2 DEMETRIO LABATE the mask of the lter, since it eights selectively the dierent time{frequency components of the signal. In our approach, the pseudodi

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1 Preprint (revised) 07/18/00 TIME{FREQUENCY ANALYSIS OF PSEUDODIFFERENTIAL OPERATORS Demetrio Labate Abstract. In this paper e apply a time{frequency approach to the study of pseudodierential operators. Both the Weyl and the Kohn{Nirenberg correspondences are considered. In order to quantify the time{frequency content of a function or distribution, e use certain function spaces called modulation spaces. We deduce a time{frequency characterization of the tisted product ] of to symbols and, and e sho that modulation spaces provide the natural setting to exactly control the time{frequency content of ] from the time{frequency content of and. As a consequence, e discuss some boundedness and spectral properties of the corresponding operator ith symbol ]. 1. Introduction A pseudodierential operator can be dened through the Weyl or the Kohn{Nirenberg correspondence by bijectively assigning to any distributional symbol 2S 0 (R 2n ) a linear operator T : S(R n )!S 0 (R n ), so that the properties of the operator are in an appropriate ay reected in the properties of the symbol. One ay to construct a pseudodierential operator is as a superposition of time{ frequency shifts. Even though this is a classical idea, going back to H. Weyl [17], this interpretation has reblossomed in recent years in the context of the study of the harmonic analysis in the Heisenberg group ([11], [6]). As a consequence, a number of methods from the time{frequency analysis have been employed to the study of pseudodierential operators (for instance: [16], [10], [14], [9], [18]). In this paper, e are interested in pseudodierential operators hose symbols satisfy certain integrability conditions in the time{frequency plane and are not necessarily smooth. The interest of these classes of operators stems partly from electrical engineering applications, in particular signal processing and time{varying ltering theory, here operators arising from the Weyl correspondence are used as models for time{frequency or time{ varying lters (cf., for instance, [5], [12], [15]). In this context, the symbol is interpreted as 1991 Mathematics Subject Classication. Primary 35S05, 47G30 Secondary 42C15, 47B10. Key ords and phrases. Kohn{Nirenberg correspondence, modulation spaces, pseudodierential operators, time{frequency analysis, trace{class operators, tisted product, Weyl correspondence. 1

2 2 DEMETRIO LABATE the mask of the lter, since it eights selectively the dierent time{frequency components of the signal. In our approach, the pseudodierential operator T is realized as a superposition of elementary rank{one operators through the time{frequency decomposition of the associated symbol. In order to exactly quantify the time{frequency content of the symbol, certain function spaces, called modulation spaces, are natural. These spaces are dened by prescribing the decay properties of the Short{Time Fourier Transform (STFT) of a given function or distribution, and they contain a large class of objects, including some classical function spaces (e.g, L 2, Sobolev spaces). The precise denition of modulation spaces, together ith the basic time{frequency tools employed in this paper, are revieed in Section 2. In Section 3 e apply this time{frequency approach to study the composition of pseudodierential operators. Let T and T be pseudodierential operators having symbols and respectively. Then T ] = T T is a pseudodierential operator ith symbol ], here ] is called the tisted product of and. The direct computation of the tisted product (cf. [6, Section 2.3]) leads to a complicated integral formula, hich is usually asymptotically expanded into a poer series and approximated. Hoever, such an expansion requires and to be arbitrarily smooth. Our time{frequency approach yields a characterization of the tisted product hich does not require smoothness assumptions on the symbols. The STFT of ] behaves essentially as a matrix multiplication of the STFTs of and, from hich the modulation space norm of ] can be controlled by the modulation space norms of and. In particular, the tisted product turns out to be closed on the modulation spaces M p p ith (x y) = (1 + jxj + jyj) s, for 1 p 2. Notation. Let X, X 0, Y, Y 0 be Banachspaces. L(X Y ) is the space of all bounded linear operators from X to Y, and L(X) =L(X X). The norm of X is kk X, or simply kk if the context is clear. The dual space of X is X 0. We ritehf gi for the action of g 2 X 0 on f 2 X. We rite T for the adjointoperatoroft. A linear subspace A(X Y )ofl(x Y )isanoperator ideal if UTV 2 A(X Y ) henever U 2 L(Y Y 0 ), T 2 A(X Y ), and V 2 L(X 0 X): The Schatten class I p L(H) consists of the compact operators on a Hilbert space H hose singular values lie in `p. I p is an operator ideal and coincides ith the class of Hilbert{ Schmidt operators hen p = 2and ith the trace{class operators hen p = 1. L p q (R 2n ) is the eighted mixed{normed space of functions f on R 2n ith norm kfk L p q = ( R ( R jf(x y)j p (x y) p dx) q=p dy) 1=q : If 1 e rite L p q (R 2n ). If R n R n p = q, e have the classical space L p (R 2n ) = L p p (R 2n ). `p q ( 2n ) is the space of sequences a =(a km ) k m2 n ith norm kak`p q =(P m (P k ja kmj p (k m) p ) q=p ) 1=q : If 1

