LOCALIZATION OPERATORS, WIGNER TRANSFORMS AND PARAPRODUCTS

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1 LOCALIZATION OPERATORS, WIGNER TRANSFORMS AND PARAPRODUCTS M. W. WONG Department of Mathematics Statistics York University 47 Keele Street Toronto, Ontario M3J 1P3 Canada Abstract The connections between localization operators, Wigner transforms paraproducts are elucidated in the context of the Weyl- Heisenberg group, the affine group SU(1,1. 1. The Connections Let G be a locally compact Hausdorff group endowed with a left Haar measure µ. Let X be an infinite-dimensional, separable complex Hilbert space in which the inner product norm are denoted by (, respectively. A unitary representation π of G on X is said to be square-integrable if there exists a nonzero element ϕ in X such that (ϕ, π(gϕ dµ(g <. G This is known as the admissibility condition for the square-integrable representation π of G on X, we define the constant c ϕ by c ϕ = (ϕ, π(gϕ dµ(g. G We call c ϕ the wavelet constant associated to the admissible wavelet ϕ. Theorem 1.1 Let π be an irreducible square-integrable representation of G on X. If ϕ is an admissible wavelet for π, then (x, y = 1 (x, π(gϕ(π(gϕ, y dµ(g c ϕ for all x y in X. G

2 M. W. WONG Remark 1. In order to underst the formula in Theorem 1.1 better, let us note that it tells us informally that I = 1 (, π(gϕπ(gϕ dµ(g, c ϕ G where I is the identity operator on X. In other words, the identity operator I on X can be resolved into a sum of rank-one operators 1 c ϕ (, π(gϕπ(gϕ, which are parametrized by elements g in G. Thus, the formula in Theorem 1.1 is known as the resolution of the identity formula. Theorem 1.1 is an abridged version of Theorem 3.1 in the paper [9] by Grossmann, Morlet Paul, where the original contributions of Duflo Moore in [8] are acknowledged. Chapter 14 of the book [7] by Dixmier is devoted to squareintegrable representations. See also the paper [4] by Carey the book [16] by Wong in this connection. Let F L 1 (G L (G. Then for all x X, we define L F,ϕ x by (L F,ϕ x, y = 1 F (g(x, π(gϕ(π(gϕ, y dµ(g c ϕ G for all y in X. Then we have the following boundedness results, which are wellknown in fact very easy to prove. Theorem 1.3 Let F L 1 (G. Then L F,ϕ : X X is a bounded linear operator L F,ϕ 1 c ϕ F L 1 (G, where is the norm in the C -algebra of all bounded linear operators from X into X. Theorem 1.4 Let F L (G. Then L F,ϕ : X X is a bounded linear operator L F,ϕ F L (G. The bounded linear operators L F,ϕ : X X are dubbed localization operators in the paper [1] by He Wong. The impetus for the terminology stems from the simple observation that if F (g = 1 for all g in G, then the resolution of the identity forumla in Theorem 1.1 implies that the corresponding operator is simply the identity operator on X. Therefore the function or the symbol F is there to localize on G so as to produce a nontrivial bounded linear operator on X with various applications in the mathematical sciences. Many deeper properties of localization operators than Theorems can be found in the book [16] by Wong.

