ∈ N . Assume that (cid:80) ∞ m =1 (cid:107)L m f (cid:107) − m = ∞ .If f and all its partial derivatives vanish at some point, then f is identically zero. An immediate corollary of this theorem is the following, which can be seen as an L version of the classical Denjoy-Carleman theorem on the Heisenberg group using iterates ofsublaplacian. Corollary 1.6.
Let { M k } k be a log convex sequence. Define C ( { M k } k , L , H n ) to be the classof all smooth functions f on H n such that L m f ∈ L ( H n ) for all k ∈ N and (cid:107)L k f (cid:107) ≤ M k λ k for some constant λ (may depend on f ). Suppose that (cid:80) ∞ k =1 M − k k = ∞ . Then every memberof that class is quasi-analytic.
BAGCHI, GANGULY, SARKAR AND THANGAVELU Preliminaries
In this section, we collect the results which are necessary for the study of uncertaintyprinciples for the Fourier transform on the Heisenberg group. We refer the reader to thetwo classical books Folland [9] and Taylor [28] for the preliminaries of harmonic analysis onthe Heisenberg group. However, we will be closely following the notations of the books ofThangavelu [30] and [31].2.1.
Heisenberg group and Fourier transform.
Let H n := C n × R denote the (2 n + 1)-Heisenberg group equipped with the group law( z, t ) . ( w, s ) := (cid:0) z + w, t + s + 12 (cid:61) ( z. ¯ w ) (cid:1) , ∀ ( z, t ) , ( w, s ) ∈ H n . This is a step two nilpotent Lie group where the Lebesgue measure dzdt on C n × R serves asthe Haar measure. The representation theory of H n is well-studied in the literature. In orderto define Fourier transform, we use the Schr¨odinger representations as described below.For each non zero real number λ we have an infinite dimensional representation π λ realisedon the Hilbert space L ( R n ) . These are explicitly given by π λ ( z, t ) ϕ ( ξ ) = e iλt e i ( x · ξ + x · y ) ϕ ( ξ + y ) , where z = x + iy and ϕ ∈ L ( R n ) . These representations are known to be unitary andirreducible. Moreover, by a theorem of Stone and Von-Neumann, (see e.g., [9]) upto unitaryequivalence these account for all the infinite dimensional irreducible unitary representationsof H n which act as e iλt I on the center. Also there is another class of finite dimensionalirreducible representations. As they do not contribute to the Plancherel measure we will notdescribe them here.The Fourier transform of a function f ∈ L ( H n ) is the operator valued function obtainedby integrating f against π λ : ˆ f ( λ ) = (cid:90) H n f ( z, t ) π λ ( z, t ) dzdt. Note that ˆ f ( λ ) is a bounded linear operator on L ( R n ) . It is known that when f ∈ L ∩ L ( H n )its Fourier transform is actually a Hilbert-Schmidt operator and one has (cid:90) H n | f ( z, t ) | dzdt = (2 π ) − ( n +1) (cid:90) ∞−∞ (cid:107) (cid:98) f ( λ ) (cid:107) HS | λ | n dλ where (cid:107) . (cid:107) HS denote the Hilbert-Schmidt norm. The above allows us to extend the Fouriertransform as a unitary operator between L ( H n ) and the Hilbert space of Hilbert-Schmidtoperator valued functions on R which are square integrable with respect to the Plancherel HERNOFF AND INGHAM ON THE HEISENBERG GROUP 5 measure dµ ( λ ) = (2 π ) − n − | λ | n dλ. We polarize the above identity to obtain (cid:90) H n f ( z, t ) g ( z, t ) dzdt = (cid:90) ∞−∞ tr ( (cid:98) f ( λ ) (cid:98) g ( λ ) ∗ ) dµ ( λ ) . Also for suitable function f on H n we have the following inversion formula f ( z, t ) = (cid:90) ∞−∞ tr ( π λ ( z, t ) ∗ (cid:98) f ( λ )) dµ ( λ ) . Now by definition of π λ and ˆ f ( λ ) it is easy to see that (cid:98) f ( λ ) = (cid:90) C n f λ ( z ) π λ ( z, dz where f λ stands for the inverse Fourier transform of f in the central variable: f λ ( z ) := (cid:90) ∞−∞ e iλ.t f ( z, t ) dt. This motivates the following operator. Given a function g on C n , we consider the followingoperator valued function defined by W λ ( g ) := (cid:90) C n g ( z ) π λ ( z, dz. With these notations we note that ˆ f ( λ ) = W λ ( f λ ) . For λ = 1, W ( g ) := W ( g ) is called theWeyl transform of g . Moreover, the fourier transform bahaves well with the convolution oftwo functions defined by f ∗ g ( x ) := (cid:90) H n f ( xy − ) g ( y ) dy. Infact, for any f, g ∈ L ( H n ), directly from the definition it follows that (cid:91) f ∗ g ( λ ) = ˆ f ( λ )ˆ g ( λ ) . Special functions and Fourier transform.
For each λ (cid:54) = 0, we consider the followingfamily of scaled Hermite functions indexed by α ∈ N n :Φ λα ( x ) := | λ | n Φ α ( (cid:112) | λ | x ) , x ∈ R n where Φ α denote the n − dimensional Hermite functions (see [29]). It is well-known that thesescaled functions Φ λα are eigenfunctions of the scaled Hermite operator H ( λ ) := − ∆ R n + λ | x | with eigenvalue (2 | α | + n ) | λ | and { Φ λα : α ∈ N n } forms an orthonormal basis for L ( R n ). Asa consequence, (cid:107) (cid:98) f ( λ ) (cid:107) HS = (cid:88) α ∈ N n (cid:107) (cid:98) f ( λ )Φ λα (cid:107) . In view of this the Plancheral formula takes the following very useful form (cid:90) H n | f ( z, t ) | dzdt = (cid:90) ∞−∞ (cid:88) α ∈ N n (cid:107) (cid:98) f ( λ )Φ λα (cid:107) dµ ( λ ) . BAGCHI, GANGULY, SARKAR AND THANGAVELU
Given σ ∈ U ( n ), we define R σ f ( z, t ) = f ( σ.z, t ). We say that a function f on H n isradial if f is invariant under the action of U ( n ) i.e., R σ f = f for all σ ∈ U ( n ) . The Fouriertransforms of such radial integrable funtions are functions of the Hermite operator H ( λ ) . Infact, if H ( λ ) = (cid:80) ∞ k =0 (2 k + n ) | λ | P k ( λ ) stands for the spectral decomposition of this operator,then for a radial intrgrable function f we have (cid:98) f ( λ ) = ∞ (cid:88) k =0 R k ( λ, f ) P k ( λ ) . More explicitly, P k ( λ ) stands for the orthogonal projection of L ( R n ) onto the k th eigenspacespanned by scaled Hermite functions Φ λα for | α | = k . The coefficients R k ( λ, f ) are explicitlygiven by R k ( λ, f ) = k !( n − k + n − (cid:90) C n f λ ( z ) ϕ n − k,λ ( z ) dz. (2.1)In the above formula, ϕ n − k,λ are the Laguerre functions of type ( n − ϕ n − k,λ ( z ) = L n − k ( 12 | λ || z | ) e − | λ || z | where L n − k denotes the Laguerre polynomial of type ( n − ϕ n − k,λ . In order toget such estimates, we use the available sharp estiamtes of standard Laguerre functions asdescribed below in more general context.For any δ > −
1, let L δk ( r ) denote the Laguerre polynomials of type δ . The standardLaguerre functions are defined by L δk ( r ) = (cid:16) Γ( k + 1)Γ( δ + 1)Γ( k + δ + 1) (cid:17) L δk ( r ) e − r r δ/ which form an orthonormal system in L ((0 , ∞ ) , dr ). In terms of L δk ( r ) , we have ϕ δk ( r ) = 2 δ (cid:16) Γ( k + 1)Γ( δ + 1)Γ( k + δ + 1) (cid:17) − r − δ L δk (cid:16) r (cid:17) . Asymptotic properties of L δk ( r ) are well known in the literature, see [29, Lemma 1.5.3]. Theestimates in [29, Lemma 1.5.3] are sharp, see [15, Section 2] and [16, Section 7]. For ourconvenience, we restate the result in terms of ϕ n − k,λ ( r ) . HERNOFF AND INGHAM ON THE HEISENBERG GROUP 7
Lemma 2.1.
