Integr. equ. oper. theory 59 (2007), 99–128 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010099-30, published online June 27, 2007 DOI 10.1007/s00020-007-1504-2
Integral Equations and Operator Theory
Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols Patrik Wahlberg Abstract. We study the short-time Fourier transformation, modulation spaces, Gabor representations and time-frequency localization operators, for functions and tempered distributions that have as range space a Banach or a Hilbert space. In the Banach space case the theory of modulation spaces contains some modifications of the scalar-valued theory, depending on the Banach space. In the Hilbert space case the modulation spaces have properties similar to the scalar-valued case and the Gabor frame theory essentially works. For localization operators in this context symbols are operator-valued. We generalize two results from the scalar-valued theory on continuity on certain modulation spaces when the symbol belongs to an Lp,q space and M ∞ , respectively. The first result is true for any Banach space as range space, and the second result is true for any Hilbert space as range space. Mathematics Subject Classification (2000). Primary 47G30, 42B35; Secondary 47B38, 35S99. Keywords. Time-frequency analysis, vector-valued modulation spaces, localization operators, pseudodifferential operators.
1. Introduction The theory of the short-time Fourier transform (STFT) and modulation spaces of scalar-valued functions and tempered distributions is a very well developed theory of representation of tempered distibutions in the time-frequency (phase) space [22]. Define the modulation and translation operators by (Mξ f )(x) := ei2πξx f (x) and (Tt f )(x) := f (x− t), respectively. For a fixed so called window function g ∈ S(Rd ), the STFT f → Vg f is the localized Fourier transform defined by Vg f (t, ξ) = f, Mξ Tt g = f (x)ei2πxξ g(x − t)dx. (1.1) Rd
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The STFT is unitary L2 (Rd ) → L2 (R2d ) provided gL2 = 1, a topological isomorphism S(Rd ) → S(R2d ), and extends to a topological isomorphism S (Rd ) → S (R2d ). The inverse mapping is given by ∗ Vγ h = h(t, ξ)Mξ Tt γdtdξ, γ ∈ S(Rd ), γ, g = 1, (1.2) R2d
where Vγ∗ h in general is interpreted as the functional h(t, ξ)Mξ Tt γ, ϕdtdξ, ϕ ∈ S(Rd ). Vγ∗ h, ϕ = R2d
p,q The map (1.1) is also invertible on the family of modulation spaces Mm (Rd ), 1 ≤ p, q ≤ ∞, which were invented and developed 1983 by Feichtinger [12]. p,q (Rd ) = {f ∈ The weighted modulation spaces are Banach spaces defined by Mm d p,q 2d p,q 2d S (R ); Vg f ∈ Lm (R )} where Lm (R ) is the weighted mix-normed space of all measurable h : R2d → C such that q/p 1/q hLp,q = |h(t, ξ)m(t, ξ)|p dt dξ < ∞. m Rd
Rd
The modulation spaces thus quantifies the asymptotic decay of f ∈ S (Rd ) in the time and frequency variables. For the family of modulation spaces a discretized version of the map Vg is also invertible and there exists a so called Gabor frame expansion [6, 22] p,q f, Mβn Tαk gMβn Tαk γ, f ∈ Mm (Rd ), (1.3) f= k∈Zd n∈Zd
under certain conditions on the functions g, γ and the time-frequency lattice parameters α, β > 0 discovered by Feichtinger, Gr¨ ochenig, Leinert [15, 22, 24] and others. The inversion formula (1.2) suggests operators of the form g,γ (Aa f )(x) = a(t, ξ)Vg f (t, ξ)Mξ Tt γ(x)dtdξ R2d
which performs a multiplicative modification in the time-frequency domain, using a symbol a, before the distribution is reconstructed by integration. Such operators are called (time-frequency) localization operators, sometimes also Toeplitz or Anti-Wick operators, and they are a special case of pseudodifferential operators. The different names reflect that they have been studied in the signal analysis, mathematical and physical literature, as signal filtering operators [8, 45], pseudodifferential operators [17, 45], and quantization rules [17], respectively. They have recently attracted much attention [5, 7, 43, 44, 45], not least with respect to continuity when they act on modulation spaces. Cordero, Gr¨ochenig [7] and Toft [43] have proved that the large space of symbols M ∞ (R2d ) corresponds to operators that are bounded on all modulation spaces M p,q (Rd ), 1 ≤ p, q ≤ ∞. Also the discrete expansion (1.3) can be used to define localization operators, using discrete symbols, called Gabor multiplier operators [16]. More information on the theory
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and applications (eg in pseudodifferential calculus) of modulation spaces can be found in [14, 22, 23, 43, 44]. In this paper we generalize some of these results and replace scalar-valued functions and tempered distributions by vector-valued functions and tempered distributions taking values in a Banach space B or a Hilbert space H. First we discuss STFT and modulation space theory for vector-valued tempered distributions. It turns out that a large part of the theory of scalar-valued modulation spaces is possible to generalize to the H valued case, with certain modification in the B valued case depending on the Banach space B. This is essentially due to the fact that Parseval’s formula is true for B-valued L2 spaces if and only if B is isomorphic to a Hilbert space. The identity L2 = M 2 which is true in the scalar-valued case generalizes to the vector-valued case if and only if the vector space is a Hilbert space. In the general Banach space case this identity is replaced by the embeddings Lp ∩ FLp → M p , 1 ≤ p ≤ r, which depends on the Fourier type r ≤ 2 of the Banach space B. There is a generalization of the duality result (M p,q ) = M p ,q , 1 ≤ p, q < ∞ ([22, Thm. 11.3.6]), which is true when B is a reflexive Banach space and p = q = 1 or 1 < p < ∞ and 1 ≤ q < ∞. For a corresponding discussion, of greater depth, on the properties of vector-valued Sobolev and Besov spaces we refer to [38]. Furthermore we state that vector-valued modulation spaces can be characterized as the Fourier transform of Wiener amalgam spaces with discrete global component [10, 12, 13], similarly to the scalar-valued case. We discuss the concept of weak stationarity (with application eg to stochastic processes) in the Hvalued case and obtain the result that each weakly stationary H-valued tempered ∞,1 . distribution belongs to a weighted modulation space of the type M1⊗m We also discuss Gabor frame expansions for H-valued functions and tempered distributions and conclude that the Gabor frame theory for scalar-valued modulation spaces is possible to generalize to H-valued modulation spaces almost without modification. Finally we treat localization, Weyl and Gabor multiplier operators in this context, using an operator-valued symbol. We formulate generalizations of recent results on continuity of localization operators acting on modulation spaces, developed in the context of scalar-valued distributions and symbols. First we state that a result of Boggiatto’s [5] on continuity between certain modulation spaces of a localization operator, if the symbol belongs to Lp,q , generalizes to B-valued modulation spaces, regardless of B. By an example we show that there exists Banach spaces B1 , B2 and a symbol in L∞ such that the localization operator is not bounded from the B1 -valued L2 space to the B2 -valued L2 space, which can not happen in the scalar-valued case. Secondly we generalize the above mentioned result by Cordero and Gr¨ ochenig [7] on continuity of a localization operator, acting on any modulation space, provided the symbol belongs to M ∞ . We prove that the result is valid for H-valued modulation spaces. We also obtain a generalization of a result of Gr¨ ochenig and Heil [22, 23] on continuity of Weyl operators on all
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modulation spaces provided the symbol belongs to the Sj¨ ostrand space M ∞,1 , to the context of H-valued modulation spaces and operator-valued symbols. The motivating background of this work consists of two directions is contemporary analysis. On the one hand, pseudodifferential calculus for scalar-valued functions, which classically is studied with symbols in Fr´echet spaces of smooth functions [17, 27, 40], has since the work of Sj¨ostrand [41] been extended to symbols in modulation spaces, which are Banach spaces of limited regularity (see eg [5, 7, 22, 23, 41, 43, 44]). One of Sj¨ostrand’s results says that symbols in M ∞,1 give rise to bounded operators on L2 , which was extended by Gr¨ ochenig and Heil who proved boundedness on all modulation spaces M p,q , 1 ≤ p, q ≤ ∞, for the same class of symbols. On the other hand, there has independently of this trend appeared several papers dealing with new versions of classical Fourier multiplier theorems, where the operator act on scalar-valued function spaces, to vector-valued function spaces and operator-valued symbols [2, 19, 20, 29]. For example, versions of Mihlin’s multiplier theorem on conditions on a symbol that are sufficient for continuity on Lp spaces, have been proved for vector-valued Besov spaces by Amann, Girardi, Weis and Hyt¨onen [2, 20, 29]. Fourier multiplier operators are a special case of pseudodifferential operators, which also have been studied in the vector-valued context [30, 36]. These results have applications eg in PDE. Embeddings between scalar-valued Besov and modulation spaces have been proved by Toft [43]. It is natural to combine the above mentioned research directions and investigate pseudodifferential operators with operator-valued symbols in modulation spaces, acting on vector-valued modulation spaces, which is the topic of the present study. For this purpose we first need to examine the properties of vector-valued modulation spaces. There is also a motivation from the point of view of certain applications in engineering. In fact, pseudodifferential operators are used as mathematical models of mobile radio channels [42]. Such channels are often assumed to be stochastic [34]. If one assumes that the signals to be transmitted also are stochastic processes, then the framework presented here may be a candidate for a model of signal transmission, since stochastic processes may be seen as vector-valued functions where the vector space is a space of stochastic variables. 1.1. Definitions and notation We denote Lebesgue measure by µ, the Schwartz space by S(Rd ) and the smooth functions of compact support by Cc∞ (Rd ). The Fourier transformation of f ∈ S(Rd ) is defined by F f (ξ) = f(ξ) = f (t)e−i2πtξ dt. Rd
Partial Fourier transformation with respect to variable j is denoted Fj . The joint modulation-translation operator is sometimes denoted π(z) = Mξ Tt , z = (t, ξ) ∈ R2d . Discrete subgroups (lattices) of R2d of the form {(αk, βn)}k,n∈Zd , α, β > 0,
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will be denoted Λ. The (extended) Wigner distribution [17, 22] of a function f ∈ S(R2d ) is defined by f (t + τ /2, t − τ /2)e−i2πξτ dτ = F2 (f ◦ κ)(t, ξ) (1.4) W (f )(t, ξ) = Rd
with κ(t, τ ) = (t+τ /2, t−τ /2). When f = g⊗g we write W (g⊗g) = W (g) (ambiguous but clear from context) which is the proper Wigner distribution [17, 22]. We denote coordinate reflection by f(x) = f (−x). In estimates we denote by C a positive constant which may change value over inequalities. The tempered distributions are denoted S (Rd ), and are in this paper assumed to be antilinear (ie conjugate linear), in order to be compatible with the L2 (Rd ) inner product. The bracket ·, · will denote, depending on context, (i) action of an antilinear scalar-valued tempered distribution, (ii) action of an antilinear Banach space valued tempered distribution, or (iii) the Lebesgue or Bochner integral f, g = Rd f (x)g(x)dx where in general f is vector-valued and g is scalar-valued. The consistent definition of the := f, ϕ, ϕ ∈ S(Rd ). SomeFourier transformation of f ∈ S (Rd ) is then f, ϕ times we emphasize the space on which a distribution acts by writing eg ·, ·S(Rd ) . The map (1.4) extends by duality to a continuous map W : S (R2d ) → S (R2d ). For the range space of vector-valued distributions we use B to denote a Banach space and H to denote a Hilbert space. The topological dual of a Banach (or more generally a linear topological) space B is denoted B , the duality (·, ·)B ,B = (·, ·)B , and the Hilbert space inner product (·, ·)H , linear in the first and antilinear in the second argument. In order to be compatible with the Hilbert space inner product, (·, ·)B is defined to be linear in the first and antilinear in the second argument. For an exponent p, 1 ≤ p ≤ ∞, we denote the conjugate exponent p , which fulfills 1/p + 1/p = 1. The set of bounded linear transformations from a Banach space B1 to another Banach space B2 is denoted L(B1 , B2 ) and L(B, B) := L(B). Capital letters will often denote Banach space valued
functions or distributions. A function F is simple if it is a finite sum F = j xj χAj where xj ∈ B, χA denotes indicator function, Aj ∈ B(Rd ) (the Borel σ-algebra) and µ(Aj ) < ∞ for all j. A function F is strongly measurable if there exists a sequence of simple functions Fn such that limn→∞ Fn (t) − F (t)B = 0 for almost all t ∈ Rd . It is weakly measurable if t → (x , F (t))B is measurable for each x ∈ B , and it is almost separably valued if there exists a null set NF ⊂ Rd such that the range space F (Rd \ NF ) ⊂ B is separable. According to Pettis’s measurability theorem [9] F is strongly measurable if and only if F is weakly measurable and almost separably valued. The Bochner integral of a B-valued function F is well defined provided it is strongly measurable and Rd F (t)B dt < ∞ [9]. For each 1 ≤ p ≤ ∞ the Lebesgue-Bochner space Lp (Rd , B) [9] is defined as the set of strongly measurable functions F : Rd → B such that F Lp(Rd ,B) := ( Rd F (t)pB dt)1/p < ∞, with the standard modification when p = ∞. For F : Rd → B and G : Rd → B one defines the covariance function by (1.5) σF G (t, s) = F (t), G(s) B , t, s ∈ Rd .
