Almost Diagonalization of Pseudodifferential Operators
aa r X i v : . [ m a t h . F A ] A p r Almost Diagonalization of PseudodifferentialOperators
S. Ivan Trapasso
Abstract
In this review we focus on the almost diagonalization of pseudodifferen-tial operators and highlight the advantages that time-frequency techniques providehere. In particular, we retrace the steps of an insightful paper by Gr¨ochenig, whosucceeded in characterizing a class of symbols previously investigated by Se¨ostrandby noticing that Gabor frames almost diagonalize the corresponding Weyl operators.This approach also allows to give new and more natural proofs of related results suchas boundedness of operators or algebra and Wiener properties of the symbol class.Then, we discuss some recent developments on the theme, namely an extension ofthese results to a more general family of pseudodifferential operators and similaroutcomes for a symbol class closely related to Sj¨ostrand’s one.This is a precopyedited version of a contribution published in
Landscapes ofTime-Frequency Analysis (Boggiatto P. et al. (eds)) published by Birkh¨auser, Cham.The definitive authenticated version is available online via https://doi.org/10.1007/978-3-030-05210-2 14.
Key words:
Almost diagonalization, τ -Wigner distribution, τ -pseudodifferentialoperators, Wiener algebras, Wiener amalgam spaces, modulation spaces : 47G30, 35S05, 42B35, 81S30 Salvatore Ivan TrapassoDipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di TorinoCorso Duca degli Abruzzi 24, 10129 Torino (Italy)e-mail: [email protected]
The wide range of problems that one can tackle by means of Time-frequency Anal-ysis bears witness to the relevance of this quite modern discipline stemmed fromboth pure and applied issues in harmonic analysis. There is no way to provide herea comprehensive bibliography on the theme, which would encompass studies inquantum mechanics and partial differential equations. We confine ourselves to listsome references to be used as points of departure for a walk through the topic: see[1, 4, 6, 10, 23, 26]. Besides the countless achievements as tool for other fields, Ga-bor analysis is a fascinating subject in itself and it may happen to shed new light onestablished facts in an effort to investigate the subtle problems underlying its foun-dation. We report here the case of Gr¨ochenig’s work [16]: the author retrieved andextended well-known outcomes obtained by Sj¨ostrand within the realm of “hard”analysis - cf. [24, 25], and this was achieved using techniques from phase spaceanalysis. We will give a detailed account in the subsequent sections, but let us brieflyintroduce here the main characters of this story.The (cross-)Wigner distribution is a quadratic time-frequency representation of sig-nals f , g in suitable function spaces (for instance f , g ∈ S ( R d ) , the Schwartz class)defined as W ( f , g )( x , ω ) = Z R d e − π iy ω f (cid:16) x + y (cid:17) g (cid:16) x − y (cid:17) d y . (1)It is possible to associate a pseudodifferential operator to this representation, namelythe so-called Weyl transform - it is a quite popular quantization rule in Physics com-munity. Given a tempered distribution σ ∈ S ′ ( R d ) as symbol (also observable , inphysics vocabulary), the corresponding Weyl transform maps S ( R d ) into S ′ ( R d ) and can be defined via duality by h Op W ( σ ) f , g i = h σ , W ( g , f ) i , f , g ∈ S ( R d ) . (2)The Weyl transform has been thoroughly studied in [14, 30] among others. Inhis aforementioned works, Sj¨ostrand proved that Weyl operators with symbols ofspecial type satisfy a number of interesting properties concerning their boundednessand algebraic structure as a set. In terms that will be specified later in Section 3,we can state that the set of such operators is a spectral invariant *-subalgebra of B ( L ( R d )) , the ( C ∗ -)algebra of bounded operators on L ( R d ) .To be precise, given a Schwartz function g ∈ S ( R d ) \ { } , we provisionallydefine the Sj¨ostrand’s class as the space of tempered distributions σ ∈ S ′ ( R d ) such that Z R d sup z ∈ R d |h σ , π ( z , ζ ) g i| d ζ < ∞ . As a rule of thumb, notice that a symbol in M ∞ , ( R d ) locally (i.e. for fixed z ∈ R d )coincides with the Fourier transform of a L function. Furthermore, it can be provedthat this somewhat exotic symbol class contains classical H¨ormander’s symbols oftype S , , together with non-smooth ones. lmost Diagonalization of Pseudodifferential Operators 3 The crucial remark here is that Sj¨ostrand’s class actually coincides with a func-tion space of a particular type, namely the modulation space M ∞ , ( R d ) . In moregeneral terms, modulation spaces (and also related Wiener amalgam spaces of spe-cial type) are Banach spaces defined by means of estimates on time-frequency con-centration and decay of its elements - see Section 2 for the details. They were in-troduced by Feichtinger in the ’80s (cf. the pioneering papers [7, 8]) and soon es-tablished themselves as the optimal environment for time-frequency analysis. Nev-ertheless, they also provide a fruitful context to set problems in harmonic analysisand PDEs - see for instance [11, 13, 29].Gr¨ochenig deeply exploited this connection with time-frequency analysis byproving that Sj¨ostrand’s results extend to more general modulation spaces and, moreimportantly, he was able to completely characterize symbols in these classes bymeans of a property satisfied by the corresponding Weyl operators, namely approx-imate diagonalization. This is a classical problem in pure and applied harmonicanalysis - a short list of references is [2, 19, 21, 22]. We will thoroughly examineGr¨ochenig’s results in Section 3. Here, we limit ourselves to heuristically argue thatthe choice of a certain type of symbols assures that the corresponding Weyl oper-ators preserve the time-frequency localization, since their “kernel” with respect tocontinuous or discrete time-frequency shifts satisfies a convenient decay condition.In the subsequent Section 4 we report some results on almost diagonalizationobtained by the author in a recent joint work with Elena Cordero and Fabio Nicola- see [5]. Mimicking the scheme which leads to define the Weyl transform, in [1]the authors consider a one-parameter family of time-frequency representations ( τ -Wigner distributions ) and also define the corresponding pseudodifferential operatorsOp τ via duality. Precisely, for τ ∈ [ , ] , the (cross-) τ -Wigner distribution is givenby W τ ( f , g )( x , ω ) = Z R d e − π iy ζ f ( x + τ y ) g ( x − ( − τ ) y ) d y , f , g ∈ S ( R d ) , (3)whereas the corresponding τ -pseudodifferential operator is defined by h Op τ ( a ) f , g i = h a , W τ ( g , f ) i , f , g ∈ S ( R d ) . (4)For τ = / τ = , τ -pseudodifferentialoperators for any τ ∈ [ , ] . While this is not surprising for reasons that will bediscussed later, it seems worthy of interest to get a similar result for symbols be-longing to a function space closely related to M ∞ , , namely the Wiener amalgamspace W ( F L ∞ , L ) . The connection between these spaces is established by Fouriertransform: in fact, the latter exactly contains the Fourier transforms of symbols in S. Ivan Trapasso the Sj¨ostrand’s class. It is important to remark that even if the spirit of the result isthe same, numerous differences occur and we try to clarify the intuition behind thissituation in Section 4.To conclude, we take advantage of this characterization in regards to bounded-ness results. We were able to study the boundedness of τ -pseudodifferential opera-tor covering several possible choices among modulation and Wiener amalgam spacefor symbols classes and spaces on which they act. We mention that in a number ofthese outcomes we have benefited from a strong linkage with the theory of Fourierintegral operators. Besides, the latter condition also made possible to establish (ordisprove) the algebraic properties considered by Sj¨ostrand for special classes of τ -pseudodifferential operators. Notation.