3 LOCALIZATION OPERATORS 3 As a prime example, localization operators on the Weyl-Heisenberg group are analyzed. To recall, we let (W H n = R n R n R/πZ. Then we define the binary operation on (W H n by (q 1, p 1, t 1 (q, p, t = (q 1 + q, p 1 + p, t 1 + t + q 1 p for all (q 1, p 1, t 1 (q, p, t in (W H n, where q 1 p is the Euclidean inner product of q 1 p in R n ; t 1, t t 1 +t +q 1 p are cosets in the quotient group R/πZ in which the group law is addition modulo π. It is easy to prove that with respect to the multiplication, (W H n is a non-abelian group in which (,, is the identity element the inverse element of (q, p, t is ( q, p, t + q p for all (q, p, t in (W H n. In fact, (W H n, equipped with the multiplication, is a unimodular group on which the left ( right Haar measure is the Lebesque measure on R n R n [,. It is known as the Weyl-Heisenberg group. Let U(L (R n be the set of all unitary operators on L (R n. Then it becomes a group with respect to the usual composition of mappings. We let π : (W H n U(L (R n be the mapping defined by (π(q, p, tf(x = e i(q x+ 1 q p+t f(x + p for all x in R n, (q, p, t in (W H n f in L (R n. Then it can be shown that π is an irreducible square-integrable representation of (W H n on L (R n, which is called the Schrödinger representation. It is a fact that every function ϕ in L (R n with ϕ L (R n = 1 is an admissible wavelet for π. The wavelet constant c ϕ for every admissible wavelet ϕ is equal to (π n+1. The following result can be found in the book [15] by Wong. Theorem 1.5 Let F be a function in L 1 ((W H n such that F (q, p, t = σ(q, p for all (q, p, t in (W H n. Let ϕ be the function on R n defined by ϕ(x = π n 4 e x, x R n. Then the localization operator L F,ϕ : L (R n L (R n associated to the symbol F the admissible wavelet ϕ is the pseudo-differential operator associated to the symbol σ Λ given by (L F,ϕ u, v L (R n = (π n (σ Λ(x, ξw (u, v(x, ξ dx dξ R n R n

4 4 M. W. WONG for all u v in L (R n, where σ Λ is the convolution of σ Λ, Λ(x, ξ = π n e ( x + ξ, x, ξ R n, W (u, v is the Wigner transform of u v defined by ( W (u, v(x, ξ = (π n e iξ p u x + p v R n for all x ξ in R n. ( x p dp In the case when u = v, W (u, u(x, ξ is the substitute used by Wigner in [14] for the joint probability distribution of position x momentum ξ in the state u in quantum mechanics. That localization operators can be studied in the context of anti-wick quantization is carried out in the book [1] by Boggiatto, Buzano Rodino, the paper [] by Boggiatto Cordero also the book [1] by Shubin. Another canonical example is provided by localization operators on the affine group, which we now recall. Let U be the upper half plane given by U = {(b, a : b R, a > }. Then we define the binary operation on U by (b 1, a 1 (b, a = (b 1 + a 1 b, a 1 a for all (b 1, a 1 (b, a in U. With respect to this multiplication, U is a nonabelian group in which (, 1 is the identity element the inverse element of (b, a is ( b a, a 1 for every (b, a in U. The left right Haar measures on U are given by db da dµ = a db da dν = a respectively. Therefore U is a non-unimodular group, which is called the affine group. Let H +(R be the subspace of L (R defined by H +(R = {f L (R : supp( ˆf [, }, where supp( ˆf is the support of the Fourier transform ˆf of f. The Fourier transform ˆf of a function f in L (R is chosen to be the one defined by 1 R ˆf(ξ = lim e ixξ f(x dx R π R

5 LOCALIZATION OPERATORS 5 for all ξ in R, where the convergence is understood to take place in L (R. We define H (R to be the subspace of L (R by H (R = {f L (R : supp( ˆf (, ]}. H +(R H (R are known as the Hardy space the conjugate Hardy space respectively. They are closed subspaces of L (R. Only the Hardy space H +(R is considered in this paper. Let U(H+(R be the group of all unitary operators on H+(R. Let π : U U(H+(R be the mapping defined by (π(b, af(x = 1 ( x b f a a for all x in R, (b, a in U f in H +(R. Then it can be proved that π is an irreducible square-integrable representation of U on H +(R. The following characterization of admissible wavelets for π is a well-known result in wavelet theory. Theorem 1.6 The set of admissible wavelets for π consists of all functions ϕ in H +(R for which ϕ L (R = 1 ˆϕ(ξ dξ <. ξ Moreover, the wavelet constant c ϕ for every admissible wavelet ϕ is given by c ϕ = π ˆϕ(ξ dξ. ξ Proofs of Theorem 1.6 can be found in the book by Daubechies [6] the book by Wong [16]. Theorem 1.7 Let F be a function in L (U such that F (b, a = β(b for all (b, a in U. Let ϕ be an admissible wavelet for π. Then the localization operator L F,ϕ : H+(R H+(R associated to the symbol F the admissible wavelet ϕ is given by (L F,ϕ u, v L (R = 1 c ϕ for all u v in H +(R, where ψ(x = ϕ( x, x R, β(yp ψ (u, v(y dy