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The Schwartz space of functions F : Rd → B such that supt∈Rd tα ∂ β F (t)B < ∞ for all multi-indices α and β is denoted S(Rd , B). There exists a theory of distributions which take values in a locally convex topological vector space, developed by L. Schwartz [39]. We will however restrict to tempered distributions taking values in a Banach space [1, 18, 28, 31, 32]. Such a distribution, denoted F ∈ S (Rd , B), is here defined as a bounded antilinear map F : S(Rd ) → B, ie there exists positive constants C, N, M such that F, ϕB ≤ C ϕα,β (1.6) |α|≤N
|β|≤M
where α, β are multi-indices and ϕα,β := supt∈Rd |∂ α (tβ ϕ(t))| are seminorms on S(Rd ). We have the duality result S(Rd , B) = S (Rd , B ), where the duality is induced by f ⊗ x , ϕ ⊗ x := f, ϕ(x , x)B , f ∈ S (Rd ), ϕ ∈ S(Rd ), x ∈ B , x ∈ B [1]. If B is reflexive then S(Rd , B) is reflexive as well [1]. For F ∈ S (Rd , B ) and G ∈ S (Rd , B) the covariance distribution σF G is defined by (1.7) σF G , ϕ ⊗ ψ = F, ϕ, G, ψ B , ϕ, ψ ∈ S(Rd ), for rank-one elements ϕ ⊗ ψ ∈ S(R2d ). According to the Schwartz kernel theorem [18, 37] one can extend σF G to a distribution σ ∈ S (R2d ). A function d F ∈ Lp (Rd , B), 1 ≤ p ≤ ∞, defines an element in S (R , B) by the Bochner integral F, ϕ = Rd F (t)ϕ(t)dt, which admits consistency between (1.5) and (1.7). The Fourier transform of a function F ∈ L1 (Rd , B) is defined by a Bochner Fourier integral and the Fourier transform F ∈ L∞ (Rd , B). The Hausdorff-Young inequality is however in general not valid [19, 20, 35]. A Banach space B is said to be of Fourier type r, 1 ≤ r ≤ 2, if the Fourier transform F : Lr (Rd , B) → Lr (Rd , B). This notion was introduced by Peetre [35]. It follows by interpolation that if B is of Fourier type r then it is also of Fourier type p for all p in the interval 1 ≤ p ≤ r. Every Banach space has Fourier type at least one. By a result of Kwapie´ n [33] a Banach space B has Fourier type 2 if and only if it is isomorphic to a Hilbert space. Hence Parseval’s formula holds for L2 (Rd , B) if and only if B is isomorphic to a Hilbert space.
2. The STFT of vector-valued tempered distributions
The STFT of F ∈ Lp (Rd , B) with respect to a window function g ∈ Lp (Rd ) is defined by the Bochner integral F (s)ei2πξs g(s − t)ds = F (F Tt g)(ξ) = F, Mξ Tt g. (2.1) Vg F (t, ξ) = Rd
Following the definition of the STFT of scalar-valued tempered distributions [22], the STFT of a vector valued F ∈ S (Rd , B) is for g ∈ S(Rd ) defined by the right hand side of (2.1) where ·, · denotes distribution action instead of Bochner
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integral. The following two lemmas generalize results for scalar-valued tempered distributions and will be useful in later sections. Lemma 2.1. If F ∈ S (Rd , B) and g ∈ S(Rd ) then Vg F ∈ C ∞ (R2d , B), Vg F is strongly measurable, and there exists an integer N > 0 and C > 0 such that Vg F (t, ξ)B ≤ C(1 + |t| + |ξ|)N , (t, ξ) ∈ R2d .
(2.2)
Proof. By [26, Thm. 2.1.3] (slightly modified) Vg F ∈ C ∞ (R2d , B), which implies that Vg F is strongly measurable. By (1.6) there exists positive C, M1 and M2 such that F, Mξ Tt gB ≤ C Mξ Tt gα,β . (2.3) |α|≤M1
|β|≤M2
Lemma 11.2.1 in [22] gives the bound Mξ Tt gα,β ≤ C(1 + |t| + |ξ|)2 max(|α|,|β|) , which in combination with (2.3) gives (2.2). Lemma 2.2. If F1 ∈ S(Rd , B ), F2 ∈ S(Rd , B) and g1 , g2 ∈ S(Rd ) then (F1 (x), F2 (x))B dx g1 , g2 L2 (Rd ) . Vg1 F1 (z), Vg2 F2 (z) B dz = R2d
Rd
Proof. We obtain from (2.1), since σF1 F2 ∈ S(R2d ), Vg1 F1 (z), Vg2 F2 (z) B dz R2d = ··· σF1 F2 (x, y)e−i2πξ(x−y) g1 (x − t)g2 (y − t)dxdydtdξ R4d = σF1 F2 (x, x)dx g1 , g2 L2 (Rd ) . Rd
Density arguments [1] now gives the following generalization of the orthogonality relations of the STFT [22] from scalar-valued L2 functions to H-valued L2 functions. The inner product in the Hilbert space L2 (Rd , H) is (F, G)L2 (Rd ,H) = Rd (F (t), G(t))H dt. Corollary 2.3. If F1 , F2 ∈ L2 (Rd , H) and g1 , g2 ∈ L2 (Rd ) then (Vg1 F1 , Vg2 F2 )L2 (R2d ,H) = (F1 , F2 )L2 (Rd ,H) g1 , g2 L2 (Rd ) . Given F ∈ L2 (R2d , H) and γ ∈ L2 (Rd ) we define for G ∈ L2 (Rd , H) (Uγ F, G) := (F (z), Vγ G(z))H dz. R2d
Then Uγ F is a continuous antilinear functional on L2 (Rd , H), since by Cor. 2.3 |(Uγ F, G)| ≤ F L2 (R2d ,H) GL2 (Rd ,H) γL2 (Rd ) . Since L2 (Rd , H) is a Hilbert space there exists an element, denoted R2d F (z)π(z)γdz ∈ L2 (Rd , H), such that (Uγ F, G) = F (z)π(z)γdz, G L2 (Rd ,H) . R2d
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The next result generalizes [22, Cor. 3.2.3] and says that F ∈ L2 (Rd , H) can be reconstructed from Vg F . Proposition 2.4. If g, γ ∈ L2 (Rd ) fulfill γ, g = 0 then −1 F = γ, g Vg F (z)π(z)γdz, F ∈ L2 (Rd , H).
(2.4)
R2d
Proof. Let G ∈ L2 (Rd , H). Then γ, g−1 Vg F (z)π(z)γdz, G L2 (Rd ,H) R2d = γ, g−1 (Vg F (z), Vγ G(z))H dz = (F, G)L2 (Rd ,H) R2d
2
by Cor. 2.3. Since G ∈ L (Rd , H) is arbitrary (2.4) follows.