We write t = t · t , for t ∈ R d , and xy = x · y is the scalar product on R d . The Schwartz class is denoted by S ( R d ) , the space of tempered distributionsby S ′ ( R d ) . The brackets h f , g i denote both the duality pairing between S ′ ( R d ) and S ( R d ) and the inner product h f , g i = R f ( t ) g ( t ) d t on L ( R d ) . In particular, weassume it to be conjugate-linear in the second argument. The symbol . means thatthe underlying inequality holds up to a positive constant factor C > f . g ⇒ ∃ C > f ≤ Cg . . The Fourier transform of a function f on R d is normalized as F f ( ξ ) = Z R d e − π ix ξ f ( x ) d x . Given x , ω ∈ R d , the modulation M ω and translation T x operators acts on a func-tion f (on R d ) as M ω f ( t ) = e π it ω f ( t ) , T x f ( t ) = f ( t − x ) . We write a point in phase space as z = ( x , ω ) ∈ R d , and the corresponding phase-space shift acting on a function or distribution as π ( z ) f ( t ) = e π i ω t f ( t − x ) , t ∈ R d . (5)Denote by J the canonical symplectic matrix in R d : J = (cid:18) d × d I d × d − I d × d d × d (cid:19) ∈ Sp ( d , R ) , where the symplectic group Sp ( d , R ) is defined by lmost Diagonalization of Pseudodifferential Operators 5 Sp ( d , R ) = n M ∈ GL ( d , R ) : M ⊤ JM = J o . Observe that, for z = ( z , z ) ∈ R d , we have Jz = J ( z , z ) = ( z , − z ) , J − z = J − ( z , z ) = ( − z , z ) = − Jz , and J = − I d × d . Short-time Fourier transform.
Let f ∈ S ′ ( R d ) and g ∈ S ( R d ) \ { } . The short-time Fourier transform (STFT) of f with window function g is defined as V g f ( x , ω ) = h f , π ( x , ω ) g i = F ( f T x g )( ω ) = Z R d f ( y ) g ( y − x ) e − π iy ω d y . (6)We remark that the last expression has to be intended in formal sense, but it trulyrepresents the integral corresponding to the inner product h f , π ( x , ω ) g i whenever f , g ∈ L ( R d ) .Recall the fundamental property of time-frequency analysis: V g f ( x , ω ) = e − π ix ω V ˆ g ˆ f ( J ( x , ω )) . (7) Gabor frames.
Let Λ = A Z d , with A ∈ GL ( d , R ) , be a lattice in the time-frequency plane. The set of time-frequency shifts G ( ϕ , Λ ) = { π ( λ ) ϕ : λ ∈ Λ } fora non-zero ϕ ∈ L ( R d ) (the so-called window function) is called Gabor system. AGabor system G ( ϕ , Λ ) is said to be a Gabor frame if the lattice is such thick that theenergy content of a signal as sampled on the lattice by means of STFT is comparablewith its total energy, that is: there exist constants A , B > A k f k ≤ ∑ λ ∈ Λ |h f , π ( λ ) ϕ i| ≤ B k f k , ∀ f ∈ L ( R d ) . (8) Weight functions.
Let us call admissible weight function any non-negative contin-uous function v on R d such that:1. v ( ) = v is even in each coordinate: v ( ± z , . . . , ± z d ) = v ( z , . . . , z d ) . v is submultiplicative, that is v ( w + z ) ≤ v ( w ) v ( z ) ∀ w , z ∈ R d . v satisfies the Gelfand-Raikov-Shilov (GRS) condition:lim n → ∞ v ( nz ) n = ∀ z ∈ R d . (9) S. Ivan Trapasso
Examples of admissible weights are given by v ( z ) = e a | z | b ( + | z | ) s log r ( e + | z | ) ,with real parameters a , r , s ≥ ≤ b <
1. Functions of polynomial growth suchas v s ( z ) = h z i s = (cid:16) + | z | (cid:17) s , z ∈ R d , s ≥ v will denote an admissible weight func-tion unless otherwise specified. We remark that the GRS condition is exactly thetechnical tool required to forbid an exponential growth of the weight in some direc-tion. For further discussion on this feature, see [17].Given a submultiplicative weight v , a positive function m on R d is called v-moderate weight if there exists a constant C ≥ m ( z + z ) ≤ Cv ( z ) m ( z ) , z , z ∈ R d . The set of all v -moderate weights will be denoted by M v ( R d ) .In order to remain in the framework of tempered distributions, in what follows weshall always assume that weight functions m on R d under our consideration satisfythe following condition: m ( z ) ≥ , ∀ z ∈ R d or m ( z ) & h z i − N , (11)for some N ∈ N . The same holds with suitable modifications for weights on R d . Modulation spaces.
