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Kernel Theorems for Modulation Spaces 3 KERNEL THEOREMS FOR MODULATION SPACES ELENA CORDERO AND FABIO NICOLA Abstract. We deal with kernel theorems for modulation spaces. We completely characterize the continuity of a linear operator on the modulation spaces M p for every 1 p , by the membership of its kernel to (mixed) modulation spaces. Whereas≤ Feichtinger’s≤ ∞ kernel theorem (which we recapture as a special case) is the modulation space counterpart of Schwartz’ kernel theorem for temperate distributions, our results do not have a couterpart in distribution theory. This reveals the superiority, in some respects, of the modulation space formalism upon distribution theory, as already emphasized in Feichtinger’s manifesto for a post-modern harmonic analysis, tailored to the needs of mathematical signal processing. The proof uses in an essential way a discretization of the problem by means of Gabor frames. We also show the equivalence of the operator norm and the modulation space norm of the corresponding kernel. For operators acting on M p,q a similar characterization is not expected, but sufficient conditions for boundedness can be sated in the same spirit. 1. Introduction Schwartz’ kernel theorem certainly represents one of the most important results in modern Functional Analysis. It states, in the framework of temperate distribu- tions, that every linear continuous operator A : (Rd) ′(Rd) can be regarded as an integral operator in a generalized sense, namelyS → S Af,g = K,g f f,g (Rd), h i h ⊗ i ∈ S in the sense of distributions, for some kernel K ′(R2d), and viceversa [20]. We write, formally, ∈ S arXiv:1702.03201v1 [math.FA] 10 Feb 2017 (1) Af(x)= K(x, y) f(y)dy f (Rd). Rd ∈ S Z This result provides a general big framework including most operators of interest in Harmonic Analysis. However, for most applications in Time-frequency Analysis and Mathematical Signal Processing the spaces (Rd) and ′(Rd) can be fruit- fully replaced by the modulation spaces M 1(Rd) andS M ∞(Rd),S respectively, intro- duced by H. Feichtinger in [13] and nowadays widely used as a fundamental tool in 2010 Mathematics Subject Classification. 42B35, 42C15, 47G30, 81Q20. Key words and phrases. Time-frequency analysis, Gabor frames, modulation spaces, 1 2 ELENACORDEROANDFABIONICOLA Time-frequency Analysis (see [18] and also Feichtinger’s manifesto for a postmod- ern Harmonic Analysis [17]). One of the obvious advantages is that M 1(Rd) and M ∞(Rd) are Banach spaces; secondly their norm gives a direct and transparent information on the time-frequency concentration of a function or temperate distri- bution. The same holds, more generally, for a whole scale of intermediate spaces M p(Rd), 1 p , as well as the more general mixed norm spaces M p,q(Rd), 1 p, q ≤. ≤ ∞ ≤In short,≤∞ for (x, ξ) Rd Rd, define the time-frequency shifts π(x, ξ) by ∈ × π(x, ξ)f(t)= e2πitξf(t x), f ′(Rd). − ∈ S Then M p,q(Rd) is the space of temperate distributions f in Rd such that the function (x, ξ) f, π(x, ξ)g , 7→ h i for some (and therefore any) window g (Rd) 0 , is in Lq(Rd; Lp(Rd)), endowed ∈ S \{ } ξ x with the obvious norm (see Section 2 below for details). We also set M p(Rd) = M p,p(Rd). Weighted versions are also used in the literature but here we limit ourselves, for simplicity, to the unweighted case. Now, a kernel theorem in the framework of modulation spaces was announced in [9] and proved in [14]; see also [18, Theorem 14.4.1]. It states that linear continuous operators A : M 1(Rd) M ∞(Rd) are characterized by the memenbership of thier distribution kernel K to→M ∞(R2d). In this note we provide a similar characterization of linear continuous operators acting on the following spaces: M 1(Rd) M p(Rd), for a fixed 1 p • M p(Rd) → M ∞(Rd), for a fixed 1≤ p≤∞ • M p(Rd) → M p(Rd), for every 1 ≤p ≤∞. • → ≤ ≤∞ In all cases the characterization is given in terms of the membership of their distribution kernels to certain mixed modulation spaces, first introduced and studied in [3]. The proof uses in an essential way a discretization of the problem by means of Gabor frames. We also show the equivalence of the operator norm and the modulation space norm of the corresponding kernel. These results seem remarkable, because no such a characterization is known, for example, for linear continuous operators from (Rd) into itself and similarly from ′(Rd) into itself. This again shows the superiority,S in some respects, of the modulationS space framework upon distribution theory. For operators acting on mixed-norm modulation spaces M p,q(Rd) M p,q(Rd) such a characterization is not expected, as we will see in Section 3 below.