Let ν ( k ) = 2(2 k + n ) and C k,n = (cid:16) k !( n − k + n − (cid:17) . For λ (cid:54) = 0 , we have the estimates C k,n | ϕ n − k,λ ( r ) | ≤ C ( r (cid:112) | λ | ) − ( n − ( ν ( k ) r | λ | ) ( n − / , ≤ r ≤ √ √ ν ( k ) | λ | ( ν ( k ) r | λ | ) − , √ √ ν ( k ) | λ | ≤ r ≤ √ ν ( k ) √ | λ | ν ( k ) − ( ν ( k ) + | ν ( k ) − | λ | r | ) − , √ ν ( k ) √ | λ | ≤ r ≤ √ ν ( k ) √ | λ | e − γr | λ | , r ≥ √ ν ( k ) √ | λ | , where γ > is a fixed constant and C is independent of k and λ . The sublaplacian and Sobolev spaces on H n . We let h n stand for the HeisenbergLie algebra consisting of left invariant vector fields on H n . A basis for h n is provided by the2 n + 1 vector fields X j = ∂∂x j + 12 y j ∂∂t , Y j = ∂∂y j − x j ∂∂t , j = 1 , , ..., n and T = ∂∂t . These correspond to certain one parameter subgroups of H n . The sublaplacianon H n is defined by L := − ∞ (cid:88) j =1 ( X j + Y j )which can be explicitly calculated as L = − ∆ C n − | z | ∂ ∂t + N ∂∂t where ∆ C n stands for the Laplacian on C n and N is the rotation operator defined by N = n (cid:88) j =1 (cid:18) x j ∂∂y j − y j ∂∂x j (cid:19) . This is a sub-elliptic operator and homogeneous of degree 2 with respect to the non-isotropicdilations given by δ r ( z, t ) = ( rz, r t ) . The sublaplacian is also invariant undrer rotation i.e., R σ ◦ L = L ◦ R σ , σ ∈ U ( n ) . It is convenient for our purpose to represent the sublaplacian interms of another set of vector fields defined as follows: Z j := 12 ( X j − iY j ) = ∂∂z j + i z j ∂∂t , ¯ Z j := 12 ( X j + iY j ) = ∂∂ ¯ z j + i z j ∂∂t where ∂∂z j = (cid:16) ∂∂x j − i ∂∂y j (cid:17) and ∂∂ ¯ z j = (cid:16) ∂∂x j + i ∂∂y j (cid:17) . Now an easy calculation yields L = − n (cid:88) j =1 (cid:0) Z j ¯ Z j + ¯ Z j Z j (cid:1) . The action of Fourier transform on Z j f , ¯ Z j f and T f are well-known and are given by (cid:100) Z j f ( λ ) = i (cid:98) f ( λ ) A j ( λ ) , (cid:100) ¯ Z j f ( λ ) = i (cid:98) f ( λ ) A j ( λ ) ∗ and (cid:99) T f ( λ ) = − iλ (cid:98) f ( λ ) (2.2) BAGCHI, GANGULY, SARKAR AND THANGAVELU where A j ( λ ) and A ∗ j ( λ ) are the annihilation and creation operators given by A j ( λ ) = (cid:16) − ∂∂ξ j + iλξ j (cid:17) , A ∗ j ( λ ) = (cid:16) ∂∂ξ j + iλξ j (cid:17) . These along with the above representation of the sublaplacian yield the relation (cid:99) L f ( λ ) = (cid:98) f ( λ ) H ( λ ) . We can define the spaces W s, ( H n ) for any s ∈ R as the completion of C ∞ c ( H n ) underthe norm (cid:107) f (cid:107) ( s ) = (cid:107) ( I + L ) s/ f (cid:107) where the fractional powers ( I + L ) s/ are defined usingspectral theorem. To study these spaces, it is better to work with the following expressionof the norm (cid:107) f (cid:107) ( s ) for f ∈ C ∞ c ( H n ) . In view of Plancherel theorem for the Fourier transform (cid:107) f (cid:107) s ) = (2 π ) − n − (cid:90) ∞−∞ (cid:107) (cid:98) f ( λ )(1 + H ( λ )) s/ (cid:107) HS | λ | n dλ which is valid for any s ∈ R . Here we have made use of the fact that (cid:99) L f ( λ ) = (cid:98) f ( λ ) H ( λ ) . Computing the Hilbert-Schmidt norm in terms of the Hermite basis, we have the moreexplicit expression: (cid:107) f (cid:107) s ) = (2 π ) − n − (cid:90) ∞−∞ (cid:88) α ∈ N n (cid:88) β ∈ N n (1 + (2 | α | + n ) | λ | ) s |(cid:104) (cid:98) f ( λ )Φ λα , Φ λβ (cid:105)| | λ | n dλ. Consider (cid:98) H n = R ∗ × N n × N n equipped with the measure µ × ν where ν is the counting measureon N n × N n . The above shows that, for f ∈ C ∞ c ( H n ) the function m ( λ, α, β ) = (cid:104) (cid:98) f ( λ )Φ λα , Φ λβ (cid:105) . belongs to the weighted space W s, ( (cid:98) H n ) = L ( (cid:98) H n , w s d ( µ × ν ))where w s ( λ, α ) = (1 + (2 | α | + n ) | λ | ) s . As these weighted L spaces are complete, we canidentify W s, ( H n ) with W s, ( (cid:98) H n ) . It is then clear that for any s > W s, ( (cid:98) H n ) ⊂ W , ( (cid:98) H n ) ⊂ W − s, ( (cid:98) H n )and the same inclusion holds for W s, ( H n ) . It is clear that any m ∈ W s, ( (cid:98) H n ) can be writtenas m ( λ, α, β ) = (1 + (2 | α | + n ) | λ | ) − s/ m ( λ, α, β ) where m ∈ W , ( (cid:98) H n ) = L ( (cid:98) H n ) forany s ∈ R . Consequently, any f ∈ W s, ( H n ) can be written as f = ( I + L ) − s/ f , where f ∈ L ( H n ) is the function which corresponds to m which is given explicitly by f ( z, t ) = (cid:90) (cid:98) H n m ( λ, α, β ) e − λα,β ( z, t ) dν ( α, β ) dµ ( λ ) . Thus we see that f ∈ W s, ( H n ) if and only if there is an f ∈ L ( H n ) such that f =( I + L ) − s/ f . The inner product on W s, ( H n ) is given by (cid:104) f, g (cid:105) s = (cid:104) ( I + L ) s/ f, ( I + L ) s/ g (cid:105) = (cid:104) f , g (cid:105) HERNOFF AND INGHAM ON THE HEISENBERG GROUP 9 where (cid:104) f, g (cid:105) is the inner product in L ( H n ) . This has the following interesting consequence.Given f ∈ W s, ( H n ) and g ∈ W − s, ( H n ) , let f , g ∈ L ( H n ) be such that f = ( I + L ) − s/ f and g = ( I + L ) s/ g . The duality bracket ( f, g ) defined by( f, g ) = (cid:104) ( I + L ) − s/ f , ( I + L ) s/ g (cid:105) = (cid:104) f , g (cid:105) allows us to identify the dual of W s, ( H n ) with W − s, ( H n ) . This is also clear from theidentification of W s, ( H n ) with W s, ( (cid:98) H n ) . Thus for every g ∈ W − s, ( H n ) there is a linearfunctional Λ g : W s, ( H n ) → C given by Λ g ( f ) = (cid:104) f , g (cid:105) . The following observation is also very useful in applications. For s > f ∈ W s, ( H n ) defines a distribution on H n . The same is true for every g ∈ W − s, ( H n ) aswell. To see this, consider the map taking f ∈ C ∞ c ( H n ) into the duality bracket ( f, g ) whichsatisfies | ( f, g ) | ≤ (cid:107) f (cid:107) ( s ) (cid:107) g (cid:107) ( − s ) ≤ (cid:107) g (cid:107) ( − s ) (cid:107) ( I + L ) m f (cid:107) where m > s/ g ( f ) = ( f, g ) is a distribution.If g ∈ W − s, ( H n ) is such a distribution, it is possible to define its Fourier transform asan unbounded operator valued function on R ∗ . Indeed, let g ∈ L ( H n ) be such that g =( I + L ) s/ g then we define (cid:98) g ( λ ) = (cid:98) g ( λ )(1 + H ( λ )) s/ which is a densely defined operatorwhose action on Φ λα is given by (cid:98) g ( λ )Φ λα = (1 + (2 | α | + n ) | λ | ) s/ (cid:98) g ( λ )Φ λα . Thus we see that when g ∈ W − s, ( H n ) we have (cid:90) ∞−∞ (cid:88) α ∈ N n (cid:88) β ∈ N n (1 + (2 | α | + n ) | λ | ) − s |(cid:104) (cid:98) g ( λ )Φ λα , Φ λβ (cid:105)| dµ ( λ ) = (cid:90) H n | g ( z, t ) | dzdt < ∞ . (2.3) Remark . When g ∈ W − s, ( H n ) is a compactly supported distribution, then we alreadyhave a definition of (cid:98) g ( λ ) given by (cid:104) (cid:98) g ( λ )Φ λα , Φ λβ (cid:105) = ( g, e λα,β ) , the action of g on the smoothfunction e λα,β ( z, t ) . The two definitions agree as e λα,β are eigenfunctions of L with eigenvalues(2 | α | + n ) | λ | . Chernoff’s theorem on the Heisenberg group