Given a non-zero window g ∈ S ( R d ) , a v -moderate weightfunction m on R d satisfying (11), and 1 ≤ p , q ≤ ∞ , the modulation space M p , qm ( R d ) consists of all tempered distributions f ∈ S ′ ( R d ) such that V g f ∈ L p , qm ( R d ) (weightedmixed-norm space). The norm on M p , qm is k f k M p , qm = k V g f k L p , qm = Z R d (cid:18) Z R d | V g f ( x , ω ) | p m ( x , ω ) p d x (cid:19) q / p d ω ! / q , with suitable modifications if p = ∞ or q = ∞ . If p = q , we write M pm instead of M p , pm , and if m ( z ) ≡ R d , then we write M p , q and M p for M p , qm and M p , pm .It can be proved (see [14]) that M p , qm ( R d ) is a Banach space whose definitionis independent of the choice of the window g - meaning that different windowsprovide equivalent norms on M p , qm . The window class can be extended to M v , cf.[14, Thm. 11.3.7]. Hence, given any g ∈ M v ( R d ) and f ∈ M p , qm we have k f k M p , qm ≍ k V g f k L p , qm . (12)We recall the inversion formula for the STFT (see [14, Proposition 11.3.2]). If g ∈ M v ( R d ) \ { } , f ∈ M p , qm ( R d ) , with m satisfying (11), then f = k g k Z R d V g f ( z ) π ( z ) g d z , (13) lmost Diagonalization of Pseudodifferential Operators 7 and the equality holds in M p , qm ( R d ) .The adjoint operator of V g , defined by V ∗ g F ( t ) = Z R d F ( z ) π ( z ) g d z , maps the Banach space L p , qm ( R d ) into M p , qm ( R d ) . In particular, if F = V g f the inver-sion formula (13) reads Id M p , qm = k g k V ∗ g V g . (14) Wiener Amalgam Spaces.
Fix g ∈ S ( R d ) \{ } and consider even weight functions u , w on R d satisfying (11). The Wiener amalgam space W ( F L pu , L qw )( R d ) is thespace of distributions f ∈ S ′ ( R d ) such that k f k W ( F L pu , L qw )( R d ) : = Z R d (cid:18) Z R d | V g f ( x , ω ) | p u p ( ω ) d ω (cid:19) q / p w q ( x ) d x ! / q < ∞ with obvious modifications for p = ∞ or q = ∞ .Using the fundamental identity of time-frequency analysis (7), we have | V g f ( x , ω ) | = | V ˆ g ˆ f ( ω , − x ) | = | F ( ˆ f T ω ˆ g )( − x ) | and (since u ( x ) = u ( − x ) ) k f k M p , qu ⊗ w = (cid:18) Z R d k ˆ f T ω ˆ g k q F L pu w q ( ω ) d ω (cid:19) / q = k ˆ f k W ( F L pu , L qw ) . Hence the Wiener amalgam spaces under our consideration are simply the imageunder Fourier transform of modulation spaces with weights of tensor product type,namely m ( x , ω ) = u ⊗ w ( x , ω ) = u ( x ) w ( ω ) : F ( M p , qu ⊗ w ) = W ( F L pu , L qw ) . (15)For this reason among others, their inventor H. Feichtinger suggested to call themmodulation spaces too - although in a generalized sense, see [9] for an intriguingconceptual account on the theme. τ -Pseudodifferential Operators Let us introduce the τ -pseudodifferential operators as it is customary in time-frequency analysis, i.e. by means of superposition of time-frequency shifts:Op τ ( σ ) f ( x ) = Z R d ˆ σ ( ω , u ) e − π i ( − τ ) ω u ( T − u M ω f ) ( x ) d u d ω , x ∈ R d , (16)for any τ ∈ [ , ] . The symbol σ and the function f belong to suitable functionspaces, to be determined in order for the previous expression to make sense. As an S. Ivan Trapasso example, minor modifications to [14, Lem. 14.3.1] give that Op τ ( σ ) maps S (cid:0) R d (cid:1) to S ′ (cid:0) R d (cid:1) whenever σ ∈ S ′ (cid:0) R d (cid:1) .Assuming that (16) is a well-defined absolutely convergent integral (for instance,it is enough to assume ˆ σ ∈ L (cid:0) R d (cid:1) ), easy computations lead to the usual integralform of τ -pseudodifferential operators, namelyOp τ ( σ ) f ( x ) = Z R d e π i ( x − y ) ω σ (( − τ ) x + τ y , ω ) f ( y ) d y d ω . We finally aim to represent Op τ ( σ ) as an integral operator of the formOp τ ( σ ) f ( x ) = Z R d k ( x , y ) f ( y ) d y . Let us introduce the operator T τ acting on functions on R d as T τ F ( x , y ) = F ( x + τ y , x − ( − τ ) y ) , T − τ F ( x , y ) = F (( − τ ) x + τ y , x − y ) , and denote by F i , i = ,
2, the partial Fourier transform with respect to the i -th d − dimensional variable (it is then clear that F = F F ).Since the operators T τ and F i are continuous bijections on S (cid:0) R d (cid:1) , the kernel k is well-defined (as a tempered distribution) also for symbols in S ′ (cid:0) R d (cid:1) and wefinally recover the representation by duality given in the Introduction according to[1]. Proposition 1.