Next we extend Prop. 2.4 and show that the STFT is invertible on S (Rd , B). The proof is similar to the corresponding result for S (Rd ) [22]. Define for a strongly measurable F : R2d → B and γ ∈ S(Rd ) the B-valued map F (z)π(z)γ, ϕdz, ϕ ∈ S(Rd ), (2.5) Vγ∗ F, ϕ := R2d
and denote this Vγ∗ F =
F (z)π(z)γdz. R2d
We can define F0 := γ, g−1 Vγ∗ Vg F , ie F0 , ϕ = γ, g−1 R2d Vg F (z)π(z)γ, ϕdz, since Vg F is strongly measurable by Lemma 2.1. In order to show that F0 ∈ S (Rd , B) we estimate using (2.2) Vg F (z)B |Vγ ϕ(z)| dz F0 , ϕB ≤ |γ, g|−1 R2d (2.6) ≤ C sup (1 + |z|)N +2d+1 |Vγ ϕ(z)| . z
According to [22, Cor. 11.2.6] the last expression is a seminorm on ϕ ∈ S(Rd ), and hence F0 ∈ S (Rd , B). Proposition 2.5. If g, γ ∈ S(Rd ) fulfill γ, g = 0 then F = F0 := γ, g−1 Vγ∗ Vg F, F ∈ S (Rd , B). (2.7) Proof. For ϕ ∈ S(Rd ) we have ϕ = g, γ−1 R2d Vγ ϕ(z)π(z)gdz. Thus, by a generalization of the tensor product rule f ⊗ 1 = 1 ⊗ f for f ∈ S (Rd ) [26] to f ∈ S (Rd , B) [38], we have −1 F, π(z)gVγ ϕ(z)dz F, ϕ = g, γ R2d = γ, g−1 Vg F (z)π(z)γ, ϕdz = F0 , ϕ. R2d
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p,q 3. Modulation spaces Mm (Rd , B)
The concept of weighted mix-normed spaces of scalar-valued functions [3] can be generalized to weighted mix-normed spaces of vector-valued functions F : R2d → 2d B. For 1 ≤ p, q ≤ ∞, F ∈ Lp,q m (R , B) if F is strongly measurable and q/p 1/q F Lp,q = F (t, ξ)pB m(t, ξ)p dt dξ < ∞, 2d ,B) m (R Rd
Rd
with the standard modification when p or q equals infinity. Here m is a positive weight function which is assumed to be v-moderate [22], ie m fulfills m(x + y) ≤ Cv(x)m(y), x, y ∈ R2d , for a positive function v which is submultiplicative, ie v(x + y) ≤ v(x)v(y), x, y ∈ R2d . In this paper we restrict to polynomially bounded functions v. A standard class of polynomially increasing submultiplicative functions is vs (x) = (1 + |x|2 )s/2 , x ∈ R2d , s ≥ 0. In Section 4 we need also the corresponding class where the weight is constant in the first variable, denoted τs (x) = (1 + |x2 |2 )s/2 , s ≥ 0, x1 ∈ Rd , x2 ∈ Rd , x = (x1 , x2 ). Analogously to the case of scalar-valued distributions [12, 22] we define modulation spaces, p,q 2d 1 ≤ p, q ≤ ∞, by F ∈ Mm,g (Rd , B), if F ∈ S (Rd , B) and Vg F ∈ Lp,q m (R , B) where g ∈ S. By Lemma 2.1 Vg F is always strongly measurable. The modulation space norm is d p,q F Mm,g 2d ,B) , g ∈ S(R ). (Rd ,B) := Vg F Lp,q m (R
(3.1)
Modulation spaces of Banach space valued distributions have already been considered by Toft [44]. 3.1. Retract property, admissible windows, Wiener amalgam space characterization, properties Prop. 2.5 says that γ, g−1 Vγ∗ Vg = idS (Rd ,B) . The following result corresponds to [22, Prop. 11.3.2] and [14, Cor. 4.6], has a similar proof, and shows that 2d p,q d p,q d Vγ∗ : Lp,q m (R , B) → Mm,g (R , B), ie Mm,g (R , B) is a Banach space retract of p,q 2d Lm (R , B) [4]. Proposition 3.1. Suppose g, γ ∈ S(Rd ) fulfill γ, g = 0, m is a v-moderate weight p,q p,q function, and 1 ≤ p, q ≤ ∞. Then Mm,g (Rd , B) → S (Rd , B), and Mm,g (Rd , B) is p,q 2d −1 ∗ p,q a Banach space retract of Lm (R , B), ie γ, g Vγ Vg = idMm,g (Rd ,B) and p,q 2d Vg : Mm,g (Rd , B) → Lp,q m (R , B), 2d p,q d Vγ∗ : Lp,q m (R , B) → Mm,g (R , B). p,q 2d Proof. By (3.1) Vg is an isometric linear map from Mm,g (Rd , B) to Lp,q m (R , B), −1 ∗ −1 ∗ p,q Vγ Vg = idS (Rd ,B) . If G ∈ and γ, g Vγ Vg = idMm,g (Rd ,B) follows from γ, g 2d (R , B) then G is strongly measurable and by H¨ o lder’s inequality Lp,q m Vγ∗ G, ϕB ≤ G(z)B |Vγ ϕ(z)| dz R2d (3.2) ≤ GLp,q sup (v (z) |V ϕ(z)|) , 2d ,B) v−N p ,q N γ (R 2d m L (R ) 1/m
z
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,q where N is chosen large enough to guarantee v−N ∈ Lp1/m (R2d ), which is possible since 1/m ≤ Cv [22] and v is polynomially bounded. Thus Vγ∗ G ∈ S (R2d , B). By Lemma 2.1 Vg (Vγ∗ G) is strongly measurable. By the commutation rule Tx My f = e−i2πxy My Tx f G(s, η)Vγ (Mξ Tt g)(s, η)dsdη Vg (Vγ∗ G)(t, ξ) = Vγ∗ G, Mξ Tt g = R2d (3.3) −i2πs(ξ−η) = G(s, η)Vg γ(t − s, ξ − η)e dsdη. R2d
Hence
Vg (Vγ∗ G)(z)B ≤ G(·)B ∗ |Vg γ| (z)
(3.4)
which, using an inequality of Young type for mix-normed spaces, [22, Prop. 11.1.3], admits the estimate ∗ p,q Vγ∗ GMm,g (Rd ,B) = Vg (Vγ G)(z)B Lp,q 2d ) m (R (3.5) ≤ C G(·)B Lp,q (R2d ) Vg γL1v (R2d ) . m
2d p,q d Thus Vγ∗ defines a bounded linear map from Lp,q m (R , B) to Mm,g (R , B). If we p,q put G = Vg F with F ∈ Mm,g (Rd , B) and use Prop. 2.5, we can also conclude from d p,q (3.2) that F, ϕB ≤ C(ϕ)F Mm,g (Rd ,B) where C(ϕ) is a seminorm on S(R ), p,q d d ie Mm,g (R , B) → S (R , B).
For later use we specialize the inequality (3.4) to G = Vγ F where F ∈ S (Rd , B), ie Vg F (z)B = γ−2 Vg Vγ∗ Vγ F (z)B ≤ γ−2 Vγ F (·)B ∗ |Vg γ| (z).
(3.6)
Likewise, with arguments similar to the proof of the corresponding result for scalar-valued modulation spaces [22, Thm. 11.3.7] (we omit the details), one can prove the following result. Proposition 3.2. Let m be a v-moderate weight function. p,q p,q (i) If g, γ ∈ S(Rd ) then · Mm,γ (Rd ,B) and · Mm,g (Rd ,B) are equivalent norms. d 1 d p,q p,q (ii) If g ∈ S(R ) and γ ∈ Mv (R ) then · Mm,γ (Rd ,B) and · Mm,g (Rd ,B) are equivalent norms. (iii) Prop. 3.1 holds with g, γ ∈ Mv1 (Rd ). p,q (Rd , B) are independent of g ∈ Mv1 (Rd ), and Consequently the spaces Mm,g for a fixed but arbitrary 0 = g ∈ Mv1 (Rd ) and a v-moderate weight function p,q m we can define the modulation space Mm (Rd , B) ⊂ S (Rd , B) as the set of d p,q F ∈ S (R , B) such that F Mm,g (Rd ,B) < ∞. We follow the conventional notation p p,p (Rd , B) = Mm (Rd , B) and M p,q (Rd , B) for the unweighted case, ie m ≡ 1. Mm The fact that the Fourier transform of the modulation spaces of scalarvalued distributions equals certain Wiener amalgam spaces [12] is true also in
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the Banach space valued case. In fact, if F F ∈ M p,q (Rd , B) and g ∈ Cc∞ (Rd ), then F, Mξ T−t g = F (F T−t g)(ξ) by a generalization to the vector-valued case of a result valid for scalar-valued distributions of compact support [26, 38]. From gB = F, Mξ T−t gB = F (F T−t g)(ξ)B we obtain FF, Mt Tξ 1/q , F F M p,q (Rd ,B) = F T−t gqF Lp (Rd ,B) dt = F p d q d W F L (R ,B),L (R )
Rd
where W F Lp (Rd , B), Lq (Rd ) is the Wiener amalgam space of B-valued tempered distributions with local component F Lp (Rd , B) and global component Lq (Rd ) [10, 12]. The norm of a Wiener amalgam space is equivalent to a discrete norm defined by a so called BUPU (bounded uniform partition of unity) [10, 12, 13]. A BUPU Ψ = {ψj }j∈J is a set of nonnegative functions required to fulfill, for a discrete set {xj }j∈J and a relatively compact set U with nonempty interior, ψj (x) ≡ 1, (i) j∈J
(ii) sup ψj F L1 (Rd ) < ∞, j∈J
(iii) supp ψj ⊂ xj + U ∀j ∈ J, (iv) sup |{j; (xi + U ) ∩ (xj + U ) = ∅}| < ∞. i∈J
Then if ψj ∈ M 1 (Rd ) for all j we have the norm equivalence for all 1 ≤ p, q ≤ ∞ [10, 13] 1/q F W (F Lp(Rd ,B),Lq (Rd )) F ψj qF Lp(Rd ,B) . j∈J
The proof of this result in the scalar-valued case [10, 13] extends to the Banach space valued case. The proof of the completeness of Wiener amalgam spaces W (X, Y ) under quite general hypotheses on the local norm X and the global p,q (Rd , B) are norm Y [10] also extends to W F Lp (Rd , B), Lq (Rd ) , and hence Mm Banach spaces for all 1 ≤ p, q ≤ ∞. The following proposition treats more generalizations of results valid for scalar-valued modulation spaces. We omit the proofs since they are almost identical to the proofs in [22], again after replacement of | · | with · B in the obvious places. Proposition 3.3. Suppose m is a v-moderate weight function. (i) If |m(z)| ≤ CvN (z), z ∈ R2d , for some N > 0 and 1 ≤ p, q < ∞, then p,q (Rd , B). S(Rd , B) is dense in Mm p1 ,q1 p2 ,q2 (ii) If p1 ≤ p2 , q1 ≤ q2 and m2 ≤ Cm1 then Mm (Rd , B) → Mm (Rd , B). 1 2 d ∞ d (iii) S (R , B) = s≥0 Mv−s (R , B). (iv) If m(ξ, −t) ≤ Cm(t, ξ) then the Fourier transformation is continuous on p (Rd , B). Mm
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(v) Let H be a Hilbert space. 2 If m depends only on t then L2m (Rd , H) = Mm (Rd , H) with equivalent norms. 2 d 2 If m depends only on ξ then F Lm (R , H) = Mm (Rd , H) with equivalent norms. In the following proposition we use the Fourier type of the Banach space B. It can be seen as a modification of (v) in the last proposition. Proposition 3.4. Suppose B has Fourier type r, 1 ≤ r ≤ 2, let 1 ≤ p ≤ r and let m be a v-moderate weight function.
p (i) If m depends only on t then Lpm (Rd , B) → Mm (Rd , B). p d p (ii) If m depends only on ξ then F Lm (R , B) → Mm (Rd , B). p d p d p d (iii) Thus L (R , B) ∩ FL (R , B) → M (R , B).
Proof. (i). Let F ∈ S (Rd , B) and ϕ ∈ Cc∞ (R2d ). If p > 1, using the restricted Hausdorff-Young inequality which is true by the definition of the Fourier type, p /p ≥ 1, Minkowski’s inequality and m(s − x) ≤ Cm(s)v(−x), we have F p p d = F (F Tt ϕ)(ξ)pB dξ m(t)p dt Mm (R ,B) d Rd R p /p ≤C (F Tt ϕ)(s)pB ds m(t)p dt d d R R p /p =C F (s)pB |ϕ(s − t)|p m(t)p ds dt Rd Rd (3.7) p/p p /p p p p ≤C F (s)B |ϕ(s − t)| m(t) dt ds d Rd R p /p ≤C F (s)pB m(s)p ds |ϕ(x)|p v(−x)p dx =
Rd CF pLpm (Rd ,B) .
