Multiplier theorems for the short-time Fourier transform Ferenc Weisz∗ Department of Numerical Analysis E¨otv¨os L. University P´azm´any P. s´et´any 1/C., H-1117 Budapest, Hungary e-mail: [email protected] Abstract So-called short-time Fourier transform multipliers (also called Anti-Wick operators in the literature) arise by applying a pointwise multiplication operator to the STFT before applying the inverse STFT. Boundedness results are investigated for such operators on modulation spaces and on Lp -spaces. Because the proofs apply naturally to Wiener amalgam spaces the results are formulated in this context. Furthermore, a version of the Hardy-Littlewood inequality for the STFT is derived. 2000 AMS subject classifications: Primary 42C15, Secondary 42C40, 42A38, 46B15. Key words and phrases: Wiener amalgam spaces, modulation spaces, short-time Fourier transform, time-frequency analysis, Hardy-Littlewood inequality, multipliers. 1 Introduction Marcinkiewicz [23] gave a sufficient condition for a multiplier operator of the trigonometric Fourier series to be bounded on Lp (1 < p < ∞) spaces. Later Mihlin [24] and H¨ormander [20] generalized the Marcinkiewicz condition and theorem. These results hold for Fourier transforms, too. Since then multiplier operators acting on various spaces are studied very intensively in the literature (see e.g. Larsen [22], Stein [26], Zygmund [30] and Grafakos [17]). Multipliers were investigated also for short-time Fourier transforms. The multiplier operator with symbol (or STFT multiplier) λ and windows g, γ is defined by Z Z 1 Mg,γ,λ f := λ(x, ω)Vg f (x, ω)Mω Tx γ dω dx, hγ, gi Rd Rd ∗ This paper was written while the author was researching at University of Vienna (NuHAG) supported by Lise Meitner fellowship No M733-N04. This research was also supported by the Hungarian Scientific Research Funds (OTKA) No K67642. 1 where Vg f is the short-time Fourier transform, M the modulation and T the translation 2 operator. If g(t) = γ(t) = e−πt , then Mg,γ,λ is the classical Anti-Wick operator. Mg,γ,λ was investigated in many papers, such as Berezin [1], Shubin [25], Wong [29], Feichtinger and Nowak [13], Boggiato, Cordero, Gr¨ochenig, Tabacco [2, 4, 5, 6, 7, 8]. The modulation spaces are defined by mixed Lp,q norms of the short-time Fourier transform. Cordero and Gr¨ochenig [5] proved that if λ is in the modulation space M∞ (R2d ) and g, γ in the Feichtinger’s algebra M1 (Rd ) then Mg,γ,λ is bounded on the modulation spaces Mp,q (Rd ). In particular, it is bounded on L2 (Rd ) = M2,2 (Rd ). In this paper we extend this result to the Wiener amalgam spaces W (L2 , Lq )(Rd ) (1 ≤ q ≤ ∞). Next we suppose that g, γ are in the Wiener algebra W (C, L1 )(Rd ) which is larger than M1 (Rd ) and show that if λ(x, ·) is a multiplier for Lp (Rd ) uniformly in x then it is also an STFT multiplier for W (Lp , Lq )(Rd ), independent of q. For g, γ ∈ W (L∞ , L1 )(Rd ) ⊃ W (C, L1 )(Rd ) we define the STFT multiplier operator as a limit in W (Lp , Lq )(Rd ) and verify the same result. Some conditions (such as Marcinkiewicz, Mihlin and H¨ormander conditions) are listed for a function such that it is a multiplier for Lp (Rd ) and so an STFT multiplier for W (Lp , Lq )(Rd ). Since the short-time Fourier transform is a local tool it appears as natural to present the results in the context of Wiener amalgam spaces W (Lp , Lq )(Rd ), even if some readers may be interested in the Lp -context only (at a first reading). We give an equivalent norm on W (Lp , Lq )(Rd ) and generalize the classical HardyLittlewood inequality on Fourier transforms for short-time Fourier transforms. Some new results are obtained in this way for modulation spaces. The weighted versions of the theorems are also proved. 