For any symbol σ ∈ S ′ (cid:0) R d (cid:1) and any real τ ∈ [ , ] , the map Op τ ( σ ) : S (cid:0) R d (cid:1) → S (cid:0) R d (cid:1) is defined as integral operator with distributionalkernel k = T − τ F − σ ∈ S ′ (cid:16) R d (cid:17) , meaning that, for any f , g ∈ S (cid:0) R d (cid:1) , h Op τ ( σ ) f , g i = (cid:10) k , g ⊗ f (cid:11) . In particular, since the representationW τ ( f , g ) ( x , ω ) = F T τ ( f ⊗ g ) ( x , ω ) holds for f , g ∈ S (cid:0) R d (cid:1) , we have h Op τ ( σ ) f , g i = h σ , W τ ( g , f ) i . As a consequence of the celebrated Schwartz’s kernel theorem (see for in-stance [14, Theorem 14.3.4]), we are able to relate the representations for τ -pseudodifferential operators given insofar. lmost Diagonalization of Pseudodifferential Operators 9 Theorem 1.
Let T : S (cid:0) R d (cid:1) → S ′ (cid:0) R d (cid:1) be a continuous linear operator. Thereexist tempered distributions k , σ , F ∈ S ′ (cid:0) R d (cid:1) and τ ∈ [ , ] such that T admits thefollowing representations: ( i ) as an integral operator: h T f , g i = (cid:10) k , g ⊗ f (cid:11) for any f , g ∈ S (cid:0) R d (cid:1) ; ( ii ) as a τ -pseudodifferential operator T = Op τ ( σ ) with symbol σ ; ( iii ) as a superposition (in a weak sense) of time-frequency shifts :T = Z R d F ( x , ω ) e ( − τ ) π ix ω T x M ω d x d ω . The relations among k, σ and F are the following: σ = F T τ k , F = I ˆ σ , where I denotes the reflection in the second d-dimensional variable (i.e. I G ( x , ω ) = G ( x , − ω ) , ( x , ω ) ∈ R d ). To conclude this anthology, since the algebraic properties of pseudodifferentialoperators families will be considered, recall that the composition of Weyl transformsprovides a bilinear form on symbols, the so-called twisted product :Op W ( σ ) ◦ Op W ( ρ ) = Op W ( σ ♯ ρ ) . Although explicit formulas for the twisted product of symbols can be derived (cf.[30]), we will not need them hereafter. Anyway, this is a fundamental notion in orderto establish an algebra structure on symbol spaces: it is quite natural to ask if thecomposition of operators with symbols in the same class reveals to be an operatorof the same type for some symbol in the same class. Also recall that taking theadjoint of a Weyl operator provides an involution on the level of symbols, since ( Op W ( σ )) ∗ = Op W ( σ ) . The study of pseudodifferential operators has a wide and long tradition in the fieldof mathematical analysis, starting from the monumental work of H¨ormander. It hasto be noticed that the classical symbol classes considered in these investigationsare usually defined by means of differentiability conditions. In the spirit of time-frequency analysis, we hereby employ modulation and Wiener amalgam spacesas reservoirs of symbols for pseudodifferential operator and hence the short-timeFourier transform to shape the desired properties.Recall that the Sj¨ostrand’s class is the modulation space M ∞ , ( R d ) consisting ofdistributions σ ∈ S ′ ( R d ) such that Z R d sup z ∈ R d |h σ , π ( z , ζ ) g i| d ζ < ∞ . The control on symbols can be improved by weighting the condition on theirshort-time Fourier transform, i.e. the modulation space norm. In the following wewill employ weight functions of type 1 ⊗ v , where v is an admissible weight on R d , according to the properties assumed in the Preliminaries. Weighted Sj¨ostrand’sclasses of this type are thus defined as M ∞ , ⊗ v (cid:16) R d (cid:17) = ( σ ∈ S ′ (cid:16) R d (cid:17) : Z R d sup z ∈ R d | V g σ ( z , ζ ) | v ( ζ ) d ζ < ∞ ) . A function space closely related to the previous one is the Wiener amalgamspace W ( F L ∞ , L v )( R d ) . As discussed in the previous section, we have indeed W ( F L ∞ , L v )( R d ) = F M ∞ , ⊗ v ( R d ) . Heuristically, a symbol in W ( F L ∞ , L )( R d ) locally coincides with the Fourier transform of a L ∞ ( R d ) signal and exhibitsglobal decay of L type. For instance, the δ distribution (in S ′ ( R d ) ) belongs to W ( F L ∞ , L )( R d ) .Although Sj¨ostrand’s definition of the eponym symbol class was quite differentfrom the one given here in terms of modulation spaces, in his works [24, 25] heproved three fundamental results on Weyl operators with symbols in M ∞ , . Theorem 2. ( i ) ( Boundedness ) If σ ∈ M ∞ , (cid:0) R d (cid:1) , then Op W ( σ ) is a bounded operator onL ( R d ) . ( ii ) ( Algebra property ) If σ , σ ∈ M ∞ , (cid:0) R d (cid:1) and Op W ( ρ ) = Op W ( σ ) Op W ( σ ) ,then ρ = σ ♯ σ ∈ M ∞ , (cid:0) R d (cid:1) . ( iii ) ( Wiener property ) If σ ∈ M ∞ , (cid:0) R d (cid:1) and Op W ( σ ) is invertible on L ( R d ) , then [ Op W ( σ )] − = Op W ( ρ ) for some ρ ∈ M ∞ , (cid:0) R d (cid:1) . For sake of conciseness, we can resume the preceding outcomes by sayingthat the family of Weyl operators with symbols in Sj¨ostrand’s class (denoted byOp W ( M ∞ , ) ) is an inverse-closed Banach *-subalgebra of B ( L ( R d )) .Both these results and their original proofs might appear