3 TIME{FREQUENCY ANALYSIS OF PSEUDODIFFERENTIAL OPERATORS 3 e rite `p q ( 2n ): If p = q e have the classical sequence space `p( 2n ) = `p p ( 2n ): C(R n ) is the space of continuous functions on R n, and C 0 (R n ) is the space of continuous functions on R n vanishing at innity. S(R n ) is the Schartz space of all innitely dierentiable functions on R n decaying rapidly at innity, and S 0 (R n ) is its topological dual, the space of tempered distributions. H s (R n ) is the Sobolev space of functions dened by the norm kfk 2 H s = R R n j ^f()j 2 (1 + jj 2 ) s d: The usual dot product of x, y 2 R n is denoted by juxtaposition, i.e., xy = x 1 y x n y n. The symplectic form on =( 1 2 ) =( 1 2 ) 2 R n R n is [ ] = 1 2 ; 2 1. The composition of f and g is (f g)(t) = f(g(t)): The inner product of f, g 2 L 2 (R n ) is hf gi = R R n f(t) g(t) dt the same notation is used for the extension of the inner product to S(R n ) S 0 (R n ). The Fourier transform is Ff() = ^f() = R f(t) e ;2it dt the inverse Fourier transform is f() = ^f(;). The Fourier transform maps S(R n ) onto itself, and extends to S 0 (R n ) by duality. The convolution of f and g is (f g)(x) = R R n f(x ; t)g(t) dt. 2. Background: Time{Frequency Analysis We briey revie the Schrodinger representation of the Heisenberg group as a tool for constructing and analyzing pseudodierential operators. We adopt most of the notation and conventions of Folland's book [6] The Schrodinger representation. The Schrodinger representation of the Heisenberg group H n = R n R n R is the map from H n to the group of unitary operators on L 2 (R n ) dened by (a b t)f(x) = e 2it e iab e 2ibx f(x + a): In many considerations the t-variable is unimportant, so for (a b) 2 R 2n e dene (a b)f(x)=e iab e 2ibx f(x + a). We refer to (a b)f as a time-frequency shift of f. We recall the folloing useful facts. Proposition 2.1. Let f 2 L 2 (R n ) and let a, b, a 0, b 0 2 R n. Then: (a) k(a b)fk L 2 = kfk L 2, (b) ((a b)f)^ = (;b a) ^f, (c) ((a b)) ;1 =((a b)) = (;a ;b), (d) (a b) (a 0 b 0 )f = e i(ab0 ;a 0b) (a + a 0 b + b 0 )f: The (cross-)ambiguity function, or Fourier-Wigner transform, of f, g 2 L 2 (R n ) is: A(f g)(a b) = h(a b)f gi = e iab e 2ibx f(x + a) g(x) dx: R n

4 4 DEMETRIO LABATE If f = g, e rite A(f f) = A(f). The ambiguity function extends to a map from S(R n )S(R n )into S(R 2n ) and from S 0 (R n )S 0 (R n )into S 0 (R 2n ). A slightchange in the denition yields the Short{Time Fourier Transform (STFT) of a distribution f 2S 0 (R n ) ith respect to a indo g 2S(R n ): S g f(a b) = R n f(x) g(x ; a) e ;2ixb dx = e ;iab h(a ;b)f gi = e ;iab A(f g)(a ;b): The Wigner transform of f, g 2 L 2 (R n ) is the Fourier transform of the ambiguity function of f and g: W (f g)( x) = A(f g)^( x) = e ;2ip f(x + p ) g(x ; p ) dp: (2.1) R n 2 2 We set W (f) =W (f f). Similarly to the ambiguity function, also the Wigner transform extends to a map from S(R n )S(R n )into S(R 2n )andfroms 0 (R n )S 0 (R n )into S 0 (R 2n ). The folloing facts ill be useful (cf. [6, Sec. 1.4 and 1.8]). Proposition 2.2. Let f, g 2 L 2 (R n ) and let a, b, u 1, u 2, v 1, v 2 2 R n. Then: (a) A(f g) 2 L 2 (R 2n ), ith ka(f g)k L 2 = kfk L 2 kgk L 2. (b) A(f g) 2 C 0 (R 2n ), and ka(f g)k L 1 kfk L 2 kgk L 2. (c) A(f g)(a b)=a(g f)(;a ;b). (d) A((u 1 u 2 )f (v 1 v 2 )g)(a b) = e i(u 1v 2 ;u 2 v 1 ) e i((u 2+v 2 )a;(u 1 +v 1 )b) A(f g)(a + u 1 ; v 1 b+ u 2 ; v 2 ). (e) W ((u 1 u 2 )f (v 1 v 2 )g)(a b) = e i(u 2v 1 ;u 1 v 2 ) e i((u 1;v 1 )a+(u 2 ;v 2 )b) W (f g)(a ; u 2+v 2 b+ u 1+v 1 ). 2 2 (f) (Moyal's Identity) W (f 1 g 1 ) W (f 2 g 2 ) = hf 1 f 2 ihg 2 g 1 i. (g) If f 2 L 1 (R n ), then R A(f)(a b) da = ^f(; b 2 ) ^f( b 2 ), R A(f)(a b) db = f( a 2 )f(; a 2 ): (h) S g f(a b) = e 2iab S^g ^f(b ;a): The Weyl correspondence is the 1-1 correspondence beteen a distributional symbol 2S 0 (R 2n ) and the pseudodierential operator L = (D X): S(R n )!S 0 (R n ) dened implicitly by: hl f gi = h^ A(g f)i = h W(g f)i