In this section we prove Theorem 1.5 for the sublaplacian on the Heisenberg group. For theproof we need to recall some properties of the so called Stieltjes vectors for the sublaplacian.Let X be a Banach space and A, a linear operator on X with domain D ( A ) ⊂ X. A vector x ∈ X is called a C ∞ - vector or smooth vector for A if x ∈ ∩ ∞ n =1 D ( A n ) . A C ∞ - vector x issaid to be a Stieltjes vector for A if (cid:80) ∞ n =1 (cid:107) A n x (cid:107) − n = ∞ . These vectors were first introducedby Nussbaum [19] and independently by Masson and Mc Clary [17]. We denote the set ofall Stieltjes vector for A by D St ( A ) . The following theorem summarises the interconnection between the theory of Stieltjes vectors and the essential self adjointness of certain class ofoperators.
Theorem 3.1.
Let A be a semibounded symmetric operator on a Hilbert space H . Assumethat the set D St ( A ) has a dense span. Then A is essentially self adjoint. A very nice simplified proof this theorem can be found in Simon [26]. In 1975, P.R.Chernoffused this result to prove an L -version of the classical Denjoy-Carleman theorem regardingquasi-analytic functions on R n . The above theorem talks about essential self adjointness of operators. Let us quicklyrecall some relevant definitions from operator theory. By an operator A on a Hilbert space H we mean a linear mapping whose domain D ( A ) is a subspace of H and whose range Ran ( A ) ⊂ H . We say that an operator S is an extension of A if D ( A ) ⊂ D ( S ) and Sx = Ax for all x ∈ D ( A ). An operator A is called closed if the graph of A defined by G = { ( x, Ax ) : x ∈ D ( A ) } is a closed subset of H × H . We say that an operator A is closableif it has a closed extension. Every closable operator has a smallest closed extension, calledits closure, which we denote by ¯ A. An operator A is said to be densely defined if D ( A ) isdense in H and it is called symmetric if (cid:104) Ax, y (cid:105) = (cid:104) x, Ay (cid:105) for all x, y ∈ D ( A ). A denselydefined symmetric operator A is called essentially self adjoint if its closure ¯ A is self adjoint.It is easy to see that an operator A is essentially self adjoint if and only if A has uniqueself adjoint extension. The following is a very important characterization of essentially selfadjoint operators. Theorem 3.2. ( [22] ) Let A be a positive, densely defined symmetric operator. The followingsare equivalent: (i) A is essentially self adjoint (ii) Ker ( A ∗ + I ) = { } and (iii) Ran ( A + I ) is dense in H . We apply the above theorem to study the essential self adjointness of L considered ona domain inside the Sobolev space W s, ( H n ) , s > . Let A stand for the sublaplacian L restricted to the domain D ( A ) consisting of all smooth functions f such that for all α, β ∈ N n , j ∈ N the derivatives X α Y β T j f are in L ( H n ) and vanish at the origin. Since X j , Y j agree with ∂∂x j , ∂∂y j at the origin, we can also define D ( A ) in terms of ordinary derivatives ∂ αx ∂ βy ∂ jt . Proposition 3.3.
Let A and D ( A ) be defined as above where ( n − < s ≤ ( n + 1) . Then A is not essentially self adjoint.Proof. In view of Theorem 3.2 it is enough to show that for s as in the statement of theproposition, D ( A ) is dense in W s, ( H n ) but ( I + A ) D ( A ) is not. These are proved in thefollowing lemmas. (cid:3) HERNOFF AND INGHAM ON THE HEISENBERG GROUP 11
Lemma 3.4. D ( A ) is dense in W s, ( H n ) for any ≤ s ≤ ( n + 1) . Proof.
If we let Ω = H n \ { } so that C ∞ c (Ω) ⊂ D ( A ) , it is enough to show that the smallerset is dense in W s, ( H n ) . This will follow if we can show that the only linear functional thatannihilates C ∞ c (Ω) is the zero functional (see chapter 3 of [24]). Let Λ ∈ ( W s, ( H n )) (cid:48) , thedual of W s, ( H n ), be such that Λ( C ∞ c (Ω)) = 0 . Then there exists g ∈ W − s, ( H n ) such thatΛ = Λ g and hence Λ g ( φ ) = 0 for any φ ∈ C ∞ c (Ω) Notice that for φ ∈ C ∞ c ( H n ) the linear map φ (cid:26) Λ g ( φ ) defines a distribution. Indeed, the estimate | Λ g ( φ ) | ≤ (cid:107) g (cid:107) ( − s ) (cid:107) φ (cid:107) ( s ) ≤ (cid:107) g (cid:107) ( − s ) (cid:107) ( I + L ) m φ (cid:107) for any integer m > s/ g is a finite linear combination of derivatives of Dirac δ at the origin, Λ g = (cid:80) | a |≤ N c a ∂ a δ, see e.g Chapter 6 of [24].Since X a δ = ∂ ax δ, and Y b δ = ∂ by δ in the above representation we can also use X a Y b T j . It is even more convenient to write them in terms of the complex vector fields defined by Z j = ( X j − iY j ) , Z j = ( X j + iY j ) . Thus we have g = (cid:80) | a | + | b | +2 j ≤ N c a,b,j Z a Z b T j δ. If g ∈ L ( H n ) is such that ( I + L ) − s/ g = g then by (2.3) we have (cid:90) ∞−∞ (cid:88) α ∈ N n (cid:88) β ∈ N n (1 + (2 | α | + n ) | λ | ) − s |(cid:104) (cid:98) g ( λ )Φ λα , Φ λβ (cid:105)| dµ ( λ ) < ∞ . Since g is compactly supported we can calculate the Fourier transform of g as in Remark2.1. In view of the relations 2.2 we have (cid:104) (cid:98) g ( λ )Φ λα , Φ λβ (cid:105) = (cid:88) | a | + | b | +2 j ≤ N c a,b,j λ j (cid:104) A ( λ ) a ( A ( λ ) ∗ ) b Φ λα , Φ λβ (cid:105) By defining m ( λ, α, β ) to be the expression on the right hand side of the above equation wesee that (cid:90) ∞−∞ (cid:88) α ∈ N n (cid:88) β ∈ N n (1 + (2 | α | + n ) | λ | ) − s | m ( λ, α, β ) | | λ | n dλ < ∞ . (3.1)The action of A ( λ ) a and ( A ( λ ) ∗ ) b on Φ λα are explicitly known, see ([31]). It is therefore easyto see that m ((2 | α | + n ) − λ, α, β ) = (cid:88) | a | + | b | +2 j ≤ N C a,b,j ( α, β ) λ j +( | a | + | b | ) / where the coefficients C a,b,j ( α, β ) are uniformly bounded in both variables. We also remarkthat for a given α the function C a,b,j ( α, β ) is non zero only for a single value of β. By makinga change of variables in (3.1) we see that (cid:88) α ∈ N n (cid:88) β ∈ N n (2 | α | + n ) − n − (cid:90) ∞−∞ (cid:0) (cid:88) | a | + | b | +2 j ≤ N C a,b,j ( α, β ) λ j +( | a | + | b | ) / (cid:1) | λ | n (1 + | λ | ) s dλ < ∞ . As we are assuming that 0 ≤ s ≤ ( n + 1) the above integral cannot be finite unless all thecoefficients c a,b,j = 0 . Hence g = 0 proving the density of D ( A ) . (cid:3) Lemma 3.5.