fairly technical at firstglance. Nonetheless, they unravel a deep and fascinating analogy between Weyl op-erators with symbols in the Sj¨ostrand’s class and Fourier series with ℓ coefficients.Similarities of this kind come under the multifaceted problem of spectral invariance,a topic thoroughly explored by Gr¨ochenig in his insightful lecture [17].In view of the structure of τ -pseudodifferential operators as superposition oftime-frequency shifts (cf. Equation (16)), it can be fruitful to study how operatorsinteract with time-frequency shifts. A measure of this interplay is given by the en-tries of the infinite matrix which we are going to refer to as channel matrix , ac-cording to traditional nomenclature in applied contexts like data transmission. First,fix a non-zero window ϕ ∈ M v ( R d ) (cid:0) R d (cid:1) and a lattice Λ = A Z d ⊆ R d , where A ∈ GL ( d , R ) , such that G ( ϕ , Λ ) is a Gabor frame for L (cid:0) R d (cid:1) . Therefore, theentries of the channel matrix are given by lmost Diagonalization of Pseudodifferential Operators 11 h Op W ( σ ) π ( z ) ϕ , π ( w ) ϕ i , z , w ∈ R d , or M ( σ ) λ , µ : = h Op W ( σ ) π ( λ ) ϕ , π ( µ ) ϕ i , λ , µ ∈ Λ , if we restrict to the lattice Λ . In this context, we could say that Op W is almostdiagonalized by the Gabor frame G ( ϕ , Λ ) if its channel matrix exhibits a suitableoff-diagonal decay. The key result proved by Gr¨ochenig in [16] is a characterizationof this type: a symbol belongs to the (weighted) Sj¨ostrand’s class if and only if time-frequency shifts are almost eigenvectors of the corresponding Weyl operator. Moreprecisely, the claim is the following. Theorem 3.
Let v be an admissible weight and fix a non-zero window ϕ ∈ M v ( R d ) (cid:0) R d (cid:1) such that G ( ϕ , Λ ) is a Gabor frame for L (cid:0) R d (cid:1) . The following properties are equiv-alent: ( i ) σ ∈ M ∞ , ⊗ v ◦ J − (cid:0) R d (cid:1) . ( ii ) σ ∈ S ′ (cid:0) R d (cid:1) and there exists a function H ∈ L v (cid:0) R d (cid:1) such that |h Op W ( σ ) π ( z ) ϕ , π ( w ) ϕ i| ≤ H ( w − z ) , ∀ w , z ∈ R d . ( iii ) σ ∈ S ′ (cid:0) R d (cid:1) and there exists a sequence h ∈ ℓ v ( Λ ) such that |h Op W ( σ ) π ( µ ) ϕ , π ( λ ) ϕ i| ≤ h ( λ − µ ) , ∀ λ , µ ∈ Λ . This characterization is very strong: in particular, by applying Schwartz’s kerneltheorem, we also have:
Corollary 1.
Under the hypotheses of the previous Theorem, assume that T : S (cid:0) R d (cid:1) → S ′ (cid:0) R d (cid:1) is continuous and satisfies one of the following conditions: ( i ) |h T π ( z ) ϕ , π ( w ) ϕ i| ≤ H ( w − z ) , ∀ w , z ∈ R d for some H ∈ L . ( ii ) |h T π ( µ ) ϕ , π ( λ ) ϕ i| ≤ h ( λ − µ ) , ∀ λ , µ ∈ Λ for some h ∈ ℓ .Therefore, T = Op W ( σ ) for some symbol σ ∈ M ∞ , ⊗ v ◦ J − (cid:0) R d (cid:1) . The proof of the main result heavily relies on a simple but crucial interplay be-tween the entries of the channel matrix of Op W and the short-time Fourier transformof the symbol, which will be discussed in complete generality in the subsequent sec-tion. We mention that at this point Gr¨ochenig establishes a strong link with matrixalgebra, hence heading towards a more conceptual discussion of the almost diago-nalization property. In particular, it is easy to prove that σ ∈ M ∞ , ⊗ v ◦ J − if and onlyif its channel matrix M ( σ ) belongs to the class C v ( Λ ) of matrices A = ( a λ , µ ) λ , µ ∈ Λ such that there exists a sequence h ∈ ℓ v which almost diagonalizes its entries, i.e. k a λ , µ k ≤ h ( λ − µ ) , λ , µ ∈ Λ . It can be proved that C v ( Λ ) is indeed a Banach *-algebra and this insight allows anatural extension if one considers other matrix algebras and investigates the relation between symbols and the membership of their Gabor matrices in a matrix algebra.For further investigations in more general contexts, see for instance [18].Thanks to this fresh new formulation, the proofs of Sj¨ostrand’s results providedby Gr¨ochenig are to certain extent more natural. Furthermore, they extend the previ-ous ones since weighted spaces are considered. We summarize the main outcomesin the following claims. Theorem 4 (Boundedness). If σ ∈ M ∞ , ⊗ v ◦ J − , then Op W ( σ ) is bounded on M p , qm for any ≤ p , q ≤ ∞ and anym ∈ M v . In particular, if σ ∈ M ∞ , , Op W ( σ ) is bounded on L ( R d ) and • if ≤ p ≤ , Op W ( σ ) maps L p into M p , p ′ ; • if ≤ p ≤ ∞ , Op W ( σ ) maps L p into M p . Theorem 5 (Algebra property).
If v is a submultiplicative on R d , then M ∞ , v is a Banach ∗ -algebra with respect tothe twisted product ♯ and the involution σ σ . Theorem 6 (Wiener property).
Assume that v is a submultiplicative weight on R d . Op W (cid:16) M ∞ , v (cid:17) is inverse-closed in B ( L ( R d )) (i.e. if σ ∈ (cid:16) M ∞ , v (cid:17) and Op W ( σ ) is invertible on L , then [ Op W ( σ )] − = Op W ( ρ ) for some ρ ∈ (cid:16) M ∞ , v (cid:17) ) if and only if v satisfies the GRScondition (9) . Corollary 2 (Spectral invariance on modulation spaces).