Rd
The case p = 1 follows similarly. Result (ii) follows from the estimate (3.7) and B. F, Mξ Tt ϕB = FF, M−t Tξ ϕ p,q (Rd , B) when B is reflexive 3.2. The dual of Mm The following three lemmas have statements and proofs that are slight adaptations of results in [3] where B is a space of scalars.
Lemma 3.5. If 1 ≤ p, q ≤ ∞ and F : R2d → B is strongly measurable then
F Lp,q (R2d ,B) = sup G(z), F (z) B dz R2d
where the supremum is taken over all G : R2d → B such that: (i) z → G(z), F (z) B is measurable, (ii) z → G(z)B is measurable, and (iii) GLp ,q (R2d ,B ) = 1.
(3.8)
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If B = H is a Hilbert space then the supremum can be taken over G ∈ Lp ,q (R2d , H) such that GLp ,q (R2d ,H) = 1. Note that in the case of a general Banach space B the requirements (i)–(iii) do not imply G ∈ Lp ,q (R2d , B ) since G is not necessarily strongly measurable. Proof. Inequality ≥ in (3.8) follows from H¨ older’s inequality. To prove inequality ≤ we first restrict to F Lp,q (R2d ,B) < ∞. For all (x, y) ∈ R2d there exists by the Hahn-Banach theorem a functional P (x, y) ∈ B such that P (x, y)B = 1 and P (x, y), F (x, y) B = F (x, y)B . When F (x, y) = 0 we define P (x, y) = 0. If p, q < ∞ we define q−p · F (x, y)p−1 · F (·, y)B Lp (Rd ) . G(x, y) := P (x, y) · F 1−q B Lp,q (R2d ,B) Then (i)–(iii) are fulfilled and R2d G(z), F (z) B dz = F Lp,q (R2d ,B) . Next suppose p = ∞, q < ∞ and supp(F ) is compact. Let > 0 and define q−1 G (x, y) := P (x, y) · F 1−q · F (·, y)B Lp (Rd ) · χU (x, y)/µ(U (·, y)) Lp,q (R2d ,B) if µ(U (·, y)) > 0 and G (x, y) := 0 otherwise, where U = {(x, y) ∈ R2d : F (x, y)B > (1 + )−1 F (·, y)B Lp (Rd ) } ⊂ supp(F ). Then (i)–(iii) are again fulfilled, and G (z), F (z) B dz ≥ (1 + )−1 F Lp,q (R2d ,B) . R2d
Also if supp(F ) is not compact the result follows since F Lp,q (R2d ,B) = lim F χKj Lp,q (R2d ,B) j→∞
for an exhausting nested sequence of compact sets Kj ⊂ R2d . The cases p < ∞, q = ∞ and p = q = ∞ are proved in a way similar to p = ∞, q < ∞ [3]. If F Lp,q (R2d ,B) = ∞ we define Fj := F χKj if F (z)B ≤ j and Fj := 0 otherwise. Then Fj ∈ Lp,q (R2d , B) and Fj (z)B → F (z)B monotoneously as j → ∞ for all z ∈ R2d . Hence ∞ = F Lp,q (R2d ,B) = lim Fj Lp,q (R2d ,B) j→∞
= lim sup G(z), Fj (z) B dz j→∞ G R2d
≤ sup G(z), F (z) dz . G
R2d
B
Finally, in the case B = H, P (z) = F (z)/F (z)H which means that P , and therefore also G, is strongly measurable.
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Remark 3.6. Not only in the Hilbert space case B = H the supremum in (3.8) can be taken over all (strongly measurable) G ∈ Lp ,q (R2d , B ) such that GLp ,q (R2d ,B ) = 1. In fact it can be checked that this is also true when B = lr (Z), 1 < r < ∞. Lemma 3.7. The mix–normed space Lp,q (R2d , B) is complete for 1 ≤ p, q ≤ ∞. Proof. Let {Fn }n≥1 be a Cauchy sequence in Lp,q (R2d , B) and let {Kj }j≥1 be an exhaustive nested sequence of compact sets in R2d . By H¨older’s inequality Fn (z) − Fm (z)B dz ≤ Fn − Fm Lp,q (R2d ,B) χKj Lp ,q (R2d ) . Kj
There exists F such that limn→∞ (F − Fn )χKj L1 (R2d ,B) = 0 for every j, by the completeness of L1 (R2d , B). F is strongly measurable since F χKj ∈ L1 (R2d , B) for each Kj . By a Cantor diagonal procedure we can extract a subsequence {Fnk } such that limk→∞ Fnk = F almost everywhere. Lemma 3.5 and Fatou’s lemma gives F − Fnk Lp,q (R2d ,B) ≤ sup lim G(z)B Fnj (z) − Fnk (z)B dz G R2d j→∞ G(z)B Fnj (z) − Fnk (z)B dz ≤ sup lim inf G
j→∞
R2d
≤ lim inf Fnj − Fnk Lp,q (R2d ,B) . j→∞
By the following lemma most mix–normed spaces may be looked upon as Lq –spaces of functions taking values in Lp (Rd , B). Lemma 3.8. If 1 ≤ p, q < ∞ then Lq (Rd , Lp (Rd , B)) ⊃ Lp,q (R2d , B) with equal norms. If 1 ≤ p, q ≤ ∞ then Lq (Rd , Lp (Rd , B)) ⊂ Lp,q (R2d , B) with equal norms. Proof. Suppose F ∈ Lp,q (R2d , B) and 1 ≤ p, q < ∞. Then there exists a sequence of simple functions ank χAnk (x)χBnk (y), ank ∈ B, Fn (x, y) = k
such that limn→∞ F − Fn Lp,q (R2d ,B) = 0 [3, 25]. Thus for a subsequence Fnk we have limk→∞ F (·, y) − Fnk (·, y)Lp (Rd ,B) = 0 for a.a. y. Hence y → F (·, y) ∈ Lp (Rd , B) is strongly measurable and F Lp,q (R2d ,B) = F Lq (Rd ,Lp (Rd ,B)) . Suppose on the other hand that F ∈ Lq (Rd , Lp (Rd , B)) and 1 ≤ p, q ≤ ∞. Since F is strongly measurable there exists a sequence of functions ∞ Fn (x, y) = fnk (x)χBnk (y), fnk ∈ Lp (Rd , B), Bnk ∈ B(Rd ), k=1
{Bnk }∞ k=1
where are pairwise disjoint, such that for a.a. y, limn→∞ Fn (·, y) − F (·, y)Lp (Rd ,B) = 0 uniformly [9]. Since fnk are almost separably valued in B for
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all k, Fn is almost separably valued in B, and since fnk is weakly measurable in B for all k, Fn is weakly measurable in B. Hence Fn is strongly measurable by Pettis’s theorem. Let Kj ⊂ Rd be a nested exhausting sequence of compact sets. Then χKj (y)(Fn (·, y) − Fm (·, y))Lp (Rd ,B) q d −→ 0, n, m −→ ∞, L (R )
and hence χ1×Kj Fn is a Cauchy sequence in L (R2d , B) for all j. By an argument borrowed from the proof of Lemma 3.7 there exists a strongly measurable function G : R2d → B such that Fnk → G for a.a. (x, y) for a subsequence Fnk . Thus we have for a.a. y, G(·, y) = limk→∞ Fnk (·, y) = F (·, y) in Lp (Rd , B). Therefore F = G in Lq (Rd , Lp (Rd , B)) and F Lq (Rd ,Lp (Rd ,B)) = GLp,q (R2d ,B) . p,q
If p or q is infinite, the inclusion Lq (Rd , Lp (Rd , B)) ⊂ Lp,q (R2d , B) may be strict, since for F ∈ Lp,q (R2d , B) the function y → F (·, y) ∈ Lp (Rd , B) may fail to be strongly measurable [3]. Next we use the previous lemma to give a characterization of the dual of p,q Mm (Rd , B) when B is reflexive. In contrast to the corresponding result for scalar– valued modulation spaces, Thm. 11.3.6 in [22], we exclude the cases (p = 1, 1 < q < ∞). Here Lqm (Rd , Lp (Rd , B)) denotes the set of strongly measurable F : R2d → B such that F m ∈ Lq (Rd , Lp (Rd , B)). Proposition 3.9. Suppose p = q = 1 or 1 < p < ∞ and 1 ≤ q < ∞, B is a reflexive p,q Banach space, and let ϕ ∈ S(Rd ) fulfill ϕL2 (Rd ) = 1. Then (Mm (Rd , B)) =
p ,q p ,q (Rd , B ), in the sense that every G ∈ M1/m (Rd , B ) defines an element in M1/m p,q d p,q (Mm (R , B)) under the sesquilinear form (3.9), and for every u ∈ (Mm (Rd , B)) p ,q there is a G ∈ M1/m (Rd , B ) such that p,q p,q (u, F ) = (G, F )Mm (Rd ,B) = (Rd , B). (3.9) Vϕ G(z), Vϕ F (z) B dz, F ∈ Mm R2d
p ,q Proof. H¨ older’s inequality in Lp,q (R2d ) implies that every G ∈ M1/m (Rd , B ) p,q d defines an element in (Mm (R , B)) by (3.9). Suppose on the other hand that p,q p,q 2d u ∈ (Mm (Rd , B)) . The fact that Vϕ is isometric Mm (Rd , B) → Lp,q m (R , B) justifies the factorization (u, F ) = (u1 , Vϕ F ) where u1 is an induced linear func2d tional which acts boundedly on a closed linear subspace of Lp,q m (R , B). By the p,q Hahn-Banach theorem u1 can be extended to the whole space Lm (R2d , B). In the next step we use the duality theory of the Lebesgue-Bochner spaces Lp (Rd , X) where X is a Banach space [1, 9, 21]. If 1 ≤ p < ∞ and X is reflexive, then to each ∈ (Lp (Rd , X)) there exists an F ∈ Lp (Rd , X ) such that (, Z) = (F (x), Z(x))X dx and = F Lp (Rd ,X ) . By identification of and F we Rd
thus have (Lp (Rd , B)) = Lp (Rd , B ), and hence the reflexivity of B implies that X = Lp (Rd , B) is a reflexive Banach space if 1 < p < ∞. Thus for 1 < p < ∞ and 2d q d p d 1 ≤ q < ∞ we have, using Lemma 3.8, (Lp,q m (R , B)) = (Lm (R , L (R , B)) = q p ,q L1/m (Rd , Lp (Rd , B )) ⊂ L1/m (R2d , B ). Hence if 1 < p < ∞ and 1 ≤ q < ∞, u1
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,q 2d can be represented by an element Hu1 ∈ Lp1/m (R2d , B ) and acts on Lp,q m (R , B) by 2d (u1 , P )Lp,q 2d = Hu1 (z), P (z) B dz, P ∈ Lp,q ,B) m (R , B). m (R R2d
Use of (3.3) in the form Vϕ F (z) = Vϕ Vϕ∗ Vϕ F (z) = and Fubini’s theorem finally gives
R2d
Vϕ F (w)Vϕ (π(z)ϕ)(w)dw
2d ,B) (u, F ) = (u1 , Vϕ F )Lp,q m (R p,q Vϕ Vϕ∗ Hu1 (z), Vϕ F (z) B dz, F ∈ Mm = (Rd , B).