For any s > ( n − , ( I + A ) D ( A ) is not dense in W s, ( H n ) . Proof.
For any f ∈ D ( A ) the inversion formula for the Fourier transform on H n shows that (cid:90) ∞−∞ tr ( (cid:98) f ( λ )) dµ ( λ ) = f (0) = 0 . Let g be the functions defined by (cid:98) g ( λ ) = (1 + H ( λ )) − s − we can rewrite the above as (cid:104) ( I + L ) f, g (cid:105) s = (cid:90) ∞−∞ tr ( (cid:98) f ( λ )) dµ ( λ ) = 0 . So all we need to do is to check g ∈ W s, ( H n ) , or equivalently (cid:90) ∞−∞ (cid:0) ∞ (cid:88) k =0 (1 + (2 k + n ) | λ | ) − s − (cid:107) P k ( λ ) (cid:107) HS (cid:1) | λ | n dλ < ∞ . It is known that (cid:107) P k ( λ ) (cid:107) HS = ( k + n − k !( n − ≤ C (2 k + n ) n − and so by making a change of variablesthe above integral is bounded by ∞ (cid:88) k =0 (2 k + n ) − (cid:90) ∞−∞ (1 + | λ | ) − s − | λ | n dλ. As we assume that s > ( n −
1) the integral is finite which proves that g ∈ W s, ( H n ) . Hencethe lemma. (cid:3)
We now proceed to investigate some properties of the set D St ( A ) of Stieltjes vectors forthe operator A. The following lemma about series of real numbers will be helpful in provingsome properties of Stieltjes vectors for the sublaplacian (see lemma 3.2 of [7]).
Lemma 3.6. If { M n } n is sequence of non-negetive real numbers such that (cid:80) ∞ n =1 M − n n = ∞ and ≤ K n ≤ aM n + b n , then (cid:80) ∞ n =1 K − n n = ∞ . For r > f is defined by by δ r f ( z, t ) = f ( rz, r t ) and for σ ∈ U ( n ) we define the rotation R σ f ( z, t ) = f ( σz, t ) for all ( z, t ) ∈ H n . Lemma 3.7.
Suppose f ∈ D ( A ) satisfies the condition (cid:80) ∞ m =0 (cid:107)L m f (cid:107) − m = ∞ . Then f ∈ D St ( A ) . Moreover, δ r f, R σ f are also Stieltjes vectors for A. Proof.
We first recall that
L ◦ δ r = r δ r ◦ L and L ◦ R σ = R σ ◦ L , see e.g [30]. Therefore, itfollows that if a function satisfies (cid:80) ∞ m =0 (cid:107)L m f (cid:107) − m = ∞ then the same is true of δ r f and R σ f. So we only need to prove our claim for f ; i.e., when f satisfies the above condition thenwe also have (cid:80) ∞ m =0 (cid:107)L m f (cid:107) − m ( s ) = ∞ . To see this, we use (cid:104)L m f, L m f (cid:105) ( s ) = (cid:104)L m f, (1 + L ) s f (cid:105) ≤ (cid:107) ( I + L ) s f (cid:107) (cid:107)L m f (cid:107) . HERNOFF AND INGHAM ON THE HEISENBERG GROUP 13
Thus we have (cid:107)L m f (cid:107) − m ( s ) ≥ C − m (cid:107)L m f (cid:107) − m where C = (cid:107) ( I + L ) s f (cid:107) . In view of Lemma3.6 it is enough to prove the divergence of (cid:80) ∞ m =0 (cid:107)L m f (cid:107) − m . Without loss of generality wecan assume that (cid:107) f (cid:107) = 1 . But then (cid:107)L m f (cid:107) − m is a decreasing function of m, see Lemma2.1 in [7]. Consequently, the required divergence follows from the assumption on f. (cid:3) Before stating the next lemma, let us recall some properties of the matrix coefficients e λα,β ( z, t ) = (cid:104) π λ ( z, t )Φ λα , Φ λβ ) of the Schr¨odinger representations. These are eigenfunctions ofthe sublaplacian with eigenvalues (2 | α | + n ) | λ | . Moreover, they satisfy Z j e λα,β = i (2 α j + 2) | λ | e λα + e j ,β , Z j e λα,β = i (2 α j ) | λ | e λα − e j ,β (3.2)where e j are the coordinate vectors in C n . We also recall that the sublaplacian is expressedas L = − (cid:80) nj =1 ( Z j Z j + Z j Z j ) in terms of Z j and Z j . Lemma 3.8. If f satisfies the hypothesis in Lemma 3.7 , then e λα,β f ∈ D St ( A ) , for any α, β ∈ N n and λ ∈ R ∗ . Proof.
As noted in the previous lemma, it suffices to show that (cid:80) ∞ m =1 (cid:107)L m ( e λα,β f ) (cid:107) − m = ∞ .Since L = − (cid:80) nj =1 ( Z j Z j + Z j Z j ) in terms of Z j , a simple calculation shows that L ( f g ) = ( L f ) g + f ( L g ) − n (cid:88) j =1 (cid:0) Z j f ¯ Z j g + ¯ Z j f Z j g (cid:1) . (3.3)By taking g = e λα,β and making use of (3.2) along with the estimate (cid:107) e λα,β (cid:107) ∞ ≤ (cid:107)L ( f e λα,β ) (cid:107) is bounded by (cid:107)L f (cid:107) + (2 | α | + n ) | λ |(cid:107) f (cid:107) + 1 √ n (cid:88) j =1 (cid:0)(cid:113) ( α j + 1) | λ |(cid:107) Z j f (cid:107) + (cid:113) α j | λ |(cid:107) ¯ Z j f (cid:107) (cid:1) . As the operators Z j L − / and ¯ Z j L − / are bounded on L ( H n ) with norms at most √
2, wesee that the third term above can be estimated as n (cid:88) j =1 (cid:0)(cid:113) ( α j + 1) | λ | + (cid:113) α j | λ | (cid:1) (cid:107)L / f (cid:107) ≤ n (cid:88) j =1 (cid:0) (2 α j + 1) | λ | (cid:1) (cid:107)L / f (cid:107) . Finally using the fact that (cid:107)L / (1 + L ) − f (cid:107) ≤ (cid:107) f (cid:107) we get the estimate (cid:107)L ( f e λα,β ) (cid:107) ≤ (2 | α | + n ) | λ | (2 (cid:107)L f (cid:107) + 3 (cid:107) f (cid:107) ) + (cid:107)L f (cid:107) . By defining a λ ( α ) = (2 | α | + n ) | λ | ) , b λ ( α ) = (2 | α | + n + 1) | λ | ) and c λ ( α ) = 3 b λ ( α ) + 1 , werewrite the above as (cid:107)L ( f e λα,β ) (cid:107) ≤ c λ ( α ) (cid:0) (cid:107)L f (cid:107) + (cid:107) f (cid:107) (cid:1) . In order to prove the lemma it is enough to show for any non-negative integer m thefollowing estimate holds: (cid:107)L m ( f e λα,β ) (cid:107) ≤ m − c λ ( α ) m (cid:0) (cid:107)L m f (cid:107) + (cid:107) f (cid:107) ) . (3.4)We prove this by induction. Assuming the result for any m, we write L m +1 ( f g ) = L m L ( f g )and make use of (3.3) with g = e λα,β . The first two terms L m ( L f g ) and L m ( f L g ) togethergive the estimate2 m − c λ ( α ) m (cid:0) (cid:107)L m +1 f (cid:107) + (cid:107)L f (cid:107) (cid:1) + a λ ( α )2 m − c λ ( α ) m ( (cid:107)L m f (cid:107) + (cid:107) f (cid:107) (cid:1) . The boundedness of L (1 + L m +1 ) − and L m (1 + L m +1 ) − allows us to bound the above by2 m c λ ( α ) m (1 + b λ ( α )) (cid:0) (cid:107)L m +1 f (cid:107) + (cid:107) f (cid:107) (cid:1) . (3.5)We now turn our attention to the estimation of the term1 √ n (cid:88) j =1 (cid:0)(cid:113) ( α j + 1) | λ |L m ( Z j f e λα + e j ,β ) + (cid:113) α j | λ |L m ( e λα − e j ,β ¯ Z j f ) (cid:1) . By using the induction hypothesis along with the fact that the operators L m Z j (1 + L m +1 ) − and L m ¯ Z j (1 + L m +1 ) − are bounded with norm at most √ L norm of the above isbounded by2 m − n (cid:88) j =1 (cid:0) c λ ( α + e j ) m (cid:113) ( α j + 1) | λ | + c λ ( α − e j ) m (cid:113) α j | λ | (cid:1)(cid:0) (cid:107)L m +1 f (cid:107) + (cid:107) f (cid:107) (cid:1) . Since b λ ( α + e j ) ≤ b λ ( α ) , we have c λ ( α + e j ) ≤ c λ ( α ) , and so the above sum is bounded by2 a λ ( α )2 m c λ ( α ) m (cid:0) (cid:107)L m +1 f (cid:107) + (cid:107) f (cid:107) (cid:1) . (3.6)Combining (3.5) and (3.6), using a λ ( α ) ≤ b λ ( α ) and recalling the definition of c λ we obtain(3.4), proving the lemma. (cid:3) Proposition 3.9.