6 6 M. W. WONG ψ a is the Friedrich mollifier of ψ defined by ψ a (x = 1 a ψ ( x a, x R, p ψ (u, v is the paraproduct of u v with respect to the admissible wavelet ψ given by for all y in (,. p ψ (u, v(y = (ψ a u(y(ψ a v(y da a Theorem 1.7 is due to Rochberg [11] is made precise in the paper [17] by Wong. It should be remarked that the notion of a paraproduct is rooted in Bony s work [3] on linearization of nonlinear partial differential equations. Several versions of paraproducts exist in the literature. We are using the one studied in the work [5] by Coifman Meyer. It is interesting to note that the formula for localization operators on the affine group is like the one for the Weyl-Heisenberg group. For the affine group, we have the inner product of β with the paraproduct p ψ (u, v. For the Weyl-Heisenberg case, we have the inner product of σ Λ with the Wigner transform. Thus, the paraproduct can be thought of as the Wigner transform on the affine group. The aim of this paper is to consolidate this theme with the Lie group SU(1,1. We first give in Section a recall of the Lie group SU(1,1 its discrete series representations. The reason for using the discrete series representations of SU(1,1 is that they are infinite-dimensional, irreducible square-integrable. That this is indeed the case is proved in Section 3. In Section 4, we show that localization operators associated to specific symbols on SU(1,1 a specific admissible wavelet, i.e., the vacuum state, for the discrete series representations, are given in terms of a paraproduct. This paraproduct can be used as the Wigner transform on SU(1,1.. Discrete Series Representations of SU(1,1 Let SU(1,1 be the set of all matrices α β, where α β are complex β α numbers such that α β = 1. With respect to matrix multiplication, it is a non-abelian group, which we call the pseudo-unitary group. According to the Cartan decomposition of SU(1,1, every element g in SU(1,1 is of the form g = u φ a t u ψ, φ < 4π, t, ψ < π,

7 LOCALIZATION OPERATORS 7 where u φ = u ψ = ei φ e i φ ei ψ e i ψ, a t = cosh ( t sinh ( t sinh ( t cosh (. t In fact, for φ < 4π, t ψ < π, we have α = e i(φ+ψ/ cosh β = e i(φ ψ/ sinh ( t ( t. The Cartan decomposition of SU(1,1 as given is Proposition 5. in Chapter 5 of the book [13] by Sugiura. In order to compute the Haar measure on SU(1,1, we recall the special linear group SL(, R, which consists of all matrices a b with real entries c d such that ad bc = 1. It is well-known that SU(1,1 acts on the unit disk with center at the origin, while SL(, R acts on the upper half plane. In fact, they are related by the equation SU(1, 1 = C SL(, RC 1, (.1 where C is the unitary matrix given by C = 1 1 i 1 i. According to the Iwasawa decomposition of SL(, R, every element g in SL(, R is of the form g = k θ α t n ξ, θ < 4π, t R, ξ R,

8 8 M. W. WONG where k θ = α t = cos ( θ sin ( θ sin ( ( θ cos θ e t e t, n ξ = 1 ξ 1. In fact, for θ < 4π, t R ξ R, we have e i θ = a ic a + c, e t = a + c ξ = ab + cd a + c. It follows from the Iwasawa decomposition that we can think of SL(, R as the set [, 4π R R, which becomes a non-abelian group when equipped with the multiplication induced by the matrix multiplication in SL(, R. It is proved on page 66 of the book [13] by Sugiura that SL(, R is a unimodular group on which the left right Haar measure ν is given by SL(,R f(g dν(g = 1 4π f(θ, t, ξe t dθ dt dξ, 4π for every continuous function f on SL(, R with compact support. Using this Haar measure on SL(, R, the identification of SU(1,1 with SL(, R via (.1, the Cartan decomposition, it is proved on pages that SU(1,1 is also a unimodular group the left right Haar measure µ on it is given by SU(1,1 f(g dµ(g = 1 π 4π 4π f(φ, t, ψ sinh t dφ dt dψ for every continuous function f on SU(1,1 with compact support. For the sake of computations, it is helpful to think of SU(1,1 as the set [, 4π [, [, π, which becomes a non-abelian group on which the multiplication is the one that