(3.10)
R2d
p ,q Thus we obtain (3.9) with G = Vϕ∗ Hu1 ∈ M1/m (Rd , B ) by Prop. 3.1. Concerning 1 2d 2d the case p = q = 1, the dual of Lm (R , B) is L∞ 1/m (R , B ). Thus in this case 2d ∗ ∞ d Hu1 ∈ L∞ 1/m (R , B ). Hence G = Vϕ Hu1 ∈ M1/m (R , B ) and (3.10) holds also in this case.
Remark 3.10. In the Hilbert space case B = H the sesquilinear form (3.9) is by Cor. 2.3 an extension of the L2 (Rd , H) inner product from S(Rd , H) × S(Rd , H) p ,q p,q (Rd , H) × Mm (Rd , H). It is independent of ϕ and unique except in the to M1/m cases (p, q) = (1, ∞) and (p, q) = (∞, 1) [43]. In Section 4 we will need the following lemma. We restrict to the Hilbert space valued case and denote M p ,q := M p ,q (Rd , H). Lemma 3.11. For any 1 ≤ p, q ≤ ∞ we have the norm equivalence
(G, F )M p,q (Rd ,H) . F M p,q (Rd ,H) sup G M p ,q ≤1
older’s inequality implies Proof. Let ϕ ∈ S fulfill ϕL2 (Rd ) = 1. If GM p ,q ≤ 1 H¨ that
(G, F )M p,q (Rd ,H) = Vϕ G(z), Vϕ F (z) H dz ≤ F M p,q (Rd ,H) . R2d
On the other hand, using Lemma 3.5 and Vϕ∗ P M p ,q ≤ CP Lp,q where Lp ,q := Lp ,q (R2d , H),
sup (Vϕ Vϕ∗ P (z), Vϕ F (z))H dz F M p,q (Rd ,H) = Vϕ F Lp,q (R2d ,H) = ≤C
sup G M p ,q ≤1
R2d
P Lp ,q ≤1
R2d
Vϕ G(z), Vϕ F (z) H dz .
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3.3. Weakly stationary Hilbert space valued tempered distributions In this subsection we discuss tempered distributions taking values in a Hilbert space H. Such a distribution F is said to be weakly stationary (with a term borrowed from generalized stochastic processes) if the covariance distribution σF F := σ is translation invariant in the sense of [18, 28, 31, 32] σ, ϕ ⊗ ψ = σ, Tx ϕ ⊗ Tx ψ, x ∈ Rd , ϕ, ψ ∈ S(Rd ). We denote the set of weakly stationary elements of S (Rd , H) by Ss (Rd , H). It can be shown that the covariance distribution of a weakly stationary distribution acting on a separable function ϕ ⊗ ψ equals σ, ϕ ⊗ ψ = σs , ϕ ∗ ψ ∗
(3.11)
where σs ∈ S (Rd ) and ψ ∗ (x) := ψ(−x) [18, 26]. Since σs is positive definite it has a Fourier transform which is a tempered, non-negative measure σ s [18], ie there exists u ≥ 0 such that v−u (ξ) σs (dξ) < ∞. Rd
2 Proposition 3.12. If F ∈ Ss (Rd , H) then Vϕ F (t, ξ)2H = σ s ∗ ϕ (ξ) and ∞,1 M1⊗v−u (Rd , H). Ss (Rd , H) ⊂ u≥0
Proof. Combination of (1.7) and (3.11) gives Vϕ F (t, ξ)2H = σs , Mξ Tt ϕ ∗ (Mξ Tt ϕ)∗
2 2 = σ s , |F (Mξ Tt ϕ)| = σ s , Tξ M−t ϕ
2 2 = σ s , s ∗ ϕ (ξ − ·) = σ ϕ (ξ). Thus Vϕ F (t, ξ)H is independent of t and by Lemma 2.1 there exists N > 0 such that Vϕ F (t, ξ)H ≤ CvN (ξ). Thus with u = N + d + 1 F M ∞,1 (Rd ,H) ≤ C vN −N −d−1 (ξ)dξ < ∞. 1⊗v−u
Rd
3.4. Gabor frame theory for Hilbert space valued modulation spaces A frame for L2 (Rd ) [6, 22] is a countable set {hk }k∈I ⊂ L2 (Rd ) such that for constants 0 < A ≤ B < ∞ |f, hk |2 ≤ Bf 2L2 (Rd ) , f ∈ L2 (Rd ). (3.12) Af 2L2 (Rd ) ≤ k∈I 2 d The frame
operator is a positive and invertible operator on L (R ) defined by Sf = k∈I f, hk hk .
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Lemma 3.13. Let H be a Hilbert space and let {hk }k∈I be a frame for L2 (Rd ) with frame constants A and B. Then AF 2L2 (Rd ,H) ≤ F, hk 2H ≤ BF 2L2 (Rd ,H) , F ∈ L2 (Rd , H), (3.13) k∈I
where F, hk is a Bochner integral, and we have F = F, hk S −1 hk , F ∈ L2 (Rd , H),
(3.14)
k∈I
with unconditional convergence in L2 (Rd , H). Proof. For a given F ∈ L2 (Rd , H) there exists by Pettis’s theorem a null set NF such that {F (t); t ∈ Rd \ NF } ⊂ H is separable. Let {ej }∞ j=1 be an ONB for this space. Since fj (t) := (F (t), ej )H ∈ L2 (Rd ) for all j, we have ∞ ∞ F 2L2 (Rd ,H) = F (t)2H dt = |fj (t)|2 dt = fj 2L2 (Rd ) . Rd \NF
Rd \NF j=1
Likewise F, hk 2H =
Rd \NF
F (t)hk (t)dt2H
∞
= j=1
j=1
Rd \NF
∞
2 fj (t)hk (t)dt = |fj , hk |2 , j=1
and hence we obtain using the upper frame bound (3.12) k∈I
F, hk 2H ≤ B
∞
fj 2L2 (Rd ) = BF 2L2 (Rd ,H) ,
j=1
and likewise we obtain the lower bound in (3.13). If we define the Hilbert space
2 1/2 valued sequence norm cl2 (I,H) := , the right inequality (3.13) k∈I ck H says that the coefficient operator CF = {F, hk }k∈I is bounded from L2 (Rd , H)
2 to l (I, H), and implies that the synthesis operator Dc = k∈I ck hk is bounded S can be extended to from l2 (I, H) to L2 (Rd , H) [22]. Thus the frame operator
act on L2 (Rd , H), by (Se F )(t) := (DCF )(t) = F, hk hk (t). By a slight k∈I modification of the proof of [22, Cor. 5.1.2], replacing sequences in l2 (I, C) by
2 sequences in l (I, H), the synthesis operator Dc = k∈I ck hk can be proved to be unconditionally convergent in L2 (Rd , H) when c ∈ l2 (I, H). Since bounded operators acting on unconditionally convergent series can be applied inside the sum [22], we obtain F, hk hk , F = F, hk 2H . Se F, F L2 (Rd ,H) = k∈I
k∈I
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By condition (3.13) Se is a positive invertible operator on L2 (Rd , H). Finally we prove (3.14). If x ∈ H then by [22, Cor. 5.1.3] we have for all k Se (S −1 hk ⊗ x) =
∞ S −1 hk , h h ⊗ x = hk ⊗ x
(3.15)
=1
with unconditional convergence in L2 (Rd , H). Again by the commutativity of bounded linear operators and unconditionally convergent series summation, we obtain Se−1 F, hk hk = F, hk S −1 hk F = Se−1 Se F = k∈I
k∈I −1
in the last equality using (3.15). Since S hk is a frame for L2 (Rd ) [22] and {F, hk }k∈I ∈ l2 (I, H) the convergence is unconditional in L2 (Rd , H). p,q By a Gabor frame [15, 22] for the Banach space Mm (Rd , H) we understand a discrete set of functions {MβnTαk g}n,k∈Zd = {π(λ)g}λ∈Λ , where g ∈ Mv1 (Rd ) and α, β > 0, such that for constants 0 < A ≤ B < ∞ q/p 1/q p,q F, Mβn Tαk gpH m(αk, βn)p AF Mm (Rd ,H) ≤ (3.16) n∈Zd k∈Zd p,q ≤ BF Mm (Rd ,H) .
In the proof of the next proposition we need the amalgam space W (Lp,q m ) [22] which p,q (Z2d ), is defined as the set of measurable functions f : R2d → C such that a ∈ lm where akn = ess supt,ξ∈[0,1]d |f (t + k, ξ + n)|, k, n ∈ Zd . Proposition 3.14. Let H be a Hilbert space, and suppose g ∈ Mv1 (Rd ), α, β > 0 and {π(λ)g}λ∈Λ is a Gabor frame for L2 (Rd ) with frame operator S. Then {π(λ)g}λ∈Λ p,q is also a Gabor frame for Mm (Rd , H), for all 1 ≤ p, q < ∞, and the expansion p,q F = F, π(λ)gπ(λ)γ, F ∈ Mm (Rd , H), (3.17) λ∈Λ
where γ = S −1 g, holds with unconditional convergence. Proof. By a fundamental result of Gr¨ ochenig and Leinert γ = S −1 g ∈ Mv1 (Rd ) [24]. By Lemma 3.13 {π(λ)g}λ∈Λ is a frame for L2 (Rd , H) and Dγ Cg = idL2 (R2 ,H) , where Cg is the coefficient operator defined by {π(λ)g}λ∈Λ and Dγ is the synthesis operator defined by {S −1 π(λ)g}λ∈Λ = {π(λ)γ}λ∈Λ [22]. In the following we adapt p,q p,q → lm , where the proof of [22, Thm. 12.2.3] on the boundedness of Cg : Mm s p,q ms (k, n) = m(αk, βn), to the Hilbert space valued case. If F ∈ Mm (Rd , H) we have by (3.6) with γ = g Vg F (z)H ≤ g−2 |Vg g| ∗ Vg F (·)H (z), hence by an inequality of Young type for amalgam spaces [13], [22, Thm. 11.1.5], Vg F (·)H p,q 2d . (3.18) p,q ≤ CVg gW (L1 ) Vg F (·)H v W (L ) L (R ) m
m
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By [22, Prop. 12.1.11] Vg g ∈ W (L1v ) and thus Vg F (·)H ∈ W (Lp,q m ). Next we need to prove that Vg F is continuous. From the formula (3.3) and Prop. 2.5 with ϕ ∈ S(Rd ) and ϕL2 = 1 we get ∗ Vg F (t, ξ) = Vg (Vϕ Vϕ F )(t, ξ) = Vϕ F (s, η)Vϕ g(s − t, η − ξ)ei2πt(η−ξ) dsdη R2d
which gives Vg F (t + t1 , ξ + ξ1 ) − Vg F (t, ξ)H
≤ Vϕ F (s, η)H Vϕ g(s − t − t1 , η − ξ − ξ1 )ei2π(t1 (η−ξ−ξ1 )−tξ1 ) R2d
− Vϕ g(s − t, η − ξ) dsdη. Let (t, ξ) ∈ R2d be fixed arbitrary. The integrand approaches zero everywhere as |t1 |, |ξ1 | → 0 since Vϕ g is continuous [22]. If we prove that the integrand is also bounded by an L1 (R2d ) function uniformly over t1 , ξ1 ∈ U where U ⊂ Rd is a compact neighbourhood of zero, then by the dominated convergence theorem Vg F is continuous at (t, ξ), and hence everywhere. Since Vϕ F (·)H m ∈ L∞ (R2d ), 1/m(s, η) ≤ Cv(s, η) ≤ Cv(s − t, η − ξ)v(t, ξ) [22] and (Vϕ g)v ∈ L1 (R2d ), it is by H¨older’s inequality enough to prove
sup Vϕ g(· − t1 , · − ξ1 ) v ∈ L1 (R2d ). (3.19) t ,ξ ∈U 1
1
The integral of (3.19) can be estimated from above, using the submultiplicativity of v and Q := [0, 1]d , by
sup v(s, η) sup Vϕ g(s − t1 , η − ξ1 ) t1 ,ξ1 ∈U
s∈k+Q
k,n∈Zd η∈n+Q
≤C
v(k, n) sup Vϕ g(k + t1 , n + ξ1 ) t1 ,ξ1 ∈K
k,n∈Zd
where K ⊂ Rd is a compact neighbourhood of zero. The inclusion K ⊂ where J is finite, implies the estimate of (3.20)
C v(k, n) sup Vϕ g(k + l + t1 , n + p + ξ1 ) k,n∈Zd
≤C
k,n∈Zd
(3.20)
|l|,|p|≤J
|l|≤J
l+Q
t1 ,ξ1 ∈Q
v(k, n) sup Vϕ g(k + t1 , n + ξ1 ) < ∞, t1 ,ξ1 ∈Q
where the last expression is finite due to Vϕ g ∈ W (L1v ) [22]. Thus by the dominated convergence theorem Vg F is continuous everywhere. Now we obtain by [22, Prop. 11.1.4] and (3.18)
Vg F H p,q Cg F lp,q 2d ,H) = ms (Z Λ lms ≤ C Vg F (·)H W (Lp,q ) m
p,q ≤ CF Mm (Rd ,H) .