5 TIME{FREQUENCY ANALYSIS OF PSEUDODIFFERENTIAL OPERATORS 5 here f g 2S(R n ). L is the Weyl transform of. The Kohn{Nirenberg correspondence assigns to a symbol the operator K = (D X) KN dened implicitly by: hk f gi = h^ e ix A(g f)i: (2.2) K is the Kohn{Nirenberg transform of. Equation (2.2) shos that the operators L in the Weyl correspondence and K in the Kohn{Nirenberg correspondence are equal if and only if their symbols are related by ^( x) = ^( x) e ;ix. Therefore, statements invariant under multiplication by e ;ix ill be valid for one correspondence if and only if they are valid for the other. 2.2 Modulation Spaces. The modulation spaces measure the joint time-frequency distribution of f 2S 0 (R d ). For background and detailed information on their properties e refer to [1], [2], [3], [8]. Let be a subadditive positive eight function on R 2n, i.e., 1 () < 1 and ( + ) () () for all, 2 R 2n. We assume that has at most polynomial groth, i.e., for all 2 R 2n ehave () Cjj N for some C, N 0. Let 1 p, q 1. Given a indo function g 2 S(R n ), denote by M p q (R n ) the space of all distributions f 2S 0 (R n ) for hich the norm kfk M p q (R n ) = ks g fk L p q (R 2n ) = R n R n js g f(x y)j p (x y) p dx q=p dy 1=q is nite, ith obvious modications if p or q = 1. If 1 then e rite M p q (R n ). The space M p q (R n ) is a Banach space hose denition is independent of the choice of indo g, i.e., dierent choices of indos g yield equivalent norms. The assumptions on the eight guarantees that the modulation spaces are dened in the realm of tempered distributions, and that S is dense in all modulation spaces M p q for all 1 p q 1 (cf. [1], [8, Section 11.1]). For 1 p, q < 1, the dual space of M p q (R n ) is (M p q (R n )) 0 = M p0 q 0 (R n ), here p 0 q 0 satisfy p 1 + p 1 0 = q 1 + q 1 0 =1. We recall the folloing invariance properties of the modulation spaces. If (x y) = (y x), it follos from Proposition 2.2(h) that M p p (R n ) is invariant under the Fourier transform. Moreover, the modulation spaces are invariant under the metaplectic representation (cf. [3, Theorem 29]). In particular, multiplication by e ;ixy leaves the space M p q (R n ) invariant for each 1 p q 1, i.e., kfk M p q = ke ;ixy fk M p q. We ill use this property in Section 3 to transfer statements beteen Weyl and Kohn{Nirenberg correspondences. Among the modulation spaces the folloing ell-knon function spaces occur. (a) M 2 2 (R n )=L 2 (R n ).