2 Wiener amalgam spaces Let us fix d ≥ 1, d ∈ N. For a set Y 6= ∅ let Yd be its Cartesian product Y × . . . × Y taken with itself d-times. For x = (x1 , . . . , xd ) ∈ Rd and u = (u1 , . . . , ud ) ∈ Rd set u · x := d X uk x k , x2 := x · x and |x| := max |xk |. k=1,...,d k=1 In the sequel v will always be a continuous, positive, even, submultiplicative weight function, i.e., v(0) = 1, v(z) = v(−z) (z ∈ Rd ), and v(z1 + z2 ) ≤ v(z1 )v(z2 ) (z1 , z2 ∈ Rd ). A positive, even weight function m on Rd is called v-moderate if m(z1 + z2 ) ≤ Cv(z1 )m(z2 ) (z1 , z2 ∈ Rd ). It is easy to see that if m is a v-moderate weight then 1 m(z) ≤ m(z − t) ≤ Cv(t)m(z) C v(t) 2 (z, t ∈ Rd ). (1) We consider weights parallel for Rd and R2d . The standard class of weights on R2d are weights of polynomial type vs (x, ω) := (1 + x2 + ω 2 )s/2 , τs (x, ω) := (1 + ω 2 )s/2 , (x, ω ∈ Rd ). In this paper the constants C and Cp may vary from line to line and the constants Cp are depending only on p. d m d We briefly write Lm p (R ) instead of the weighted Lp (R , λ) space equipped with the norm (or quasi-norm) Z kf kLm := ( |f m|p dλ)1/p (0 < p ≤ ∞) p Rd with the usual modification for p = ∞, where λ is the Lebesgue measure. If m = 1 then we write simply Lp (Rd ) and k · kp . The space of continuous functions with the supremum norm is denoted by C(Rd ) and S(Rd ) denotes the Schwartz function class. The Fourier transform of a tempered distribution is denoted by Ff = fˆ. If f ∈ L1 (Rd ) then Z ˆ Ff (x) := f (x) := f (t)e−2πıx·t dt (x ∈ Rd ), √ Rd where ı = −1. Translation and modulation of a function f are defined, respectively, by Tx f (t) := f (t − x) and Mω f (t) := e2πıω·t f (t) (x, ω ∈ Rd ). If B is a normed space of tempered distributions then FB denotes the space of those tempered distributions f , for which there exists g ∈ B such that gˆ = f . The space FB is equipped with the norm kf kFB := kgkB . Let X(Rd ) be a translation invariant Banach space of functions or distributions on Rd such that FL1 (Rd ) · X(Rd ) ⊂ X(Rd ) with kϕf kX ≤ kϕkFL1 kf kX . Given a v-moderate weight m and ψ ∈ FL1 (Rd ) with compact support such that X Tk ψ = 1, k∈Zd d we define the (weighted) Wiener amalgam space W (X, Lm q )(R ) (1 ≤ q ≤ ∞) with local m d component X and global component Lq (R ) as the space of functions or distributions for which the norm ³Z ´1/q q q kf kW (X,Lm := kf · Tx ψkX m(x) dx q ) Rd is finite, with the obvious modification for q = ∞. It is known that the norm ´1/q ³X q q kf k = kf · Tk ψkX m(k) k∈Zd d is an equivalent norm on W (X, Lm q )(R ) (see Heil [19], Feichtinger and Zimmermann [16]). It can be shown that different choices of ψ ∈ F L1 (Rd ) generate the same space 3 and yields equivalent norms. If X = Lp (Rd ) then ψ can be chosen also equal to 1[0,1)d . d The closed subspace of W (L∞ , Lm q )(R ) containing continuous functions is denoted by d W (C, Lm q )(R ) (1 ≤ q ≤ ∞). d m d It is easy to see that W (Lp , Lm p )(R ) = Lp (R ), d m d W (Lp1 , Lm q )(R ) ←- W (Lp2 , Lq )(R ) (p1 ≤ p2 ) (2) (q1 ≤ q2 ), (3) and d m d W (Lp , Lm q1 )(R ) ,→ W (Lp , Lq2 )(R ) (1 ≤ p1 , p2 , q1 , q2 ≤ ∞). d As the following theorems show, the function ψ in the definition of W (Lp , Lm q )(R ) can be changed by an arbitrary function from