Let A be as in Proposition 3.3 where we have assumed that ( n − < s ≤ ( n + 1) . Assume that D St ( A ) contains a nonzero element f such that δ r f and R σ f are alsoin D St ( A ) for all r > and σ ∈ U ( n ) . Then the linear span of D St ( A ) is dense in W s, ( H n ) . Proof.
Let f be a nonzero member of D St ( A ) . We will show that the closed linear span of D St ( A ) equals W s, ( H n ) . To prove this, let us take g ∈ W s, ( H n ) which is orthogonal to D St ( A ) . By Lemma 3.8 we know that f e λα,β ∈ D St ( A ) for all α, β ∈ N n and λ ∈ R ∗ . Thus, (cid:104) ( I + L ) s g, e λαβ f (cid:105) L = (cid:104) g, e λαβ f (cid:105) ( s ) = 0 . By defining p ( z, t ) = f ( z, t )( I + L ) s g ( z, t ), the above translates into (cid:104) (cid:98) p ( λ )Φ λα , Φ λβ (cid:105) = (cid:90) H n p ( z, t )( π λ ( z, t )Φ λα , Φ λβ ) dzdt = 0 . HERNOFF AND INGHAM ON THE HEISENBERG GROUP 15
By the inversion formula for the Fourier transform on H n we conclude that p = 0 whichmeans (1 + L ) s g vanishes on the support of f . Under the assumption on f it follows that(1 + L ) s g vanishes identically which forces g = 0 as the operator (1 + L ) s is invertible. Thisproves the density. (cid:3) Finally, we are in a position to prove the analogue of Chernoff’s theorem for the sublapla-cian on the Heisenberg group.
Proof of Theorem 1.5
Consider the operator A defined in Proposition 3.3. We havealready shown that it is not essentially self adjoint. Suppose there exists a nontrivial f satisfying the hypothesis of Theorem 1.5. Then by Lemma 3.7 we know that f along with δ r f and R σ f belong to D St ( A ) . But then by Proposition 3.9 we know that the linear spanof D St is dense in W s, ( H n ) . By Theorem 3.1 this allows us to conclude that A is essentiallyself-adjoint. As this is not the case, f has to be trivial which proves the theorem.4. Ingham’s theorem on the Heisenberg group
In this section we prove Theorem 1.3 using Chernoff’s theorem for the sublaplacian. Wefirst show the existence of a compactly supported function f on H n whose Fourier transformhas a prescribed decay as stated in Theorem 1.3. This proves the sufficiency part of thecondition on the function Θ appearing in the hypothesis. We then use this part of thetheorem to prove the necessity of the condition on Θ . We begin with some preparations.4.1.
Construction of F . The Koranyi norm of x = ( z, t ) ∈ H n is defined by | x | = | ( z, t ) | =( | z | + t ) . In what follows, we work with the following left invariant metric defined by d ( x, y ) := | x − y | , x, y ∈ H n . Given a ∈ H n and r >
0, the open ball of radius r with centreat a is defined by B ( a, r ) := { x ∈ H n : | a − x | < r } . With this definition, we note that if f, g : H n → C are such that supp( f ) ⊂ B (0 , r ) andsupp( g ) ⊂ B (0 , r ), then we havesupp( f ∗ g ) ⊂ B (0 , r ) .B (0 , r ) ⊂ B (0 , r + r ) , where f ∗ g ( x ) = (cid:82) H n f ( xy − ) g ( y ) dy is the convolution of f with g. Suppose { ρ j } j and { τ j } j are two sequences of positive real numbers such that both theseries (cid:80) ∞ j =1 ρ j and (cid:80) ∞ j =1 τ j are convergent. We let B C n (0 , r ) stand for the ball of radius r centered at 0 in C n and let χ S denote the characteristic function of a set S. For each j ∈ N , we define functions f j on C n and τ j on R by f j ( z ) := ρ − nj χ B C n (0 ,aρ j ) ( z ) , g j ( t ) := τ − j χ [ − τ j / ,τ j / ( t ) where the positive constant a is chosen so that (cid:107) f j (cid:107) L ( C n ) = 1 . We now consider the functions F j : H n → C defined by F j ( z, t ) := f j ( z ) g j ( t ) , ( z, t ) ∈ H n . In the following lemma, we record some useful, easy to prove, properties of these functions.
Lemma 4.1.
Let F j be as above and define G N = F ∗ F ∗ ..... ∗ F N . Then we have(1) (cid:107) F j (cid:107) L ∞ ( H n ) ≤ ρ − nj τ − j , (cid:107) F j (cid:107) L ( H n ) = 1 , (2) supp ( F j ) ⊂ B C n (0 , aρ j ) × [ − τ j / , τ j / ⊂ B (0 , aρ j + cτ j ) , where c = 1 . (3) For any N ∈ N , supp ( G N ) ⊂ B (0 , a (cid:80) Nj =1 ρ j + c (cid:80) Nj =1 τ j ) . (4) Given x ∈ H n and N ∈ N , F ∗ F ..... ∗ F N ( x ) ≤ ρ − n τ − . We also recall a result about Hausd¨orff measure which will be used in the proof of thenext theorem. Let H n ( A ) denote the n -dimensional Hausdorff measure of A ⊂ R n . Hausd¨orffmeasure coincides with the Lebesgue measure for Lebesgue measurable sets. For sets in R n with sufficiently nice boundaries, the ( n − A ∆ B stand for thesymmetric difference between any two sets A and B. See [25] for a proof of the followingtheorem.
Theorem 4.2.
Let A ⊂ R n be a bounded set. Then for any ξ ∈ R n H n ( A ∆( A + ξ )) ≤ | ξ |H n − ( ∂A ) where A + ξ is the translation of A by ξ and ∂A is the boundary of A. Theorem 4.3.
The sequence defined by G k = F ∗ F ∗ ..... ∗ F k converges to a compactlysupported F ∈ L ( H n ) . Proof.