6 6 DEMETRIO LABATE (b) (Weighted L 2 -spaces) If (x y) = (1 + jxj) s, then M 2 2 (R n ) = L 2 (R n ): (c) (Sobolev spaces) If (x y) = (1 + jyj) s, then M 2 2 (R n ) = H s (R n ): (d) If (x y) =(1+jxj + jyj) s, then M 2 2 (R n ) = L 2 s(r n ) \ H s (R n ): (e) (Feichtinger's algebra) M 1 1 (R n )=S 0 (R n ). We recall that the space S 0 is contained in L 2, and that it is an algebra under both convolution and pointise multiplication. The space S 0 plays an important role in abstract harmonic analysis (cf. [4]). 2.3 Time{Frequency Expansion of the Weyl Operator. By realizing the symbol of the Weyl operator L as a superposition of time{frequency shifts, it is possible to express L in terms of elementary rank{one operators. The fundamental result needed for the time{frequency analysis is the folloing inversion formula, hich is proved in an abstract context in [1]: Theorem 2.3. If 2 S(R 2n ) ith kk L 2 then: = = 1, and 2 M p q (R 2n ) ith 1 p q < 1, R 4n h ( )i ( ) d d (2.3) here the integral converges in the norm of M p q (R 2n ). If p = 1 or q = 1 or if 2 S 0 (R 2n ), then (2.3) holds ith eak convergence of the integral. The folloing consequence of Theorem 2.3 is proved in [9, Lemma 3.2]. Theorem 2.4. Let 2S(R n ) ith kk L 2 =1, and let =W ( ). Let 2 M p q (R 2n ), ith 1 p q 1. Denote by N: R 2n R 2n! R 4n the linear transformation N( ) = N( ) = ; ; ; 1 2 ; 2 (2.4) here =( 1 2 ), =( 1 2 ) 2 R n R n. Then, for f 2S(R n ) e have: L f = This integral converges as in Theorem 2.3. R 4n S (N( )) e i[ ] hf ())i ()dd: (2.5) From no on e ill let denote an arbitrary but xed function in S(R n ) such that (t) = (;t) and kk L 2 = 1. For example, e could take (x) = 2 n=4 e ;x2. We set = W (). We ill let N denote the linear transformation dened in (2.4), and N ~ the linear transformation N( ~ ) =N( ).

7 TIME{FREQUENCY ANALYSIS OF PSEUDODIFFERENTIAL OPERATORS 7 3. Composition of Pseudodifferential Operators Suppose that L L are Weyl operators that map S(R n )into itself. Then L ] = L L maps S(R n ) into itself, and its symbol ] 2 S 0 (R 2n ) is called the tisted product of and. 3.1 Formula for the tisted product. Using the time{frequency approach of Section 2.3, e deduce a useful characterization of the tisted product. We begin by assuming that, 2S(R 2n ). In this case the tisted product ] can be expressed [6, Sec.2.3] as: (])(!) = 2 2n R 4n (u) (v) e 4i[!;v!;u] du dv: Using Theorem 2.3 to expand and, e obtain: (])(!) = 2 2n R 8n h ( )ih ( )i ((( ))](( )))(!) d d d d (3.1) Recall that, if F 2S(R 2n ), then S F (a b) =e ;iab hf (;a b)i. By Proposition 2.2(e), (M(x y)) = W ((x) (y)), here M: R 2n R 2n! R 4n is the linear transformation for hich (;a b) = M(x y) i (a b) = N(x y). It follos that the change of variables (a b) =N(x y) yields: S F (N(x y)) = e ;iab hf (;a b)i = e ;i[x y] hf W((x) (y))i (3.2) Therefore, the change of variables (; ) =N( ), (; )=N( ) in (3.1) yields: (])(!) = 2 R 2n S (N( )) S (N( )) e ip 8n ; W (() ())]W (() ()) (!) d d d d (3.3) here P =[ ]+[ ]. By direct calculations, e have: W (() ())]W (() ()) = h() ()i W (() ()): (3.4) Consequently, using (3.2) and (3.4) into (3.3), e obtain: S (])(N( )) = 2 2n e ;i[ ] R 8n S (N( )) S (N( )) h() ()i e ip W (() ()) W(() ()) d d d d: (3.5)

8 8 DEMETRIO LABATE Finally, from(3.5), Moyal's identity (Proposition 2.2(f)) implies: S (])(N( )) = 2 2n e ;i[ ] R 8n S (N( )) S (N( )) h() ()i e ip h() ()ih() ()i d d d d: (3.6) Set S 0 ( ) = S 0 ( ) = js (N( ))jja()( ; )j d R 2n js (N( ))jja()( ; )j d: R 2n These functions are smoothed versions of the STFTs of and. The folloing result shos that the STFT of ] is controlled by the(continuous) matrix multiplication of S 0 and S 0. Proposition 3.1. Let, 2S(R 2n ). Then: js (])(N( ))j 2 2n S 0 ( ) ja()j S 0 ( ) : Proof. By Proposition 2.1, h(a) (b))i = e i[a b] A()(a ; b), so (3.6) implies: js (])(N( ))j 2 2n R 8n js (N( ))jjs (N( ))jja()( ; )j ja()( ; )jja()( ; )j d d d d = 2 2n R 4n S 0 ( ) ja()( ; )j S 0 ( ) d d = 2 2n S 0 ( ) ja()j S 0 ( ) : 3.2 Main theorem. Proposition 3.1 shos that the time{frequency content of] is controlled by the time{ frequency content of and. The modulation spaces provide the natural setting to exactly characterize this relationship. Our main results in this direction are collected in the folloing theorem and corollary, and ill be proved in Section 3.3.