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p,q p,q Thus Cg is continuous from Mm (Rd , H) to lm (Z2d , H). s We can also modify the proof of [22, Thm. 12.2.4] to prove that the synthep,q p,q sis operator Dγ : lm (Z2d , H) → Mm (Rd , H) is continuous, by replacement of s scalar-valued sequences by Hilbert space valued sequences and the sequence norm p,q p,q (Z2d , C) by lm (Z2d , H). Since p, q < ∞ the convergence is unconditional in lm s s p,q (Rd , H) [22]. As Dγ and Cg are bounded operators and Dγ Cg = idL2 (Rd ,H) Mm p,q we obtain by Prop. 3.3 (i) Dγ Cg = idMm (Rd ,H) , ie (3.17) and also (3.16), since p,q p,q p,q F Mm (Rd ,H) ≤ Dγ Cg F lms (Z2d ,H) ≤ Dγ Cg F Mm (Rd ,H) .
4. Localization operators, Weyl operators and Gabor multipliers with operator-valued symbols 4.1. Operators on Banach space valued function spaces with symbols in Lp,q The formula (2.7) of Prop. 2.5 suggests the definition of localization operators [5, 7, 8, 43, 44, 45], ie operators from B1 -valued distributions to B2 -valued distributions (B1 , B2 Banach spaces) of the form F )(s) := a(z)Vg F (z)(π(z)γ)(s)dz, F ∈ S (Rd , B1 ), (4.1) (Ag,γ a R2d
2d
where the symbol a : R → L(B1 , B2 ) is assumed to be strongly measurable. With an estimate similar to (2.6) in Section 2 one confirms that Ag,γ a , defined as before by action under the integral, is a bounded operator from S (Rd , B1 ) to S (Rd , B2 ) provided g, γ ∈ S(Rd ) and a is strongly measurable and polynomially bounded. First we notice that if we replace |a(z)Vg F (z)| by a(z)Vg F (z)B2 ≤ a(z) Vg F (z)B1 in the proof of [5, Prop. 3.2], then one obtains the following generalization of this result for scalar-valued modulation spaces to vector-valued spaces and operator-valued symbols. Proposition 4.1. Suppose g, γ ∈ Mv1 (Rd ), p0 , p1 , p2 , q0 , q1 , q2 ∈ [1, ∞] fulfill 1/p0 + 1/p1 = 1/p2 , 1/q0 + 1/q1 = 1/q2 , m1 and m2 are v-moderate weight functions, and 0 a ∈ Lpm02,q/m (R2d , L(B1 , B2 )). 1 p1 ,q1 d p2 ,q2 d Then Ag,γ a : Mm1 (R , B1 ) → Mm2 (R , B2 ) with operator norm bound p ,q Ag,γ a ≤ CgMv1 γMv1 aL 0 0
m2 /m1
(R2d ,L(B1 ,B2 )) .
The following two corollaries of Prop. 4.1, the second of which is proved with interpolation techniques, are stated and proved in [5]. Corollary 4.2. Suppose p0 , p1 , q0 , q1 ∈ [1, ∞] fulfill 1/p0 +1/p1 ≤ 1, 1/q0 +1/q1 ≤ 1, suppose a ∈ Lp0 ,q0 (R2d , L(B)) and m is v-moderate. Then Ag,γ is bounded on a p1 ,q1 (Rd , B) with operator norm bound Mm Ag,γ a ≤ CgMv1 γMv1 aLp0,q0 (R2d ,L(B)) .
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Corollary 4.3. Suppose 1 ≤ p, q, r ≤ ∞, m1 and m2 are v-moderate, and a ∈ p,q p,q Lrm2 (R2d , L(B1 , B2 )). Then Ag,γ : Mm (Rd , B1 ) → Mm (Rd , B2 ) with operator a 1 1 m2 g,γ norm bound Aa ≤ CgMv1 γMv1 aLrm (R2d ,L(B1 ,B2 )) . 2
Corollary 4.3 is not true if we replace the modulation spaces between which acts by L2 spaces. The following example shows that Ag,γ is not necessarAg,γ a a ily bounded from L2 (Rd , B1 ) to L2 (Rd , B2 ) when a ∈ L∞ (R2d , L(B1 , B2 )). This phenomenon can not occur in the theory of scalar-valued modulation spaces since L2 = M 2 then.
Example 4.4. Let the dimension d = 1, 1 < r < 2, B1 = lr (Z), B2 = lr (Z). Note that B1 and B2 both have Fourier type r [35]. Define the operator a0 acting on
ek where ek denotes the vector of zeros in all lr by a0 (β) = k βk |βk |r −2 β2−r lr positions except position k where it equals one. Then a0 L(lr ,lr ) = 1 and hence g ⊂ (−1/4, 1/4) a(z) ≡ a0 ∈ L∞ (R2 , L(lr , lr )). Let f, g ∈ S(R) fulfill supp f, supp and f L2 = gL2 = 1, let N > 0 be a given number, choose K such that
(2K + 1)1−2/r ≥ N . Define F (x) = |k|≤K Mk f (x)ek . Then F 2L2 (R,lr ) = (2K +
1)2/r and Vg F (t, ξ) = f ∗ M−t g∗ (ξ − k)ek , ie for fixed t the summands |k|≤K
have non-overlapping support in the ξ variable due to supp g, f ⊂ (−1/4, 1/4).
Hence Vg F (t, ξ)2ls = |k|≤K |F((Mk f )(Tt g))(ξ)|2 independently of s ∈ [1, ∞). Parseval’s formula gives R2 Vg F (t, ξ)2ls dz = 2K + 1. Thus g,g Vg F (z)2lr dz ≥ N F 2L2(R,lr ) , (F, Aa F )L2 (R,lr ) = R2
which by Lemma 3.5 implies that Ag,g / L L2 (R, lr ), L2 (R, lr ) . a ∈ Large parts of the Weyl and the Kohn-Nirenberg pseudo-differential calculi can be modified to treat the case of functions taking values in a separable Hilbert space H and symbols with values in L(H) as outlined in [17, pp 135–37]. Many results are true also when B1 , B2 are reflexive Banach spaces and the symbol takes values in L(B1 , B2 ) [27, p 79]. If B1 , B2 are perfectly general Banach spaces and a ∈ S(R2d , L(B1 , B2 )) the localization operator Ag,γ can be formulated as a Weyl a operator [17, 27] s + t g,γ , ξ F (t)ei2πξ(s−t) dtdξ Aa F (s) = Lρ F (s) := ρ (4.2) 2 2d R provided g, γ ∈ M 1 (Rd ), where the Weyl symbol is [17, 40] ρ = a ∗ W (γ ⊗ g).
(4.3)
4.2. Operators on Hilbert space valued function spaces with symbols in M ∞ In this subsection we shall prove a version of one of the main results of [7] (see also [43]) for Hilbert space valued modulation spaces. This result says that Ag,γ a can be extended to a uniformly bounded operator on M µ1 ,µ2 (Rd ) for all 1 ≤ µ1 , µ2 ≤ ∞, ∞ (R2d ), g, γ ∈ Mv1s (Rd ) and s ≥ 0. provided the symbol a ∈ M1/τ s
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If F, G ∈ L2 (Rd , H) then, following [17], an operator-valued Wigner distribution can be defined by (WF G (t, ξ)u, v)H = (F (t + τ /2), v)H (G(t − τ /2), u)H e−i2πτ ξ dτ, u, v ∈ H. Rd
Then WF G (t, ξ) ∈ L(H) for all (t, ξ) ∈ R2d follows from F, G ∈ L2 (Rd , H). It is also true that WF G (t, ξ) ∈ S1 (H) which denotes the set of trace-class operators, with norm [18, 37] ∞ T S1(H) = sup |(T ej , fj )H | j=1
where the supremum is taken over all pairs of orthonormal sequences {ej }j≥1 and {fj }j≥1 in H. In fact, ∞ ∞ |(WF G (t, ξ)ej , fj )H | ≤ |(F (t + τ /2), fj )H | |(G(t − τ /2), ej )H | dτ Rd j=1
j=1
≤
Rd d
F (t + τ /2)H G(t − τ /2)H dτ
≤ 2 F L2(Rd ,H) GL2 (Rd ,H) . If ψ1 , ψ2 ∈ S(Rd ) we have WF G (t, ξ)W (ψ1 ⊗ ψ2 )(t, ξ)dtdξ u, v)H R2d = (F (t + τ /2), v)H (G(t − τ /2), u)H ψ1 (t + τ /2)ψ2 (t − τ /2)dtdτ
(4.4)
R2d
= F, ψ1 , v H G, ψ2 , u H , u, v ∈ H, where F, ψ1 denotes a Bochner integral. The formula (4.4) can be generalized to F, G ∈ S (Rd , H) by the definition WF G , Φu, v H := F, ψ1 , v H G, ψ2 , u H , u, v ∈ H, if Φ = W (ψ1 ⊗ψ2 ). Thus WF G , W (ψ1 ⊗ψ2 ) is a rank-one operator with trace-class norm WF G , W (ψ1 ⊗ ψ2 )S1 (H) = F, ψ1 H G, ψ2 H (4.5) ≤C ψ1 α,β ψ2 α,β , |α|≤N1
|β|≤M1
|α|≤N2
|β|≤M2
where · α,β are seminorms on S(Rd ). The estimate (4.5) says that (ψ1 , ψ2 ) → WF G , W (ψ1 ⊗ ψ2 ) is a bilinear continuous map S(Rd ) × S(Rd ) → S1 (H). By the Schwartz kernel theorem [37] it can be extended to a continuous linear map S(R2d ) Ψ → WF G , W (Ψ) ∈ S1 (H). Since the partial Fourier transformation and the coordinate transformation which defines the Wigner distribution (1.4) are continuous operations on S(R2d ), we have WF G ∈ S (R2d , S1 (H)).