9 LOCALIZATION OPERATORS 9 comes from the matrix multiplication in SU(1,1 the left right Haar measure µ is given by dµ = 1 sinh t dφ dt dψ. 4π Thus, SU(1,1 is a unimodular, locally compact Hausdorff group. We can now introduce infinite-dimensional, irreducible unitary representations of the pseudo-unitary group SU(1,1. To this end, we let D be the open unit disk with center at the origin in the complex plane. For n 1 Z with n 1, we let X n be the set of all analytic functions u on D such that n 1 π D u(z (1 z n dz <, where dz is the Lebesque measure on C. Thus, X n is a Hilbert space in which the inner product (, Xn is given by (u, v Xn = n 1 π D u(zv(z(1 z n dz for all u v in X n. We now let U(X n be the group of all unitary operators on X n let π n : SU(1, 1 U(X n be the mapping defined by ( αz + β (π n (gu(z = (βz + α n u β + α for all g in SU(1,1 with g 1 = α β β α all u in X n. It is well-known that π n is an irreducible unitary representation of SU(1,1 on X n. The representations π n, n 1 Z, n 1, are known as the discrete series representations of SU(1,1 on X n. We assume that n 1 in this paper. The irreducible unitary representations of SU(1,1 are computed in Chapter 5 of the book [13] by Sugiura. 3. Square-Integrability Let 1 be the function on D defined by 1(z = 1 for all z in D. Then we have the following results. Theorem 3.1 For n 1 Z with n 1, we have 1 X n = 1.

10 1 M. W. WONG Proof Using the inner product in X n polar coordinates, we get 1 X n = n 1 1(z (1 z n dz π = n 1 π = n 1 π D π 1 π 1 (1 r n r dr dθ r n dr dθ = 1. Theorem 3. For n 1 Z with n 1, we have 1 π 4π 4π (1, π n (φ, t, ψ1 Xn sinh t dφ dt dψ = 4π n 1. Proof For φ < 4π, t ψ < π, we have where (π n (φ, t, ψ1, 1 Xn = n 1 ( ( ( e i φ t sinh z + e i φ t n cosh (1 z π D n dz = n 1 ( ( ( t t n cosh n 1 e iφ tanh z (1 z n dz π D = n 1 ( t ( ( n t cosh n e ilφ tanh l z l (1 z n dz π D l l= = n 1 ( t π cosh n (1 z n dz D = n 1 ( t π 1 cosh n (1 r n r dr dθ π ( t = cosh n, ( n = l ( n( n 1 ( n l + 1. l! Thus, if we denote the integral in Theorem 3. by I, then I = π cosh 4n sinh t dt

11 LOCALIZATION OPERATORS 11 = 4π = 8π the proof is complete. 1 cosh 4n+1 ( t t 4n+1 dt = ( t sinh dt 4π n 1 In light of Theorems , we see that for n 1 Z with n 1, the discrete series representations π n : SU(1, 1 U(X n are square-integrable, the function 1 is an admissible wavelet for each π n : SU(1, 1 U(X n the 4π n 1 wavelet constant for 1 is equal to. The admissible wavelet 1 is the first element in the orthonormal basis {z k : k =, 1,...} for X n. Hence we also call 1 the vacuum state associated to the discrete series representations of SU(1,1 on X n. 4. Paraproducts Let F L 1 (SU(1, 1. Then the localization operator L F,1 : X n X n associated to the symbol F the admissible wavelet 1 is defined by (L F,1 u, v Xn = n 1 F (g(u, π(g1 Xn (π(g1, v Xn dµ(g 4π SU(1,1 for all functions u v in X n. We can give a formula for L F,1 : X n X n in terms of a paraproduct when the symbol F is a function of t only. Theorem 4.1 Let F be a function in L 1 (SU(1, 1 such that F (φ, t, ψ = τ(sinh t, (φ, t, ψ SU(1, 1. Then the localization operator L F,1 : X n X n associated to the symbol F the vacuum state 1 is given by (L F,1 u, v Xn = for all u v in X n given by u(z = τ(tp πn,1(u, v(t dt a k z k, z D, k= v(z = b k z k, z D, k=