9 TIME{FREQUENCY ANALYSIS OF PSEUDODIFFERENTIAL OPERATORS 9 Theorem 3.2. Let ( ) = (1 + jj 2 + jj 2 ) s=2 1( ) = (1 + jj 2 ) s=2 2( ) = (1 + jj 2 ) s=2 ith s 0. (a) If, 2 M p p (R 2n ), ith 1 p 2, then ] 2 M p p k]k M p C kk M p kk M p (R 2n ), ith here C =2 3 2 s 2 3n ka()k L 1 1 ka()k L 1 2 : In particular, if s =0, then C =2 5n. (b) If 2 M p p (R 2n ) and 2 M p0 p 0 M p p (R 2n ),ith k]k M p (R 2n ), ith p 2 and 1 p + 1 p 0 = 1 then ] 2 C kk M p kk M p0 here C =2 3 2 s 2 3n ka()k L 1 1 ka()k L 1 2 : In particular, if s =0, then C =2 5n. Corollary 3.3. (a) If, 2 S 0 (R 2n ), then ] 2 S 0 (R 2n ), and kl ] k I1 2 4n kk S0 kk S0 : (b) If, 2 L 2 s(r 2n ) \ H s (R 2n ), here ( ) = (1 + jj 2 + jj 2 ) s=2 ith s 0, then ] 2 L 2 s(r 2n ) \ H s (R 2n ). If, in addition, s > n, then kl ] k I1 C kk L 2 s \H skk L 2 s \Hs here C is a constant hich does not depend on or. (c) If, 2 L 2 (R 2n ), then ] 2 L 2 (R 2n ),andkl ] k I2 2 5n kk L 2kk L 2. Remark 3.4. (i) Theorem 3.2 shos that the modulation spaces M p p are algebras under tisted product hen 1 p 2. As special cases, Corollary 3.3 shos that the spaces S 0, L 2 and L 2 s \ H s are also algebras under tisted product. In addition, in these cases e have the folloing property. Recall that if 2 S 0 or 2 L 2 s \ H s, then L lies in a closed subspace of I 1 (cf. [8, Theorem 3], [10, Proposition 5.4]). As a consequence, the composition of operators L and L ith symbols in S 0 or L 2 s \ H s is closed not only on I 1 (as follos from the ideal property of trace{class operators, already), but also on the closed subspaces of I 1 dened hen the symbols lie in these modulation spaces. (ii) Using the observations of Section 2.2, it is easy to transfer the results of Corollary 3.3 to the Kohn{Nirenberg correspondence. Indeed, by equation (2.2), K! = L T!, here (T!)^ = e ;ix^!. Since k(t!)^k M p q = k^!k M p q (cf. Section 2.2), it follos that Corollary 3.3 holds for the Kohn{Nirenberg correspondence as ell ithout any changes. (iii) Part (c) of Corollary 3.3 is knon (cf. [6, Proposition 1.33]) but it is reported for completeness. Observe that the operator L ] is not only in I 2, but also in I 1, being the composition of to Hilbert{Schmidt operators.