In order show that ( G k ) is Cauchy in L ( H n ) we first estimate (cid:107) G k +1 − G k (cid:107) L ∞ ( H n ) . As all the functions F j have unit L norm, for any x ∈ H n we have G k +1 ( x ) − G k ( x ) = (cid:90) H n G k ( xy − ) F k +1 ( y ) dy − G k ( x )( x ) (cid:90) H n F k +1 ( y ) dy = (cid:90) H n (cid:0) G k ( xy − ) − G k ( x ) (cid:1) F k +1 ( y ) dy. As F j are even we can change y into y − in the above and estimate the same as | G k +1 ( x ) − G k ( x ) | ≤ (cid:90) H n | G k ( xy ) − G k ( x ) | F k +1 ( y ) dy. (4.1)By defining H k − = F ∗ F ...... ∗ F k so that G k = F ∗ H k − , we have G k ( xy ) − G k ( y ) = (cid:90) H n (cid:0) F ( xyu − ) − F ( xu − ) (cid:1) H k − ( u ) du HERNOFF AND INGHAM ON THE HEISENBERG GROUP 17
Using the estimate in Lemma 4.1 (4) we now estimate | G k ( xy ) − G k ( x ) | ≤ ρ − n τ − (cid:90) H n (cid:12)(cid:12) F ( xyu − ) − F ( xu − ) (cid:12)(cid:12) du. (4.2)The change of variables u → ux transforms the integral in the right hand side of the aboveequation into (cid:90) H n (cid:12)(cid:12) F ( xyu − ) − F ( xu − ) (cid:12)(cid:12) du = (cid:90) H n (cid:12)(cid:12) F ( xyx − u − ) − F ( u − ) (cid:12)(cid:12) du. Since the group H n is unimodular, another change of variables u → u − yields (cid:90) H n (cid:12)(cid:12) F ( xyx − u − ) − F ( u − ) (cid:12)(cid:12) du = (cid:90) H n (cid:12)(cid:12) F ( xyx − u ) − F ( u ) (cid:12)(cid:12) du. Let x = ( z, t ) = ( z, , t ) , y = ( w, s ) = ( w, , s ) . As (0 , t ) and (0 , s ) belong to thecenter of H n , an easy calculation shows that xyx − = ( w, , s + (cid:61) ( z · ¯ w )) . With u = ( ζ, τ )we have xyx − u = ( w + ζ, , τ + s + (cid:61) ( z · ¯ w ) − (1 / (cid:61) ( ζ · ¯ w )) . Since F ( z, t ) = f ( z ) g ( t ) we see that the integrand F ( xyx − u ) − F ( u ) in the above integraltakes the form f ( w + ζ ) g ( τ + s + (cid:61) ( z · ¯ w ) − (1 / (cid:61) ( ζ · ¯ w )) − f ( ζ ) g ( τ ) . By setting b = b ( s, z, w, ζ ) = s + (cid:61) ( z · ¯ w ) − (1 / (cid:61) ( ζ · ¯ w ) we can rewrite the above as (cid:0) f ( w + ζ ) − f ( ζ ) (cid:1) g ( τ + b ) + f ( ζ ) (cid:0) g ( τ + b ) − g ( τ ) (cid:1) . (4.3)In order to estimate the contribution of the second term to the integral under considerationwe first estimate the τ integral as follows: (cid:90) ∞−∞ | g ( τ + b ) − g ( τ ) | dτ = τ − | ( − b + K τ )∆ K τ | where K τ = [ − τ , τ ] is the support of g . For ζ in the support of f , we have | ζ | ≤ aρ and hence | ( − b + K τ )∆ K τ | ≤ | b ( z, w, ζ ) | ≤ (2 | s | + | z || w | + aρ | w | ) . Thus we have proved the estimate (cid:90) H n f ( ζ ) | g ( τ + b ) − g ( τ ) | dζdτ ≤ C (cid:0) | s | + ( aρ + | z | ) | w | (cid:1) (4.4)As g integrates to one, the contribution of the first term in (4.3) is given by (cid:90) C n | f ( w + ζ ) − f ( ζ ) | dζ = ρ − n H n (( − w + B C n (0 , aρ ))∆ B C n (0 , aρ )) . By appealing to Theorem 4.2 in estimating the above, we obtain (cid:90) H n | f ( w + ζ ) − f ( ζ ) | g ( τ + b ) dζdτ ≤ C | w | . (4.5) Using the estimates (4.4) and (4.5) in (4.2) we obtain | G k ( xy ) − G k ( x ) | ≤ Cρ − n τ − (cid:0) | s | + ( c + c | z | ) | w | ) (cid:1) . This estimate, when used in (4.1), in turn gives us | G k +1 ( z, t ) − G k ( z, t ) | ≤ C (cid:90) H n (cid:0) | s | + ( c + c | z | ) | w | ) (cid:1) F k +1 ( w, s ) dw ds (4.6)where the constants c , c and C depend only on n. Recalling that on the support of F k +1 ( w, s ) = f k +1 ( w ) g k +1 ( s ), | w | ≤ ρ k +1 and | s | ≤ τ k +1 , the above yields the estimate | G k +1 ( z, t ) − G k ( z, t ) | ≤ C (cid:0) τ k +1 + ( c + c | z | ) ρ k +1 (cid:1) . (4.7)It is easily seen that the support of G k +1 − G k is contained in B (0 , aρ + cτ ) where ρ = (cid:80) ∞ j =1 ρ j and τ = (cid:80) τ j . Consequently, from the above we conclude that (cid:107) G k +1 − G k (cid:107) ≤ (cid:107) G k +1 − G k (cid:107) ∞ | B (0 , aρ + cτ ) | ≤ C (cid:0) τ k +1 + c ρ k +1 (cid:1) . From the above, it is clear that G k is Cauchy in L ( H n ) and hence converges to a function F ∈ L ( H n ) whose support is contained in B (0 , aρ + cτ ) . (cid:3) Estimating the Fourier transform of F . Suppose now that Θ is an even, decreas-ing function on R for which (cid:82) ∞ Θ( t ) t − dt < ∞ . We want to choose two sequences ρ j and τ j in terms of Θ so that the series (cid:80) ∞ j =1 ρ j and (cid:80) ∞ j =1 τ j both converge. We can then con-struct a function F as in Theorem 4.3 which will be compactly supported. Having donethe construction we now want to compute the Fourier transform of the constructed function F and compare it with e − Θ( √ H ( λ )) √ H ( λ ) . This can be achieved by a judicious choice of thesequences ρ j and τ j . As Θ is given to be decreasing it follows that (cid:80) ∞ j =1 Θ( j ) j < ∞ . It is thenpossible to choose a decreasing sequence ρ j such that ρ j ≥ c n e j ) j (for a constant c n to bechosen later) and (cid:80) ∞ j =1 ρ j < ∞ . Similarly, we choose another decreasing sequence τ j suchthat (cid:80) ∞ j =1 τ j < ∞ . In the proof of the following theorem we require good estimates for the Laguerre coefficientsof the function f j ( z ) = ρ − nj χ B C n (0 ,aρ j ) ( z ) where a chosen so that (cid:107) f j (cid:107) = 1 . These coefficientsare defined by R n − k ( λ, f j ) = k !( n − k + n − (cid:90) C n f j ( z ) ϕ n − k,λ ( z ) dz. (4.8) Lemma 4.4.
There exists a constant c n > such that | R n − k ( λ, f j ) | ≤ c n (cid:0) ρ j (cid:112) (2 k + n ) | λ | (cid:1) − n +1 / . Proof.