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The following lemma is needed in the main proposition. Its statement is analogous to [7, Lemma 2.2]. Lemma 4.5. Let H be a Hilbert space. If F, G ∈ S (Rd , H), ϕ ∈ S(Rd ) and Φ = W (ϕ), then with z = (z1 , z2 ), ζ = (ζ1 , ζ2 ) ∈ R2d , VΦ WF G (z, ζ)S1 (H) = Vϕ F (z1 − ζ2 /2, z2 + ζ1 /2)H Vϕ G(z1 + ζ2 /2, z2 − ζ1 /2)H . Proof. Using W (ϕ)(t − z1 , ξ − z2 ) = W (Mz2 Tz1 ϕ)(t, ξ) it can be verified that Mζ Tz Φ(t, ξ) = W (ψ1 ⊗ ψ2 )(t, ξ) where ψ1 = eiπz2 ζ2 Mz2 +ζ1 /2 Tz1 −ζ2 /2 ϕ, ψ2 = e−iπz2 ζ2 Mz2 −ζ1 /2 Tz1 +ζ2 /2 ϕ.
(4.6)
Thus the result follows from VΦ WF G (z, ζ) = WF G , Mζ Tz Φ and insertion of (4.6) into (4.5). The definition of a Weyl operator (4.2) works for symbols ρ ∈ S(R2d , L(H)) but we need to extend the definition to more general symbols. Weyl operators can be defined for ρ ∈ S (R2d , L(H)) as follows. Given any continuous bilinear multiplication with norm bounded by one, B1 × B2 (x, y) → x · y ∈ B3 , where Bj , j = 1, 2, 3 are Banach spaces, there exists according to [1, Thm. 1.7.2] a unique natural continuous bilinear extension of S(Rd , B1 ) × S(Rd , B2 ) (ρ, F ) → ρ(x) · F (x)dx ∈ B3 Rd
to (ρ, F ) ∈ S (Rd , B1 ) × S(Rd , B2 ). We use three such bilinear multiplications, (i) multiplication with scalars H × C → H, (ii) operator evaluation L(H) × H → H, and (iii) the inner product (·, ·)H H × H → C (which in fact is sesquilinear). Multiplication (i), modified with a conjugation in the second argument, has been used in the definition of S (Rd , H) in Section 1.1. Let now ρ ∈ S (R2d , L(H)) and F ∈ S(Rd , H) be fixed. By Thm. 1.3.3 (op. cit.) there exists a sequence ρn ∈ S(R2d , L(H)) such that ρn −→ ρ in S (R2d , L(H)) as n −→ ∞. Then by (4.2) Lρn F ∈ S(Rd , H) and we define the H-valued functional Lρ F by Lρ F, ϕ := ρ, W (F ⊗ ϕ)SR (R2d ,H) , ϕ ∈ S(Rd ),
(4.7)
where the multiplication of the right hand side distribution action is extended from multiplication (ii). Here SR (R2d , H) indicates that the distribution ρ acts linearly (in contrast to the antilinear convention of this paper) on S(R2d , H). It is clear that (4.7)
extends (4.2) as a definition of Lρ F and it can be verified that Lρ F, ϕH ≤ C α,β ϕα,β , ie Lρ F ∈ S (Rd , H). The definition (4.1) can be extended to symbols a ∈ S (R2d , L(H)) by Ag,γ a F := Lρ F where ρ = a ∗ W (γ ⊗ g). By the continuity asserted by Thm. 1.7.2 (op. cit.) applied to the multiplication
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(ii), limn→∞ Lρn F = Lρ F in S (Rd , H). The same result can then be applied once again, with multiplication (iii), which yields Lρ F, GS(Rd ,H) = lim (Lρn F, G)L2 (Rd ,H) ∀G ∈ S(Rd , H). n→∞
(4.8)
Since F , G and Lρn F are strongly measurable there exists a closed separable subspace H0 ⊂ H such that the range spaces of F, G, Lρn F are contained in H0 for all n a.e. Since WF G (x) : H → H0 for a.a. x ∈ R2d , and x → ρn (x) is continuous for each n, we can enlarge H0 to a closed separable subspace, still denoted H0 , such that ρn (x)WF G (x) : H0 → H0 for all n and a.a. x. If {ej }∞ j=1 ⊂ H0 is an ONB for H0 we have ∞ Lρn F, G L2 (Rd ,H) = (Lρn F (s), ej )H (G(s), ej )H ds =
j=1
=
j=1
∞
∞ j=1
R3d
R2d
Rd
(F (t + τ /2), ρn (t, ξ)∗ ej )H (G(t − τ /2), ej )H e−i2πξτ dτ dtdξ
WF G (t, ξ)ej , ρn (t, ξ)∗ ej H dtdξ =
R2d
tr ρn (x)WF G (x) H0 dx. (4.9)
Let Φ ∈ S(R2d ) be real-valued and fulfill ΦL2(R2d ) = 1. By Lemma 2.1 (z, ζ) → VΦ ρ(z, ζ) is continuous, thus we can again enlarge H0 to a closed separable subspace still denoted H0 , such that VΦ ρ(z, ζ)VΦ WF G (z, −ζ) : H0 → H0 . Finally we obtain from (4.8), (4.9), the duality S1 (H0 ) = L(H0 ) and |tr(AB)| ≤ AL(H0 ) ·BS1 (H0 ) [37], Lemma 2.2, and the dominated convergence theorem
tr ρn (x)WF G (x) H0 dx Lρ F, GS(Rd ,H) = lim n→∞ R2d
tr VΦ ρn (z, ζ)VΦ WF G (z, −ζ) H0 dzdζ = lim (4.10) n→∞ R4d
= tr VΦ ρ(z, ζ)VΦ WF G (z, −ζ) H dzdζ. 0
R4d
We are now prepared to prove the main theorem. Proposition 4.6. Let H be a Hilbert space, s ≥ 0, γ ∈ Mvrs (Rd ) where 1 ≤ r < ∞, and g ∈ Mv1s (Rd ). Suppose 1 ≤ q ≤ p ≤ ∞, q ≤ r, t = qr/(r − q), µj , νj ∈ [1, ∞], vj ≤ p, p ≤ µj , 1/µj − 1/νj = 1/q − 1/p, for j = 1, 2, (ν1 , ν2 ) = (1, ∞), p,t (ν1 , ν2 ) = (∞, 1), and a ∈ M1/τ (R2d , L(H)). Then Ag,γ can be extended to a a s µ1 ,µ2 d ν1 ,ν2 d (R , H) to M (R , H) with operator norm depending bounded map from M only on g, γ and a. p,t (R2d , L(H)) and g, γ ∈ S(Rd ). Let Φ = W (ϕ), where Proof. Suppose a ∈ M1/τ s ϕ ∈ S(Rd ), fulfill ΦL2 (R2d ) = 1. By [7, Prop. 2.5]
W (γ ⊗ g)Mτ1,r (R2d ) ≤ CgMv1s (Rd ) γMvrs (Rd ) , s
(4.11)
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and by Prop. 2.4 (op. cit.), with integrals interpreted as Bochner integrals, ρM p,q (R2d ,L(H)) ≤ CaM p,t
1/τs
(R2d ,L(H)) W (γ
⊗ g)Mτ1,r (R2d ) .
(4.12)
s
Now we extend (4.10) from F, G ∈ S(Rd , H) to a sesquilinear form, still denoted µ1 ,µ2 (Ag,γ (Rd , H) × M ν1 ,ν2 (Rd , H). Since a F, G) = (Lρ F, G), acting on (F, G) ∈ M the range spaces of Vϕ F and Vϕ G are separable subspaces of H [37], there still exists a closed separable subspace H0 ⊂ H such that VΦ ρ(z, ζ)VΦ WF G (z, −ζ) : H0 → H0 . This H0 is used in the extension of (4.10). Next we use |tr(AB)| ≤ AL(H0 ) BS1 (H0 ) , H¨ older’s inequality for mix-normed spaces, Lemma 4.5, q /p ≥ 1 (which is a consequence of q ≤ p), p /µj + p /νj = 1 + p /q for j = 1, 2 (which follows from the assumption 1/µj − 1/νj = 1/q − 1/p), and Young’s inequality for mix-normed spaces [3]. Thus
g,γ
Aa F, G = Lρ F, G ≤ VΦ ρLp,q (R4d ,L(H)) ζ2 ζ1 ζ2 ζ1 q /p 1/q · Vϕ F (z1 + , z2 − )pH Vϕ G(z1 − , z2 + )pH dz dζ 2 2 2 2 R2d R2d q /p 1/q p Vϕ F (·)pH ∗ V = ρM p,q (R2d ,L(H)) dζ ϕ G(·)H (ζ2 , −ζ1 ) R2d
1/p 1/p ≤ ρM p,q (R2d ,L(H)) Vϕ F (·)pH Lµ1 /p ,µ2 /p (R2d ) Vϕ G(·)pH ν1 /p ,ν2 /p 2d L (R ) = ρM p,q (R2d ,L(H)) Vϕ F (·)H Lµ1 ,µ2 (R2d ) Vϕ G(·)H Lν1 ,ν2 (R2d ) ≤ CaM p,t
1/τs
(R2d ,L(H)) gMv1s (Rd ) γMvrs (Rd ) F M µ1 ,µ2 (Rd ,H) GM ν1 ,ν2 (Rd ,H) ,
where (4.11). Hence last inequality we have inserted the estimates (4.12) and g,γ in the Aa F, G defines a bounded sesquilinear form on M µ1 ,µ2 × M ν1 ,ν2 , where we denote M µ1 ,µ2 := M µ1 ,µ2 (Rd , H) for brevity, which extends the canonical sesquilinear form on S(Rd , H) × S(Rd , H) defined by (·, ·)L2 (Rd ,H) . Since there is a unique extension of the latter form to M ν1 ,ν2 × M ν1 ,ν2 except in the given exceptional cases of ν1 , ν2 [43] (see Rem. 3.10), Ag,γ a F, G must equal this extended form (restricted if µj < νj ), denoted (·, ·)M p,q (Rd ,H) and defined in (3.9). Now the result follows from Lemma 3.11. The operator norm is bounded according to Ag,γ a ≤ CaM p,t
1/τs
(R2d ,L(H)) gMv1s (Rd ) γMvrs (Rd ) .