10 10 DEMETRIO LABATE 3.3 Proof of Theorem 3.2. In order to prove Theorem 3.2, e require some technical lemmas. The rst set of lemmas shos that the L p q {norm of S (]) can be controlled by the L p q {norms of S and S (Lemmas 3.5{3.7). The next lemma examines the eect of the linear transformation N and, in particular, the relationship beteen the L p q -norm of S F N and the modulation space norm of F (Lemma 3.8). Lemma 3.5. Let 1 p q r 1, ith 1 r + 1 r 0 =1, be given. Assume 2S(R 2n ). Let, 1 and 2 be nonnegative functions satisfying ( ) 1() 2(). Then: ks (]) Nk L p q 2 3n k ~ S 0 k L r0 p 1 ks 0 k L r q 2 here ~ S 0 ( ) =S 0 ( ). Proof. For simplicity of notation, dene: a() = ja()()j, s () = js (N( ))j, t () = js (N( ))j, s 0 () = (s a)() = S( 0 ), t 0 () = (t a)() = S 0 ( ). Since is even, Proposition 2.2(g) implies that kak L 1 = R A()() d = R ( p R 2n R n 2 )2 dp = 2 n. By Proposition 3.1, Holder's inequality, andyoung's inequality, e have: Consequently, js (])(N( ))j 2 2n hs 0 a t 0 i (3.7) 2 2n ks 0 ak L r0 kt 0 k L r 2 2n kak L 1 ks 0 k L r0 kt 0 k L r = 2 3n ks 0 k L r0 kt 0 k L r : q=p 1=q ks (]) Nk L p q = js (])(N( ))j p ( ) sp d d R 2n R 2n q=p 1=q 2 3n ks 0 k p L R 2n R 2n r 0 kt 0 k p L ( ) sp d d r p=r 0 1=p 2 3n js( 0 )j r0 (1()) r0 d d R 2n R 2n q=r 1=q js( 0 )j r (2()) r d d R 2n R 2n = 2 3n k ~ S 0 k L r0 p 1 ks 0 k L r q 2 :

11 TIME{FREQUENCY ANALYSIS OF PSEUDODIFFERENTIAL OPERATORS 11 Lemma 3.6. Let 1 <p 0 r q<1, ith 1 p + 1 p 0 =1and 1 r + 1 r 0 =1, be given. Assume 2S(R 2n ). Let, 1 and 2 be nonnegative functions, ith ( ) 1() 2(). Then: ks (]) Nk L p q here C =2 3n ka()k L 1 1 ka()k L 1 2 : C ks Nk L p r0 1 ks ~ NkL q r 2 Proof. We adopt the notation dened in the proof of Lemma 3.5. Since 1 p r 0 Minkoski's inequality forintegrals implies: p=r 0 1=p k S ~ 0 k L r0 p = js( 0 )j r0 (1()) r0 d d 1 R 2n R 2n r 0 =p 1=r 0 js( 0 )j p (1()) p d d = ksk 0 L p r0 : R 2n R 2n 1 It follos from Young's inequality that k(s a)k L p kak 1 L 1 ks 1 k L p 1 Consequently: ks 0 k L p r0 1 Similarly for the function S 0, since 1 q r implies: ks 0 k L r q 2 R 2n < 1 (3.8) for any 1 p 1. ka()k L 1 1 ks Nk L p r0 : (3.9) 1 < 1 Minkoski's inequality for integrals R 2n js 0 ( )j q (2()) q d r=q 1=r d = k S ~ 0 k L q r (3.10) 2 here ~ S 0 ( ) =S 0 ( ). Using Young's inequality as before, e obtain: k ~ S 0 k L q r 2 ka()k L 1 2 ks ~ NkL q r 2 : (3.11) Finally, the proof follos by substituting (3.8){(3.11) into Lemma 3.5. The folloing result is similar in nature but is not contained in Lemma 3.6. Lemma 3.7. Assume 2 S(R 2n ). Let, 1 and 2 be nonnegative functions, ith ( ) 1() 2(). Then: ks (]) Nk L n ks Nk L ks Nk L : Proof. We adopt the notation dened in the proof of Lemma 3.5. From (3.7), observing that kak L 1 1 (Proposition 2.2(b)), the Young's inequality implies that: js (])(N( ))j 2 2n ks 0 ak L 1 kt 0 k L 1 2 2n kak L 1 ks 0 k L 1 kt 0 k L 1 2 2n ks 0 k L 1 kt 0 k L 1: (3.12)

12 12 DEMETRIO LABATE By Proposition 2.2 e have R a() d =2 n,andso: R 2n ks 0 k L 1 d = s () a( ; ) d d d =2 n s () d d =2 n ks Nk L 1 1: Similarly, e can sho that R kt 0 R 2n k L 1 d =2n ks Nk L 1 1. Consequently, from equation (3.12) e obtain: ks (]) Nk L 1 1 = (R js 4n ) (])(N( ))j ( ) d d R 4n 2 R 2n ks 0 k L 1 1() d kt 0 k L 1 2() d 2n R 2n = 2 4n ks Nk L (R4n ) ks Nk L (R4n ) : No, e examine the eect of the linear transformations N and N ~ on the L p q -norm of a function of the form F N or F N. ~ Indeed, in general, kfkm p q = ks fk L p q 6= ks f Nk L p q (and similarly hen N is replaced by ~ N). The folloing lemma shos that in some special cases the L p q -norm of S f N is controlled by some modulation space norm of f. Parts (c) and (d) of the folloing lemma are not needed in the proof of Theorem 3.2, but are included for completeness. Lemma 3.8. Let 1 p 1and s 0 be given. Let ( ) = (1 + jj 2 + jj 2 ) s=2. Then: (a) 2 ;s=2 ks Nk L p p (b) ks Nk L p p (c) ks Nk L 1 1 kk M p p = kk M p p 2 2n k^k M 1 1 (d) ks Nk L 1 1 kk M 1 1: 2 s=2 ks Nk L p p Proof. (a) The change of variables ( ) =N( ) yields: ks Nk L p p = js (N( ))j R p (1 + jj 2 + jj 2 ) sp=2 d d 4n = js ( ))j p (1 + 2jj 2 + jj 2 R 4n 2 )sp=2 d d 1=p 1=p 2 s=2 R 4n js ( ))j p (1 + jj 2 + jj 2 ) sp=2 d d =2 s=2 kk M p p : The other inequality is obtained in a similar ay. 1=p