→ We have however sufficient conditions for boundedness in the same spirit. KERNEL THEOREMS FOR MODULATION SPACES 3 Let us also observe that in the literature there are plenty of sufficient conditions for operators in certain classes (Fourier multipliers, localization operators, pseudo- differential operators, Fourier integral operators) to be bounded M p(Rd) M p(Rd) for every 1 p , or M p,q(Rd) M p,q(Rd) for every 1 p, q →. For ex- ample it is known≤ ≤ that ∞ pseudodifferential→ operators with Weyl≤ symbo≤l in ∞M ∞,1 are in fact bounded M p,q(Rd) M p,q(Rd) for every 1 p, q [6, 18, 19, 26, 27] → ≤ ≤ ∞ 2 (see also [24]). On the other hand the Fourier multipier with symbol ei|ξ| is still bounded on these spaces [2], although that symbol does not belong to M ∞,1(R2d). Similary, it is easy to see that any linear continuous operator A : (Rd) ′(Rd) whose Gabor matrix enjoys decay estimates such as S → S Aπ(z)g, π(w)g C(1 + w χ(z) )−s z,w R2d |h i| ≤ | − | ∈ for some s > 2d, g (Rd) 0 , and bi-Lipschtiz mapping χ : R2d R2d, is bounded M p(Rd) ∈M Sp(Rd) for\{ every} 1 p [4] (see also [4, 7, 8]).→ But once again this condition→ is not necessary at all.≤ ≤∞ The very simple characterization provided in the present paper when p = q therefore includes, at least implicitly, all these continuity results on M p. 2. Backgrounds on time-frequency analysis 2.1. Mixed norm spaces. We summarize the main properties of mixed norm spaces, we refer to [1] for a full treatment of this subject. Consider measure spaces (Xi,µi), indices pi [1, ], with i = 1,...,d. Then 1 2 ∈ ∞ the mixed norm space Lp ,p ,...,pd (X X X ; µ µ µ ) consists 1 × 2 ×···× d 1 × 2 ×···× d of the measurable functions F : X1 X2 Xd C such that the following norm is finite: × ×···× → 1 p2 p p1 d p1 (2) F p1,p2,...,pd = F (x ,...,x ) dµ (x ) dµ (x ) k kL ··· | 1 d | 1 1 ··· d d ZXd ZX1 ! with standard modification when p = for some i. i ∞ If Xi = R and µi is the Lebesgue measure on R for every i =1,...,d, we simply p1,...,pd write L . If each Xi is a countable set and the corresponding measure µi is p1,...,pd the counting measure, we use the notation ℓ (X1 X2 Xd). p1,p2,...,pd × ×···× The mixed norm spaces L (X1 X2 Xd; µ1 µ2 µd) are Banach spaces, generalizations of the classical× ×···×Lp and ℓp spaces,× cf.×···× [1]. 2.2. Modulation spaces. For x, ξ Rd, define the translation operator T and ∈ x modulation operator Mξ by T f(t)= f(t x) M f(t)= e2πitxf(t). x − ξ For z = (x, ξ), the composition operator π(z) = MξTx is called a time-frequency shift of the phase space R2d. 4 ELENACORDEROANDFABIONICOLA For a fixed non-zero g (Rd), the short-time Fourier transform (STFT) of f ′(Rd) with respect to∈ the S window g is given by ∈ S −2πiξt (3) Vgf(x, ξ)= f(t) g(t x) e dt = f, π(z)g z =(x, ξ), Rd − h i Z where the integral is intended as the (anti-)duality between ′ and . The short- time Fourier transform gives information about the time-frequencS yS content of the signal f: indeed, roughly speaking, it can be seen as a localized Fourier transform of f in a neighbourhood of x. The STFT is the basic tool in the definition of modulation spaces. Indeed, modulation space norms are a measure of the joint time-frequency distribution of f ′. For their basic properties we refer [10, 11, 12] (see also [3, 21, 22, 23]) and the∈ textbook S [18]. d Definition 2.1. Fix g (R ), p = (p1,...,pd), q = (q1,...,qd), pi, qi [1, ], i =1,...,d. Then ∈ S ∈ ∞ p,q d ′ d M (R )= f (R ): f p,q Rd < , { ∈ S k kM ( ) ∞} where f p,q Rd = V f p1,...,pd,q1,...,qd . k kM ( ) k g kL If p = p1,...,pd, q = q1,...,qd, we come back to the classical modulation spaces of distributions f ′(Rd) such that ∈ S 1 q q p p f M p,q(Rd) = Vgf(x, ξ) dx dξ < k k Rd Rd | | ∞ Z Z ! (cf. [13]). The extension of the original modulation spaces in Definition 2.1 was widely studied and applied to the investigation of boundedness and sampling prop- erties for pseudodifferential operators in [21, 22, 23]. To state our kernel theorems we need a further extension of Definition 2.1, that was introduced by S. Bishop in [3], in her study of Schatten p-class properties for integral operators.
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