By abuse of notation we write ϕ n − k,λ ( r ) in place of ϕ n − k,λ ( z ) when | z | = r. As f j isdefined as the dilation of a radial function, the Laguerre coefficients are given by the integral R n − k ( λ, f j ) = 2 π n Γ( n ) k !( n − k + n − (cid:90) a ϕ n − k,λ ( ρ j r ) r n − dr. (4.9) HERNOFF AND INGHAM ON THE HEISENBERG GROUP 19
When a ≤ ( ρ j (cid:112) (2 k + n ) | λ | ) − we use the bound k !( n − k + n − | ϕ n − k,λ ( r ) | ≤ π n Γ( n ) k !( n − k + n − (cid:90) a ϕ n − k,λ ( ρ j r ) r n − dr ≤ π n a n +1 / Γ( n + 1) (cid:0) ρ j (cid:112) (2 k + n ) | λ | (cid:1) − n +1 / . When a > ( ρ j (cid:112) (2 k + n ) | λ | ) − we split the integral into two parts, one of which gives thesame estimate as above. To estimate the integral taken over ( ρ j (cid:112) (2 k + n ) | λ | ) − < r < a, we use the bound stated in Lemma 2.1 which leads to the estimate2 π n Γ( n ) k !( n − k + n − (cid:90) a ( ρ j √ (2 k + n ) | λ | ) − ϕ n − k,λ ( ρ j r ) r n − dr ≤ C n (cid:0) ρ j (cid:112) (2 k + n ) | λ | (cid:1) − n +1 / (cid:90) a r n − / dr = C (cid:48) n a n +1 / (cid:0) ρ j (cid:112) (2 k + n ) | λ | (cid:1) − n +1 / . Combining the two estimates we get the lemma. (cid:3)
Theorem 4.5.
Let
Θ : R → [0 , ∞ ) be an even, decreasing function with lim λ →∞ Θ( λ ) = 0 for which (cid:82) ∞ λ ) λ dλ < ∞ . Let ρ j and τ j be chosen as above. Then the Fourier transform ofthe function F constructed in Theorem 4.3 satisfies the estimate (cid:98) F ( λ ) ∗ (cid:98) F ( λ ) ≤ e − √ H ( λ )) √ H ( λ ) , λ (cid:54) = 0 . Proof.
Observe that F is radial since each F j is radial and hence the Fourier transform (cid:98) F ( λ )is a function of the Hermite opertaor H ( λ ) . More precisely, (cid:98) F ( λ ) = ∞ (cid:88) k =0 R n − k ( λ, F ) P k ( λ ) (4.10)where the Laguerre coefficients are explicitly given by (see (2.4.7) in [31]. There is a typo-the factor | λ | n/ should not be there) R n − k ( λ, F ) = k !( n − k + n − (cid:90) C n F λ ( z ) ϕ n − k,λ ( z ) dz. In the above, F λ ( z ) stands for the inverse Fourier transform of F ( z, t ) in the t variable.Expanding any ϕ ∈ L ( R n ) in terms of Φ λα it is easy to see that the conclusion (cid:98) F ( λ ) ∗ (cid:98) F ( λ ) ≤ e − √ H ( λ )) √ H ( λ ) follows once we show that( R n − k ( λ, F )) ≤ Ce − √ (2 k + n ) | λ ) √ (2 k + n ) | λ | for all k ∈ N and λ ∈ R ∗ . Now note that, by definition of g j and the choice of a, we have | (cid:98) g j ( λ ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin( τ j λ ) τ j λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ , | R n − k ( λ, f j ) | ≤ . The bound on R n − k ( λ, f j ) follows from the fact that | ϕ λk ( z ) | ≤ ( k + n − k !( n − . Since F is constructedas the L limit of the N -fold convolution G N = F ∗ F ...... ∗ F N we observe that for any N ( R n − k ( λ, F )) ≤ ( R n − k ( λ, G N )) = (Π Nj =1 R n − k ( λ, F j )) and hence it is enough to show that for a given k and λ one can choose N = N ( k, λ ) in sucha way that (Π Nj =1 R n − k ( λ, F j )) ≤ Ce − √ (2 k + n ) | λ ) √ (2 k + n ) | λ | . (4.11)where C is independent of N. From the definition of G N it follows that (cid:100) G N ( λ ) = Π Nj =1 (cid:99) F j ( λ ) = Π Nj =1 (cid:0) ∞ (cid:88) k =0 R n − k ( λ, F j ) P k ( λ ) (cid:1) and hence R n − k ( λ, G N ) = Π Nj =1 R n − k ( λ, F j ) . As F j ( z, t ) = f j ( z ) g j ( t ), we have R n − k ( λ, G N ) = (cid:0) Π Nj =1 (cid:98) g j ( λ ) (cid:1)(cid:0) Π Nj =1 R n − k ( λ, f j ) (cid:1) . As the first factor is bounded by one, it is enough to consider the product Π Nj =1 R n − k ( λ, f j ) . We now choose ρ j satisfying ρ j ≥ c n e j ) j where c n is the same constant appearing inLemma 4.4. We then take N = (cid:98) Θ(((2 k + n ) | λ | ) )((2 k + n ) | λ | ) (cid:99) and considerΠ Nj =1 R n − k ( λ, f j ) ≤ Π Nj =1 c n ( ρ j (cid:112) (2 k + n ) | λ | ) − n +1 / where we have used the estimates proved in Lemma 4.4. As ρ j is decreasingΠ Nj =1 c n ( ρ j (cid:112) (2 k + n ) | λ | ) − n +1 / ≤ c Nn (cid:0) ρ N (cid:112) (2 k + n ) | λ | (cid:1) − ( n − / N . (4.12)By the choice of ρ j it follows that ρ N (2 k + n ) | λ | ≥ c n e Θ( N ) N (2 k + n ) | λ | . As Θ is decreasing and N ≤ (cid:112) (2 k + n ) | λ | ) we have Θ( N ) ≥ Θ( (cid:112) (2 k + n ) | λ | ) and soΘ( N ) (2 k + n ) | λ | ≥ Θ (cid:0)(cid:112) (2 k + n ) | λ | (cid:1) (2 k + n ) | λ | ≥ N which proves that ρ N (2 k + n ) | λ | ≥ c n e . Using this in (4.12) we obtainΠ Nj =1 c n (cid:0) ρ j (cid:112) (2 k + n ) | λ | (cid:1) − n +1 / ≤ ( c n e ) − ( n − N e − N . Finally, as N + 1 ≥ Θ(((2 k + n ) | λ | ) )((2 k + n ) | λ | ) , we obtain the estimate (4.11). (cid:3) Ingham’s theorem.
We can now prove Theorem 1.3. Since half of the theorem hasbeen already proved, we only need to prove the following.
Theorem 4.6.
Let
Θ : R → [0 , ∞ ) be an even, decreasing function with lim | λ |→∞ Θ( λ ) = 0 and I = (cid:82) ∞ Θ( λ ) λ − dλ = ∞ . Suppose the Fourier transform of f ∈ L ( H n ) satisfies ˆ f ( λ ) ∗ ˆ f ( λ ) ≤ e − Θ( √ H ( λ )) √ H ( λ ) , λ (cid:54) = 0 . If f vanishes on a non-empty open set, then f = 0 a.e. HERNOFF AND INGHAM ON THE HEISENBERG GROUP 21
Proof.