13 TIME{FREQUENCY ANALYSIS OF PSEUDODIFFERENTIAL OPERATORS 13 (b) This case is part (a) hen s =0. (c) Denote =( 1 2 ), =( 1 2 ) 2 R n R n. By direct calculation: ks Nk L 1 1 = sup js (N( ))j d R 2n sup sup js (( 2+ 2 ; 1+ 1 ) )j d 1 2 R 2n d 2 = sup js ( )j d R 2n 2 (d) Similar to part (c). = 2 R 2n sup js ( )j d 2n = 2 R 2n sup js ^^( )j d = 2 2n k^k M 1 1: 2n Remark 3.9. Lemma 3.8 holds hen N is replaced by ~ N ithout any changes. The proof is exactly the same. No e are ready to prove Theorem 3.2 and Corollary 3.3. Proof of Theorem 3.2. We begin by assuming 2 S(R 2n ). Since the space S(R 2n ) is dense in M p q (R 2n ), for any 1 p, q < 1 (as discussed in Section 2.2), the extension to the case, 2 M p p follos by a standard continuity argument. (a) It is sucient to prove the cases p = 1 and p = 2. The theorem then follos by interpolation. In fact, by [1], [2]: (M 1 1 (R 2n ) M 2 2 (R 2n )) = M p p (R 2n ) ith p 1 =(1; )+, 2 [0 1]. 2 It is clear that ( ) 1() 2() andthat 1() 2() ( ) (3.13) for all 2 R 2n : The case p =1then follos from the folloing estimates: ks (])k L 1 2 s 2 ks (]) Nk L 1 by Lemma s 2 +4n ks Nk L 1 1 ks Nk L 1 2 by Lemma s 2 +4n ks Nk L 1 ks Nk L 1 by equation (3.13) s+4n ks k L 1 ks k L 1 by Lemma 3.8 = s+4n kk M 1 kk M 1 :

14 14 DEMETRIO LABATE For the case p =2e have: ks (])k L 2 2 s=2 ks (]) Nk L s=2 C ks Nk L 2 2 ks NkL ~ s=2 C ks Nk L s C kk M 2 2 ks ~ NkL 2 2 by Lemma 3.8 by Lemma 3.6 by equation (3.13) kk M 2 2 by Lemma 3.8 here C =2 3n ka()k L 1 1 ka()k L 1 2 : Recall that ka()k L 1 =2 n (by Proposition 2.2(b)). Furthermore, e have that ka()k L 1 1 ka()k L 1 2 ka()k L 1 =2 n : Therefore C 2 5n and C =2 5n if s =0: (b) From Lemma 3.6 and equation (3.13) e have: ks (]) Nk L p p0 C ks Nk L p p ks NkL ~ p0 p 0 : (3.14) By [6], ks Nk L p p1 k ; (S N)(k m) k m k`p p1 p1 p2 (cf. [8, Chapter 12]), this implies that: and, therefore, since `p p1 `p p2 if ks (]) Nk L p p ks (]) Nk L p p0 : (3.15) Using Lemma 3.8 and equations (3.13) and (3.15) into equation (3.14) e obtain: k]k M p p 2 s=2 ks (]) Nk L p p The proof of Corollary 3.3 follos easily. 2 s=2 ks (]) Nk L p p s C kk M p p kk M p 0 p 0 : Proof of Corollary 3.2. (a) Recall that S 0 = M 1 1. Since h ()i() is a rank{one operator, it follos that kh ()i()k I1 1. Applying this observation to equation (2.5), e obtain: kl ] k I1 kh ()i()k I1 R 4n js (])(N( ))j d d ks (]) Nk L 1 1: (3.16) Using equation (3.16) and Lemmas 3.7 and 3.8(b), e obtain: kl ] k I1 ks (]) Nk L n ks Nk L 1 1 ks Nk L 1 1 =2 4n kk S0 kk S0 :