Assume that v is an admissible weight, σ ∈ (cid:16) M ∞ , v (cid:17) and Op W ( σ ) is invertible on L .Then, Op W ( σ ) is simultaneously invertible on every modulation space M p , qm ( R d ) ,for any ≤ p , q ≤ ∞ and m ∈ M v .Remark 1. The intuition behind the last result is that the spectrum of an operatorwith suitably likable properties does not truly depend on the space on which it acts.In order to establish a link with Beals’ theorem on spectral invariance in the contextof classical pseudodifferential operators, notice that H¨ormander’s class S , ( R d ) = { σ ∈ C ∞ ( R d ) : ∂ α σ ∈ L ∞ ( R d ) ∀ α ∈ N d } can be recast as intersection of Sj¨ostrand’s classes with polynomial weights (cf.[18]), namely S , ( R d ) = \ s ≥ M ∞ , v s ( R d ) . The Wiener property of these spaces leads to the conclusion that Op W (cid:16) S , (cid:17) isinverse-closed in B ( L ) too. lmost Diagonalization of Pseudodifferential Operators 13 τ -pseudodifferential operators In a recent joint work of the author with E. Cordero and F. Nicola, an attempt hasbeen made to follow the path outlined by Gr¨ochenig. The two directions investigatedare1. the extension of the almost-diagonalization theorem to more general operators;2. the search of an almost-diagonalization-like characterization of other symbolclasses.For what concerns the first point, τ -pseudodifferential operators were investigatedinstead of those of Weyl type. We already discussed in the Introduction how thisgeneral class of operators extends in a natural way the previous one, which can berecovered as the case τ = /
2. We were able to obtain an identical result with anidentical proof - apart from the substantial modifications in the preliminary lemmas- see [5] for the details.
Theorem 7.
Let v be an admissible weight on R d . Consider ϕ ∈ M v (cid:0) R d (cid:1) \ { } and a lattice Λ ⊆ R d such that G ( ϕ , Λ ) is a Gabor frame for L (cid:0) R d (cid:1) . For any τ ∈ [ , ] , the following properties are equivalent: ( i ) σ ∈ M ∞ , ⊗ v ◦ J − (cid:0) R d (cid:1) . ( ii ) σ ∈ S ′ (cid:0) R d (cid:1) and there exists a function H τ ∈ L v (cid:0) R d (cid:1) such that |h Op τ ( σ ) π ( z ) ϕ , π ( w ) ϕ i| ≤ H τ ( w − z ) ∀ w , z ∈ R d . ( iii ) σ ∈ S ′ (cid:0) R d (cid:1) and there exists a sequence h τ ∈ ℓ v ( Λ ) such that |h Op τ ( σ ) π ( µ ) ϕ , π ( λ ) ϕ i| ≤ h τ ( λ − µ ) ∀ λ , µ ∈ Λ . This result is not surprising for at least two reasons. Looking at the mappingrelating the symbols of different τ -quantizations, namely (see for instance [20, 27])Op τ ( a ) = Op τ ( a ) ⇔ b a ( ξ , ξ ) = e − π i ( τ − τ ) ξ ξ b a ( ξ , ξ ) , we see that the map that relates a Weyl symbol to its τ -counterpart is bounded inthe Sj¨ostrand’s class. At a more fundamental level, it is instructive to give a lookat the crucial ingredient of the proof, which is the relation between the channelmatrix of the τ -pseudodifferential operator and the short-time Fourier transform ofthe symbol. Proposition 2.
Fix a non-zero window ϕ ∈ S ( R d ) and set Φ τ = W τ ( ϕ , ϕ ) for τ ∈ [ , ] . Then, for σ ∈ S ′ (cid:0) R d (cid:1) , |h Op τ ( σ ) π ( z ) ϕ , π ( w ) ϕ i| = | V Φ τ σ ( T τ ( z , w ) , J ( w − z )) | = | V Φ τ σ ( x , y ) | (17) and | V Φ τ σ ( x , y ) | = |h Op τ ( σ ) π ( z ( x , y )) ϕ , π ( w ( x , y )) ϕ i| , (18) for all w , z , x , y ∈ R d , where T τ is defined as T τ ( z , w ) = (cid:18) ( − τ ) z + τ w τ z + ( − τ ) w (cid:19) z = ( z , z ) , w = ( w , w ) ∈ R d . (19) and z ( x , y ) = (cid:18) x + ( − τ ) y x − τ y (cid:19) , w ( x , y ) = (cid:18) x − τ y x + ( − τ ) y (cid:19) . (20)The main remark here is that the controlling function H τ ∈ L v ( R d ) in the almostdiagonalization theorem can be chosen as the so-called grand symbol associated to σ ∈ M ∞ , v ◦ J − (according to [15]): for the general τ -case, we have H τ ( v ) = sup u ∈ R d | V Φ τ σ ( u , Jv ) | . The choice of the grand symbol is quite natural if one looks at the modulation normin the Sj¨ostrand’s class. However, it is clear that the dependence from τ is com-pletely confined to the window function Φ τ and does not affect the variable v ∈ R d ,which corresponds to the frequency variable for the short-time Fourier transform ofthe symbol. The proof of the general case can thus proceed exactly as the one forWeyl case. We remark that also Corollary 1 generalizes in the obvious way.It is reasonable at this stage to ask what happens if a slight modification of thegrand symbol is taken into account, that is: what happens if we look at the time de-pendence of V Φ τ σ ? This is equivalent to wonder if similar arguments extend in somefashion to Fourier transform of symbols in the Sj¨ostrand’s class, namely symbols ina suitably weighted version of Wiener amalgam space W (cid:0) F L ∞ , L (cid:1) = F M ∞ , -hereinafter referred to F -Sj¨ostrand’s class. The main outcome we got is the follow-ing. Theorem 8.