Finally we extend the last inequality by the density of S(Rd ) in Mvrs (Rd ) to g ∈ Mv1s (Rd ) and γ ∈ Mvrs (Rd ). Remark 4.7. It seems to be an open question whether Prop. 4.6 is true when H is replaced by a Banach space of a suitable kind. If we choose r = q = 1 and p = ∞ we obtain the announced generalization of [7, Thm. 3.2].
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∞ Corollary 4.8. If a ∈ M1/τ (R2d , L(H)) and g, γ ∈ Mv1s (Rd ), then Ag,γ can be a s ν1 ,ν2 d (R , H) for all 1 ≤ ν1 , ν2 ≤ ∞ except extended to a bounded map on M (ν1 , ν2 ) = (1, ∞) and (ν1 , ν2 ) = (∞, 1).
We also obtain a result for Weyl operators which is a generalization of [22, Thm. 14.5.2], see also [23, 43]. The original, restricted (continuity on L2 ) theorem was proved by Sj¨ ostrand [41]. Corollary 4.9. If ρ ∈ M ∞,1 (R2d , L(H)) then Lρ can be extended to a bounded map on M ν1 ,ν2 (Rd , H) for all 1 ≤ ν1 , ν1 ≤ ∞ except (ν1 , ν2 ) = (1, ∞) and (ν1 , ν2 ) = (∞, 1). p,q (Rd , H) by The formula (3.17) suggests the definition of operators on Mm p,q (Ag,γ a(λ)Vg F (λ)(π(λ)γ)(s), F ∈ Mm (Rd , H), a F )(s) := λ∈Λ 2d
for a : Z → L(H). Such an operator is called a Gabor multiplier, and the scalarvalued case is treated in [16]. If g ∈ Mv1 (Rd ) and {π(λ)g}λ∈Λ is a Gabor frame 2d for L2 (Rd ), then one can prove (we omit the details), replacing Lp,q m (R ) norms p,q 2d by sequence norms lms (Z ) and using Prop. 3.14, the following discrete version. Here we again denote ms (k, n) = m(αk, βn). Proposition 4.10. Let H be a Hilbert space, g ∈ Mv1 (Rd ) and let {π(λ)g}λ∈Λ be a Gabor frame for L2 (Rd ). Suppose furthermore p1 , p2 , q1 , q2 ∈ [1, ∞) and p0 , q0 ∈ [1, ∞] fulfill 1/p0 + 1/p1 = 1/p2 , 1/q0 + 1/q1 = 1/q2 , m1 and m2 are v-moderate weight functions, m = m2 /m1 and p0 ,q0 (Z2d , L(H)). a ∈ lm s p1 ,q1 d p2 ,q2 d Then Ag,γ a : Mm1 (R , H) → Mm2 (R , H) with operator norm bound g,γ p ,q Aa ≤ CgMv1 γMv1 alm0s 0 (Z2d ,L(H)) .
Acknowledgment The author would like to thank Hartmut F¨ uhr, Franz Luef, Ghassem Narimani and in particular Hans Feichtinger for nice discussions, suggestions and useful remarks which have improved the paper. He would also like to express his gratitude to the organizers of the Special Semester on Modern Methods for Time-Frequency Analysis at the Erwin Schr¨odinger Institute in Vienna, spring 2005, H. G. Feichtinger and K. Gr¨ ochenig, for a very nice and stimulating event. The paper was worked out in the hospital environment of NuHAG, Faculty of Mathematics, University of Vienna, for which again H. Feichtinger is acknowledged. This project is supported by STINT, The Swedish Foundation for International Cooperation in Research and Higher Education.
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References [1] H. Amann, Vector-valued distributions and Fourier multipliers. Unpublished manuscript, 2003. [2] H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr. 186 (1997), 5–56. [3] A. Benedek and R. Panzone, The space Lp , with mixed norm, Duke Math. J. 28 (1961), 301–24. [4] J. Bergh and J. L¨ ofstr¨ om, Interpolation spaces – an introduction. Springer, 1976. [5] P. Boggiatto, Localization operators with Lp symbols on modulation spaces. In: Advances in pseudo-differential operators, Birkh¨ auser (2004), 149–163, Eds. R. Ashino, P. Boggiatto, M. W. Wong. [6] O. Christensen, An introduction to frames and Riesz bases. Birkh¨ auser, 2003. [7] E. Cordero and K. Gr¨ ochenig, Time-frequency analysis of localization operators, J. Func. Anal. 205 (2003), 107–131. [8] I. Daubechies, Time-frequency localization operators: a geometric phase space approach, IEEE Trans. Inf. Theory 34 (1988), 605–612. [9] J. Diestel and J. J. Uhl, Jr, Vector measures. Math. Surveys 15, AMS, 1977. [10] T. Dobler, Wienertypr¨ aume auf lokalkompakten Gruppen. Diplomarbeit, Universit¨ at Wien, 1989. [11] N. Dunford and J. T. Schwartz, Linear operators, Part I. Interscience, New York and London, 1958. [12] H. G. Feichtinger, Modulation spaces on locally compact abelian groups. Technical report, University of Vienna, 1983; also in: Wavelets and their applications, Allied Publishers Pvt. Ltd. (2003), 99–140, Eds. M. Krishna, R. Radha and S. Thangavelu. [13] H. G. Feichtinger, Banach convolution algebras of Wiener type, Proc. Conf. “Functions, Series, Operators”, Budapest 1980, Coll. Math. Soc. Bolyai, North-Holland (1983), 509–23. [14] H. G. Feichtinger and K. Gr¨ ochenig, Banach spaces related to integrable group representations and their atomic decompositions – I, J. Func. Anal. 86 (1989), 307–340. [15] H. G. Feichtinger and K. Gr¨ ochenig, Gabor frames and time-frequency analysis of distributions, J. Func. Anal. 146 (1997), 464–495. [16] H. G. Feichtinger and K. Nowak, A first survey of Gabor multipliers. In: Advances in Gabor analysis, Birkh¨ auser (2003), 99–128, Eds. H. G. Feichtinger, T. Strohmer. [17] G. B. Folland, Harmonic analysis in phase space. Princeton University Press, 1989. [18] I. M. Gel’fand and N. Ya. Vilenkin, Generalized functions, vol. 4. Applications of harmonic analysis. Academic Press, 1964. [19] M. Girardi and L. Weis, Vector-valued extensions of some classical theorems in Harmonic Analysis. In: Proc. of the 3rd international ISAAC Congress, Berlin 2001, (2003), 171–185. [20] M. Girardi and L. Weis, Operator-valued Fourier multiplier theorems on Besov spaces, Math. Nachr. 251 (2003), 34–51. [21] N. E. Gretsky and J. J. Uhl, Jr, Bounded linear operators on Banach function spaces of vector-valued functions, Trans. Amer. Math. Soc. 167 (1972), 263–277.
Vol. 59 (2007) Vector-valued Modulation Spaces and Localization Operators
127
[22] K. Gr¨ ochenig, Foundations of time-frequency analysis. Birkh¨ auser, 2001. [23] K. Gr¨ ochenig and C. Heil, Modulation spaces as symbol classes for pseudodifferential operators. In: Wavelets and their applications, Allied Publishers Pvt. Ltd. (2003), 151–170, Eds. M. Krishna, R. Radha and S. Thangavelu. [24] K. Gr¨ ochenig and M. Leinert, Wiener’s lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc. 17 (2004), 1–18. [25] P. R. Halmos, Measure theory. Van Nostrand, 1950. [26] L. H¨ ormander, The analysis of linear partial differential operators I. 2nd Edition, Springer-Verlag, 1990. [27] L. H¨ ormander, The analysis of linear partial differential operators III. SpringerVerlag, 1985. [28] W. H¨ ormann, Generalized stochastic processes and Wigner distribution. PhD thesis, Universit¨ at Wien, 1989. [29] T. Hyt¨ onen, Fourier embeddings and Mihlin-type multiplier theorems, Math. Nachr. 274–275 (2004), 74–103. [30] T. Hyt¨ onen and P. Portal, Vector-valued multiparameter singular integrals and pseudodifferential operators, preprint, October 2005. [31] K. Itˆ o, Stationary random distributions, Memoirs of the College of Science, Univ. Kyoto, Ser. A, XXVIII (1953), Math. No. 3, 209–223. [32] B. Keville, Multidimensional second order generalised stochastic processes on locally compact abelian groups. PhD thesis, University of Dublin, 2003. [33] S. Kwapie´ n, Isomorphic characterizations of inner product spaces by orthogonal series with vector-valued coefficients, Studia Math. 44 (1972), 583–595. [34] G. Matz, On non-WSSUS wireless fading channels, IEEE Trans. Wireless Comm. 4 (2005), 2465–78. [35] J. Peetre, Sur la transformation de Fourier des fonctions a ` valeurs vectorielles, Rend. Sem. Mat. Univ. Padova 42 (1969), 15–26. ˇ Strkalj, ˇ [36] P. Portal and Z. Pseudodifferential operators on Bochner spaces and an application, Math. Z. 253 (2006), 805–819. [37] M. Reed and B. Simon, Methods of modern mathematical physics I. Wiley, 1975. [38] H.-J. Schmeisser, Vector-valued Sobolev and Besov spaces. In: Sem. Analysis of the Karl-Weierstrass-Institute 1985/86, Teubner Texte Math 96 (1987), 4–44. [39] L. Schwartz, Distributions a ` valeurs vectorielles, Ann. Inst Fourier I. 7 (1957), 1–141, II. 8 (1958), 1–209. [40] M. A. Shubin, Pseudodifferential operators and spectral theory. 2nd Edition, Springer, 2001. [41] J. Sj¨ ostrand, An algebra of pseudodifferential operators, Math. Res. L. 1 (1994), 185–192. [42] T. Strohmer, Pseudodifferential operators and Banach algebras in mobile communications, Appl. Comput. Harmon. Anal. 20 (2006), 237–249. [43] J. Toft, Continuity properties for modulation spaces, with applications to pseudodifferential calculus – I, J. Func. Anal. 207 (2004), 399–429.
128
Wahlberg
IEOT
[44] J. Toft, Continuity properties for modulation spaces, with applications to pseudodifferential calculus, – II, Annals of Global Analysis and Geometry 26 (2004), 73– 106. [45] M. W. Wong, Wavelet transforms and localization operators. Birkh¨ auser, 2002. Patrik Wahlberg School of Electrical Engineering & Computer Science The University of Newcastle Callaghan, NSW 2308 Australia e-mail:
[email protected] Submitted: January 30, 2006 Revised: March 26, 2007