Without loss of generality we can assume that f vanishes on B (0 , δ ) . First we assumethat Θ( λ ) ≥ | λ | − , | λ | ≥ . In view of Plancherel theorem for the group Fourier transformon the Heisenberg group we have (cid:107)L m f (cid:107) = (2 π ) − ( n +1) (cid:90) ∞−∞ (cid:107) ˆ f ( λ ) H ( λ ) m (cid:107) HS | λ | n dλ. Using the formula for Hilbert-Schmidt norm of an operator we have (cid:107)L m f (cid:107) = (2 π ) − ( n +1) (cid:90) ∞−∞ (cid:88) α ((2 | α | + n ) | λ | ) m (cid:107) ˆ f ( λ )Φ λα (cid:107) | λ | n dλ Now the given condition on the Fourier transform leads to the estimate (cid:107)L m f (cid:107) ≤ C (cid:90) ∞−∞ (cid:88) α ((2 | α | + n ) | λ | ) m e − Θ(((2 | α | + n ) | λ | ) )((2 | α | + n ) | λ | ) | λ | n dλ ≤ C ∞ (cid:88) k =0 (2 k + n ) n − (cid:90) ∞−∞ ((2 k + n ) | λ | ) m e − Θ(((2 k + n ) | λ | ) )((2 k + n ) | λ | ) | λ | n dλ Now changing the variable from λ to (2 k + n ) − λ we get (cid:107)L m f (cid:107) ≤ C ∞ (cid:88) k =0 (2 k + n ) − (cid:90) ∞ λ m + n e − Θ( √ λ ) √ λ dλ. The integral I appearing above can be estimated as follows. Under the extra assumptionΘ( λ ) ≥ | λ | − , on Θ we have I = (cid:90) m λ m + n e − Θ( √ λ ) √ λ dλ + (cid:90) ∞ m λ m + n e − Θ( √ λ ) √ λ dλ ≤ m n +1) (cid:90) m λ m − e − Θ( m ) λ dλ + 4 (cid:90) ∞ m λ m +4( n +1) − e − λ dλ. The above is dominated by a sum of two gamma integrals which can be evaluated to get I ≤ m n +1) Γ(4 m )Θ( m ) − m + 4 e − m Γ(8 m + 4( n + 1)) . Using Stirling’s formula (see Ahlfors [1]) Γ( x ) = √ πx x − / e − x e θ ( x ) / x , < θ ( x ) < x > , we observe the the second term in the estimate for I goes to zero as m tends toinfinity and the first term (and hence I itself ) is bounded by C (4 m ) m Θ( m ) − m . Thus we have proved the estimate (cid:107)L m f (cid:107) ≤ C (4 m ) m Θ( m ) − m . The hypothesis on Θnamely, (cid:82) ∞ t ) t dt = ∞ , by a change of variable implies that (cid:82) ∞ y ) y dy = ∞ . Hence byintegral test we get (cid:80) ∞ m =1 Θ( m ) m = ∞ . Therefore, it follows that (cid:80) ∞ m =1 (cid:107)L m f (cid:107) − m = ∞ . Since it vanishes on B (0 , δ ), f and all its partial derivatives vanish at the origin. Therefore,by Chernoff’s theorem for the sublaplacian we conclude that f = 0 . Now we consider thegeneral case.
The function Ψ( y ) = (1 + | y | ) − / satisfies (cid:82) ∞ y ) y dy < ∞ . By Theorem 4.3 we canconstruct a radial function F ∈ L ( H n ) supported in B (0 , δ/
2) such thatˆ F ( λ ) ∗ ˆ F ( λ ) ≤ e − Ψ( √ H ( λ )) √ H ( λ ) , λ (cid:54) = 0 . As f is assumed to vanish on B (0 , δ ) , the function h = f ∗ F vanishes on the smaller ball B (0 , δ/ . This can be easily verified by looking at f ∗ F ( x ) = (cid:90) H n f ( xy − ) F ( y ) dy = (cid:90) B (0 , δ ) f ( xy − ) F ( y ) dy. When both x, y ∈ B (0 , δ/ , d (0 , xy − ) = | xy − | ≤ | x | + | y | < δ and hence f ( xy − ) = 0proving that f ∗ F ( x ) = 0 . The same is true for all the derivatives of h. We now claim that (cid:98) h ( λ ) ∗ (cid:98) h ( λ ) ≤ e − √ H ( λ )) √ H ( λ ) where Φ( y ) = Θ( y ) + Ψ( y ) . As (cid:98) h ( λ ) = (cid:98) f ( λ ) (cid:98) F ( λ ), for any ϕ ∈ L ( R n ) we have (cid:104) (cid:98) h ( λ ) ∗ (cid:98) h ( λ ) ϕ, ϕ (cid:105) = (cid:104) (cid:98) f ( λ ) ∗ (cid:98) f ( λ ) (cid:98) F ( λ ) ϕ, (cid:98) F ( λ ) ϕ (cid:105) . The hypothesis on f gives us the estimate (cid:104) (cid:98) f ( λ ) ∗ (cid:98) f ( λ ) (cid:98) F ( λ ) ϕ, (cid:98) F ( λ ) ϕ (cid:105) ≤ C (cid:104) e − √ H ( λ )) √ H ( λ ) (cid:98) F ( λ ) ϕ, (cid:98) F ( λ ) ϕ (cid:105) . As F is radial, (cid:98) F ( λ ) commutes with any function of H ( λ ) and hence the right hand side canbe estimated using the decay of (cid:98) F ( λ ): (cid:104) (cid:98) F ( λ ) ∗ (cid:98) F ( λ ) e − Θ( √ H ( λ )) √ H ( λ ) ϕ, e − Θ( √ H ( λ )) √ H ( λ ) ϕ (cid:105) ≤ C (cid:104) e − √ H ( λ )) √ H ( λ ) ϕ, ϕ (cid:105) . This proves our claim on (cid:98) h ( λ ) with Φ = Θ + Ψ . As Φ( y ) ≥ | y | − / , by the already provedpart of the theorem we conclude that h = 0 . In order to conclude that f = 0 we proceed asfollows.Given F as above, let us consider δ r F ( z, t ) = F ( rz, r t ) . It has been shown elsewhere (seee.g. [13]) that (cid:100) δ r F ( λ ) = r − (2 n +2) d r ◦ (cid:98) F ( r − λ ) ◦ d − r where d r is the standard dilation on R n given by d r ϕ ( x ) = ϕ ( rx ) . The property of thefunction F, namely ˆ F ( λ ) ∗ ˆ F ( λ ) ≤ e − √ H ( λ )) √ H ( λ ) gives us (cid:100) δ r F ( λ ) ∗ (cid:100) δ r F ( λ ) ≤ Cr − n +2) d r ◦ e − √ H ( λ/r )) √ H ( λ/r ) ◦ d − r . Testing against Φ λα we can simplify the right hand side which gives us (cid:100) δ r F ( λ ) ∗ (cid:100) δ r F ( λ ) ≤ Cr − n +2) e − r ( √ H ( λ )) √ H ( λ ) where Ψ r ( y ) = r Ψ( y/r ) . If we let F ε ( x ) = ε − (2 n +2) δ − ε F ( x ) then it follows that F ε is anapproximate identity. Moreover, F ε is supported in B (0 , εδ ) and satisfies the same hypothesisas F with Ψ( y ) replaced by ε Ψ( εy ) which has the same integrability and decay conditions. HERNOFF AND INGHAM ON THE HEISENBERG GROUP 23
Hence, working with F ε we can conclude that f ∗ F ε = 0 for any ε > . Letting ε → f ∗ F ε converges to f in L ( H n ) we conclude that f = 0 . This completes theproof. (cid:3)
Remark . We also have a version of Ingham’s Theorem for the Weyl tranform (recall thedefinition from subsection 2.1). With the notation H = H (1) = − ∆ R n + | x | , we can provethe following theorem. Theorem . Let Θ( λ ) be a nonnegative even function on R such that Θ( λ ) decreases tozero when λ → ∞ . There exists a nonzero compactly supported continuous function f on C n whose Weyl transform W ( f ) satisfies the estimate W ( f ) ∗ W ( f ) ≤ Ce − √ H ) √ H (4.13) if and only if the function Θ satisfies (cid:82) ∞ Θ( t ) t − dt < ∞ . The proof of the theorem is very similar to that of Theorem 1.3. In fact, due to theabsence of t -variable the proof of the above theorem is easier and can be obtained by doingsome obvious modification in the proof of Theorem 1.3. Acknowledgments
The authors would like to thank Prof. Swagato K. Ray for his suggestions and discussions.The first author is supported by Inspire faculty award from D. S. T, Govt. of India. Thesecond author is supported by Int. Ph.D. scholarship from Indian Institute of Science. Thethird author is supported by a research fellowship from Indian Statistical Institute. The lastauthor is supported by J. C. Bose Fellowship from D.S.T., Govt. of India as well as a grantfrom U.G.C.
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The Journal ofAnalysis , (2018) 235-244.(S. Bagchi) Department of Mathematics and Statistics, Indian Institute of Science Educa-tion and Research Kolkata, Mohanpur-741246, Nadia, West Bengal, India.
E-mail address : [email protected] (P. Ganguly, S. Thangavelu) Department of Mathematics, Indian Institute of Science, Bangalore-560 012, India.
E-mail address : [email protected], [email protected] (J. Sarkar) Stat-Math Unit, Indian Statistical Institute, 203, B.T. Road, Kolkata-700108,India
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