Let v be an admissible weight function on R d . Consider ϕ ∈ M v (cid:0) R d (cid:1) \{ } . For any τ ∈ ( , ) , the following properties are equivalent: ( i ) σ ∈ W (cid:16) F L ∞ , L v ◦ B τ (cid:17) (cid:0) R d (cid:1) . ( ii ) σ ∈ S ′ (cid:0) R d (cid:1) and there exists a function H τ ∈ L v (cid:0) R d (cid:1) such that |h Op τ ( σ ) π ( z ) ϕ , π ( w ) ϕ i| ≤ H τ ( w − U τ z ) ∀ w , z ∈ R d , (21) where the matrices B τ and U τ are defined as B τ = (cid:18) − τ I d × d d × d d × d τ I d × d (cid:19) , U τ = − (cid:18) τ − τ I d × d d × d d × d − ττ I d × d (cid:19) ∈ Sp ( d , R ) . (22) If τ ∈ [ , ] , the estimate in (21) weakens as follows: ( ii ′ ) σ ∈ S ′ (cid:0) R d (cid:1) and there exists a function H τ ∈ L v (cid:0) R d (cid:1) such that |h Op τ ( σ ) π ( z ) ϕ , π ( w ) ϕ i| ≤ H τ ( T τ ( w , z )) ∀ w , z ∈ R d . (23) lmost Diagonalization of Pseudodifferential Operators 15 A number of differences arise with respect to its counterpart for Sj¨ostrand’s sym-bols. First, the almost diagonalization of the (continuous) channel matrix is lost, butthis is still a well-organized matrix: in the favourable case τ = ( , ) , (21) can beinterpreted as a measure of the concentration of the time-frequency representationof Op τ ( σ ) along the graph of the map U τ . If we include the endpoints, the estimateloses this meaning too.Furthermore, notice that the discrete characterization via Gabor frames is lost, themain obstruction being the following: for a given lattice Λ , the inclusion U τ Λ ⊆ Λ holds if and only if τ = /
2, i.e. U τ = U / = − I d × d . In this particular framework,the matrix B / then becomes B / = I d × d and the symmetry of Weyl operatorsis rewarded by an additional characterization: ( iii ′ ) σ ∈ S ′ (cid:0) R d (cid:1) and there exists a sequence h ∈ ℓ v ( Λ ) such that |h Op W ( σ ) π ( µ ) ϕ , π ( λ ) ϕ i| ≤ h ( λ + µ ) ∀ λ , µ ∈ Λ . We are now able to study the boundedness of τ -pseudodifferential operators cov-ering several possible choices for symbols classes and spaces on which they act. Ifone considers the action of τ -pseudodifferential operators on modulation spaces, aSj¨ostrand-type result for symbols in the Sj¨ostrand’s class can be inferred by meansof the same arguments applied in the Weyl case. Theorem 9.
Consider m ∈ M v (cid:0) R d (cid:1) satisfying (11) . For any τ ∈ [ , ] and σ ∈ M ∞ , ⊗ v ◦ J − the operator Op τ ( σ ) is bounded on M p , qm ( R d ) , and there exists a constantC τ > such that k Op τ ( σ ) k M p , qm ≤ C τ k σ k M ∞ , ⊗ v ◦ J − . (24)In order to address the problem of boundedness of τ -pseudodifferential operatorson modulation spaces with symbols in F -Sj¨ostrand’s class, a different strategy isneeded. Following [3], the idea is to recast Op τ ( σ ) as the transformation (via theshort-time Fourier transform and its adjoint) of an integral operator with the chan-nel matrix as distributional kernel. Therefore, the almost diagonalization propertyallows to obtain the desired estimates and claim the following result. Theorem 10.
Fix m ∈ M v satisfying (11) . For τ ∈ ( , ) consider a symbol σ ∈ W ( F L ∞ , L v ◦ B τ ) (cid:0) R d (cid:1) , with the matrix B τ defined in (22) . Then the operator Op τ ( σ ) is bounded from M p , qm (cid:0) R d (cid:1) to M p , qm ◦ U − − τ (cid:0) R d (cid:1) , ≤ p , q ≤ ∞ . We now turn to consider the boundedness of τ -pseudodifferential operators onWiener amalgam spaces. Looking for a big picture and given that modulation and Wiener amalgam spaces are intertwined by the Fourier transform, it is natural towonder if continuity properties of an operator acting on modulation spaces maystill hold true when it acts on the corresponding amalgam spaces. In the case of τ -pseudodifferential operators the answer is yes but heavily relies on the particu-lar way Fourier transform and τ -pseudodifferential operators commute. This phe-nomenon is a special case of the symplectic covariance property of Shubin calculus,which we briefly recall - see [12] for a comprehensive discussion on the issue. Lemma 1.
For any σ ∈ S ′ (cid:0) R d (cid:1) and τ ∈ [ , ] , F Op τ ( σ ) F − = Op − τ (cid:0) σ ◦ J − (cid:1) . This property, along with other preliminary results, allows to quickly prove thedesired claims for symbols in both Sj¨ostrand’s class and the corresponding amalgamspace.
Theorem 11.
Consider m = m ⊗ m ∈ M v (cid:0) R d (cid:1) satisfying (11) . For any τ ∈ [ , ] and σ ∈ M ∞ , ⊗ v (cid:0) R d (cid:1) , the operator Op τ ( σ ) is bounded on W (cid:0) F L pm , L qm (cid:1) (cid:0) R d (cid:1) with k Op τ ( σ ) k W ( F L pm , L qm ) ≤ C τ k σ k M ∞ , ⊗ v , for a suitable C τ > . Theorem 12.