15 TIME{FREQUENCY ANALYSIS OF PSEUDODIFFERENTIAL OPERATORS 15 (b) Recall that M 2 2 = L 2 s \ H s, M = L2 s, and M = Hs, ith 1( ) = (1 + jj 2 ) s=2, 2( ) = (1 + jj 2 ) s=2. By [10, Proposition 5.4], if s>n, then ] 2 L 2 s \ H s implies L ] 2I 1. Consequently, from Theorem 3.2 e have: kl ] k I1 c ks (])k L 2 2 c s C kk M 2 2 s kk M 2 2 s here c is a constant hich does not depend on or, and C =2 3n ka()k L 1 1 ka()k L 1 2 : (c) By a classical result of Pool ([13]), kl k I2 = kk L 2. Therefore, by Theorem 3.2 ith s =0,e have: kl ] k I2 = k]k L 2 = k]k M 2 2 = ks (])k L n kk M 2 2 kk M 2 2: A closer look at the proof of Theorem 3.2 shos that it is possible to generalize the choice of the eight function. In fact, if is any positive subadditive function ith at most polynomial groth and 1 2 are positive functions satisfying ( ) 1()2() 1() 2() ( ) and (N ;1 ( )) C( ) for some C > 0 and for all 2 R 2n then the conclusions of Theorem 3.2 hold and the proof is essentially the same. Acknoledgments. I thank C. Heil and K. Grochenig for inspiring discussions and for helpful remarks. I thank the referees for several useful comments. References [1] FEICHTINGER H G, GROCHENIG K (1989) Banach spaces related to integrable group representations and their atomic decompositions, Part I. J. Funct. Anal. 83:307{340. [2] FEICHTINGER H G, GROCHENIG K (1989) Banach spaces related to integrable group representations and their atomic decompositions, Part II. Monatsh. Math. 108:129{148. [3] FEICHTINGER H G, GR OCHENIG, K (1992) Gabor avelets and the Heisenberg group: Gabor expansions and short time Fourier transform from the group theoretical point of vie. In CHUI C K (ed) Wavelets: A Tutorial in Theory and Applications, pp 359{398. Boston: Academic Press. [4] FEICHTINGER H G, IMMERMANN G (1997) A Banach space of test functions for Gabor Analysis. In FEICHTINGER H G, STROHMER T (eds) Gabor Analysis and Algorithms: Theory and Applications, pp 123{170. Boston: Birkhauser. [5] FLANDRIN P (1988) Maximum signal energy concentration in a time{frequency domain. In Proc. \IEEE ICASSP '88", 2176{2179. [6] FOLLAND G B (1989) Harmonic Analysis on Phase Space. Princeton: Princeton University Press. [7] GR OCHENIG K (1991) Describing functions: atomic decompositions versus frames Monatsh. Math. 112:1{42. [8] GR OCHENIG K (2000) Foundations of Time{Frequency Analysis. Boston: Birkhauser.

16 16 DEMETRIO LABATE [9] GR OCHENIG K, HEIL C (1999) Modulation spaces and pseudodierential operators. Integral Equations Operator Theory 34:439{457. [10] HEIL C, RAMANATHAN J, TOPIWALA P (1997) Singular values of compact pseudodierential operators. J. Funct. Anal. 150:426{452. [11] HOWE R (1980) On the role of the Heisenberg group in harmonic analysis. Bull. Amer. Math. Soc. 3:821{843. [12] KOEK W (1992) Time{frequency signal processing based on the Wigner{Weyl frameork. Signal Processing 29:77{92. [13] POOL J C T (1966) Mathematical aspects of the Weyl correspondence. J. Math. Phys. 7:66{76. [14] ROCHBERG R, TACHIAWA K (1997) Pseudodierential operators, Gabor frames, and local trigonometric bases. In FEICHTINGER H G, STROHMER T (eds) Gabor Analysis and Algorithms: Theory and Applications, pp 171{192. Boston: Birkhauser. [15] SHENOY R G, PARKS T W (1994) The Weyl correspondence and time{frequency analysis. IEEE Trans. Signal Processing 42:318{331. [16] TACHIAWA K (1994) The boundedness of pseudodierential operators on modulation spaces. Math. Nachr. 168: 263{277. [17] WEYL H (1931) The Theory of Groups and Quantum Mechanics. London: Methuen. [18] WONG M W (1998) Weyl Transforms. Ne York: Springer-Verlag. School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia address: dlabate@math.gatech.edu