12 1 M. W. WONG in X n, respectively, where the power series converge absolutely uniformly on every compact subset of D, p πn,1(u, v is the paraproduct of u v associated to the vacuum state 1 of the representation π n of SU(1, 1 on X n is given by ( t k p πn,1(u, v(t = (n 1(1 + t n a k b k, t >. 1 + t k= Proof For k =, 1,,... g = (φ, t, ψ SU(1, 1, (π n (g1, z k Xn = (π n (φ, t, ψ1, z k Xn ( n = (n 1e inψ cosh n ( t where β is the beta function, i.e., β(k + 1, n 1 = For k =, 1,,..., ( n β(k + 1, n 1 k k e ikφ tanh k ( t Γ(k + 1Γ(n 1. Γ(n + k k (n(n + 1 (n + k 1 = ( 1 k! β(k + 1, n 1, Γ(k + 1Γ(n 1 Γ(n + k k (n(n + 1 (n + k 1Γ(n 1 = ( 1 Γ(n + k k Γ(n 1 = ( 1 = ( 1k Γ(n n 1. Therefore for φ < 4π, t ψ < π, (π(φ, t, ψ1, z k Xn = e inψ cosh n ( t for k =, 1,,..., hence (L F,1 u, v Xn = n 1 = (n 1 = τ(sinh t k= k= ( t e ikφ tanh k ( ( t t a k b k tanh k cosh 4n sinh t dt ( t k τ(t a k b k (1 + t n dt 1 + t τ(tp πn,1(u, v(t dt,

13 LOCALIZATION OPERATORS 13 as asserted. Acknowledgment This research has been partially supported by the Natural Sciences Engineering Research Council of Canada (NSERC under Grant OPG856. References 1. P. Boggiatto, E. Buzano L. Rodino, Global Hypoellipticity Spectral Theory, Academie-Verlag, P. Boggiatto E. Cordero, Anti-Wick quantization with symbols in L p spaces, Proc. Amer. Math. Soc. (13, J. -M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. 14 (1981, A. L. Carey, Square integrable representations of non-unimodular groups, Bull. Austral. Math. Soc. 15 (1976, R. R. Coifmam Y. Meyer, Au Delà des Opérateurs Pseudo-Différentiels, Astérisque 57, I. Daubechies, Ten Lectures on Wavelets, SIAM, J. Dixmier, C -Algebras, North-Holl, M. Duflo C. C. Moore, On the regular representation of a non-unimodular locally compact group, J. Funct. Anal. 1 (1976, A. Grossmann, J. Morlet T. Paul, Transforms associated to square integrable group representations I: General results, J. Math. Phys. 6 (1985, Z. He M. W. Wong, Localization operators associated to square integrable group representations, Panamer. Math. J. 6(1 (1996, R. Rochberg, The use of decomposition theorems in the study of operators, in Wavelets: Mathematics Applications, Editors: J. J. Benedetto M. W. Frazier, CRC Press, 1994, M. A. Shubin, Pseudodifferential Operators Spectral Theory, Springer-Verlag, M. Sugiura, Unitary Representations Harmonic Analysis: An Introduction, Second Edition, North-Holl, E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 4 (193, M. W. Wong, Weyl Transforms, Springer-Verlag, M. W. Wong, Wavelet Transforms Localization Operators, Birkhäuser,. 17. M. W. Wong, Localization operators on the affine group paracommutators, in Progress in Analysis, Proceedings of the 3rd International ISAAC Congress, Editors: H. G. W. Begehr, R. P. Gilbert M. W. Wong, World Scientific,