Consider m = m ⊗ m ∈ M v (cid:0) R d (cid:1) satisfying (11) . For any τ ∈ ( , ) and σ ∈ W (cid:16) F L ∞ , L v ◦ B τ ◦ J − (cid:17) (cid:0) R d (cid:1) , the operator Op τ σ is bounded fromW (cid:0) F L pm , L qm (cid:1) (cid:0) R d (cid:1) to W (cid:18) F L pm ◦ ( U − − τ ) , L qm ◦ ( U − − τ ) (cid:19) (cid:0) R d (cid:1) , ≤ p , q ≤ ∞ , where (cid:0) U − − τ (cid:1) ( x ) = − τ − τ x , (cid:0) U − − τ (cid:1) ( x ) = − − ττ x , x ∈ R d . We finally remark that even if the results with symbols in F -Sj¨ostrand’s class donot hold for the endpoint cases τ = τ =
1, it is still possible to use the weakcharacterization (23) to construct ad hoc examples of bounded operators.
Proposition 3.
Assume σ ∈ W ( F L ∞ , L )( R d ) .1. The Kohn-Nirenberg operator Op KN ( σ ) ( τ = ) is bounded on M , ∞ ( R d ) .2. The anti-Kohn-Nirenberg Op ( σ ) ( τ = ) is bounded on W ( F L , L ∞ )( R d ) . To conclude, we give a brief summary on the extension of the other properties stud-ied by Sj¨ostrand, namely algebra and Wiener property, to τ -pseudodifferential op-erators. Wiener algebras of pseudodifferential operators have been already inves-tigated by Cordero, Gr¨ochenig, Nicola and Rodino in several occasions, see for lmost Diagonalization of Pseudodifferential Operators 17 instance [2, 3]. Let us recall the definition and the relevant properties of generalizedmetaplectic operators, introduced by the aforementioned authors. Definition 1.
Given A ∈ Sp ( d , R ) , g ∈ S ( R d ) , and s ≥
0, a linear operator T : S ( R d ) → S ′ ( R d ) belongs to the class FIO ( A , v s ) of generalized metaplec-tic operators if ∃ H ∈ L v s ( R d ) such that |h T π ( z ) g , π ( w ) g i| ≤ H ( w − A z ) , ∀ w , z ∈ R d . Theorem 13.
Fix A i ∈ Sp ( d , R ) , s i ≥ , m i ∈ M v si , and T i ∈ FIO ( A i , v s i ) , i = , , .1. T is bounded from M pm (cid:0) R d (cid:1) to M pm ◦ A − i (cid:0) R d (cid:1) for any ≤ p ≤ ∞ .2. T T ∈ FIO ( A A , v s ) , where s = min { s , s } .3. If T is invertible in L (cid:0) R d (cid:1) , then T − ∈ FIO (cid:0) A − , v s (cid:1) . In short, the class
FIO ( Sp ( d , R ) , v s ) = [ A ∈ Sp ( d , R ) FIO ( A , v s ) is a Wiener sub-algebra of B ( L ( R d )) . In view of the defining property of op-erators in FIO ( A , v s ) , we immediately recognize that for any τ ∈ ( , ) , if σ ∈ W ( F L ∞ , L v s ) then Op τ ( σ ) ∈ FIO ( U τ , v s ) . Therefore, if we limit to consider admis-sible weights of polynomial type v s on R d , s ≥
0, we are able to establish a fruitfulconnection and to derive a number of properties without any effort. For instance, wehave another boundedness result.
Corollary 3. If σ ∈ W ( F L ∞ , L v s )( R d ) , s ≥ , then the operator Op τ ( σ ) is boundedon every modulation space M pv s ( R d ) , for ≤ p ≤ ∞ and τ ∈ ( , ) . For what concerns the algebra property, we in fact have a no-go result. By in-specting the composition properties of matrices U τ , we notice that there is no τ ∈ ( , ) such that U τ U τ = U τ . This implies that there is no τ -quantization rulesuch that composition of τ -operators with symbols in W (cid:0) F L ∞ , L v s (cid:1) has symbol inthe same class. We can only state weaker algebraic results, such as the followingproperty of “symmetry” with respect to the Weyl quantization. Theorem 14.
For any a , b ∈ W (cid:0) F L ∞ , L v s (cid:1) ( R d ) and τ ∈ ( , ) , there exists a sym-bol c ∈ M ∞ , ⊗ v s ( R d ) such that Op τ ( a ) Op − τ ( b ) = Op / ( c ) . Also notice that, given a ∈ W (cid:0) F L ∞ , L v s (cid:1) , b ∈ M ∞ , ⊗ v s and τ , τ ∈ ( , ) , we haveOp τ ( b ) Op τ ( a ) = Op τ ( c ) , Op τ ( a ) Op τ ( b ) = Op τ ( c ) , for some c , c ∈ W (cid:0) F L ∞ , L v s (cid:1) . This means that, for fixed quantization rules τ , τ ,the amalgam space W (cid:0) F L ∞ , L v s (cid:1) ( R d ) is a bimodule over the algebra M ∞ , ⊗ v s ( R d ) under the laws M ∞ , ⊗ v s × W (cid:0) F L ∞ , L v s (cid:1) → W (cid:0) F L ∞ , L v s (cid:1) : ( b , a ) c , W (cid:0) F L ∞ , L v s (cid:1) × M ∞ , ⊗ v s → W (cid:0) F L ∞ , L v s (cid:1) : ( a , b ) c , with c and c as before.Finally, after noticing that U − τ = U − τ for any τ ∈ ( , ) , a Wiener-like propertycomes at the price of passing to the complementary τ -quantization when invertingOp τ . Theorem 15.
For any τ ∈ ( , ) and a ∈ W (cid:0) F L ∞ , L v s (cid:1) ( R d ) such that Op τ ( a ) isinvertible on L (cid:0) R d (cid:1) , we have Op τ ( a ) − = Op − τ ( b ) for some b ∈ W (cid:0) F L ∞ , L v s (cid:1) ( R d ) . References
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