W (L∞ , Lv1 )(Rd ). The unweighted versions of Theorems 1 and 2 were proved in Feichtinger and Weisz [14]. However, for the sake of completeness, we give here short proofs for the theorems. v d Theorem 1 Suppose that m is a v-moderate weight, γ ∈ W Fx (t) = R (L∞ , Lq1 )(R )q and1/q F (x, t) is a measurable function. If 1 ≤ p, q ≤ ∞ and ( Rd kFx kp m(x) dx) < ∞ R d d then F (x, t)Tx γ(t) is integrable in x over R for a.e. t ∈ R and Rd Fx Tx γ dx ∈ d W (Lp , Lm q )(R ) with °Z ° ³Z ´1/q ° ° Fx Tx γ dx° ≤ CkγkW (L∞ ,Lv1 ) kFx kqp m(x)q dx . ° W (Lp ,Lm q ) Rd Proof. By duality, °Z ° ° ° Fx Tx γ dx° ° Rd W (Lp ,Lm q ) Rd = sup 1/m ≤1 W (Lp0 ,L 0 ) q khk ¯D Z ¯ ¯ E¯ ¯ Fx Tx γ dx, h ¯, Rd where p0 and q 0 are the dual indices to p and q. For a fixed h with khkW (L 1/m p0 ,Lq 0 ) ≤1 we get by (1) that ¯D Z E¯ ¯ ¯ Fx Tx γ dx, h ¯ ¯ Rd Z Z Z m(x) dx dt du ≤ |Fx (t)Tx γ(t)h(t)|Tx+u 1[0,1)d (t) m(x + u − u) Rd Rd Rd Z Z Z m(x) ≤ C dt dx du kγTu 1[0,1)d k∞ v(u) |Fx (t)h(t)|Tx+u 1[0,1)d (t) m(x + u) Rd Rd Rd Z ³Z ³Z ´q/p ´1/q ≤ C kγTu 1[0,1)d k∞ v(u) |Fx (t)|p dt m(x)q dx Rd Rd Rd ³Z ³Z ´q0 /p0 ´1/q0 1 0 |h(t)Tx+u 1[0,1)d (t)|p dt dx du (4) m(x + u)q0 Rd Rd Z ´1/q ³Z khkW (L 0 ,L1/m ) du = C kγTu 1[0,1)d k∞ v(u) kFx kqp m(x)q dx p q0 Rd Rd ´1/q ³Z ≤ CkγkW (L∞ ,Lv1 ) kFx kqp m(x)q dx Rd and this proves the theorem. 4 Theorem 2 Suppose that m is a v-moderate weight and g ∈ W (L∞ , Lv1 )(Rd ). If f ∈ d W (Lp , Lm q )(R ) for some 1 ≤ p, q ≤ ∞ then ³Z ´1/q kf · Tx gkqp m(x)q dx ≤ CkgkW (L∞ ,Lv1 ) kf kW (Lp ,Lm . q ) Rd Proof. Again by duality, ³Z ´1/q kf · Tx gkqp m(x)q dx = Rd sup khk ¯D E¯ ¯ ¯ ¯ kf · Tx gkp , h ¯. 1/m ≤1 L 0 q If khkL1/m ≤ 1 then q0 ¯D E¯ ¯ ¯ kf · T gk , h ¯ ¯ x p Z ³Z ¯Z m(x + u − u) ¯¯p ´1/p ¯ f (t)Tx g(t)Tx+u 1[0,1)d (t) ≤ du¯ dt |h(x)| dx ¯ m(x) Rd Rd Rd Z Z ³Z ´1/p m(x + u)v(u) p |h(x)| ≤ C |f (t)Tx g(t)| Tx+u 1[0,1)d (t) dt dx du m(x) Rd Rd Rd Z ³Z ³Z ´q/p ´1/q p q ≤ C kgTu 1[0,1)d k∞ v(u) |f (t)| Tx+u 1[0,1)d (t) dt m(x + u) dx Rd Rd Rd Z ´1/q0 ³ 1 0 dx du |h(x)|q m(x)q0 Rd ≤ CkgkW (L∞ ,Lv1 ) kkf kW (Lp ,Lm , q ) which finishes the proof of Theorem 2. Applying these theorems for Fx = f · Tx g and taking into account the equality Z ³ ´ 1 f (t) = f (t)Tx g(t) Tx γ(t) dx (t ∈ Rd ) (5) hγ, gi Rd we obtain Corollary 1 Suppose that m is a v-moderate weight and g, γ ∈ W (L∞ , Lv1 )(Rd ) such that hγ, gi 6= 0. Then ´1/q ³Z |hγ, gi| q q ≤ CkgkW (L∞ ,Lv1 ) kf kW (Lp ,Lm . kf kW (Lp ,Lm ≤ kf ·Tx gkp m(x) dx q ) q ) CkγkW (L∞ ,Lv1 ) Rd 3 Modulation spaces The short-time Fourier transform (STFT) of a tempered distribution f ∈ S 0 (Rd ) with respect to a window function g ∈ S(Rd ) is defined by Z Vg f (x, ω) := hf, Mω Tx gi = f (t)g(t − x)e−2πıω·t dt (x, ω ∈ Rd ). Rd 5 One can see by (1) that the short-time Fourier transform is also well defined if g ∈ d W (L∞ , Lv1 )(Rd ) and f ∈ W (L1 , Lm ochenig [18, Lemma ∞ )(R ). It is easy to see (Gr¨ 3.1.1]) that Vg f (x, ω) = (f · Tx g)∧ (ω) (x, ω ∈ Rd ). (6) Given a non-zero window function g ∈ S(Rd ), a v-moderate weight m on R2d and m 1 ≤ p, q ≤ ∞, the modulation space Mp,q (Rd ) consists of all tempered distributions f ∈ S 0 (Rd ) for which Vg f is in a weighted mixed-norm Lp,q space, more exactly, ³Z ³Z ´q/p ´1/q m := kf kMp,q |Vg f (x, ω)|p m(x, ω)p dx dω < ∞. Rd Rd m In this section we assume always that m is of polynomial growth. Then Mp,q (Rd ) is a Banach space whose definition is independent of the choice of the window g. It can be shown that different choices of g ∈ M1v (Rd ) \ {0} yield equivalent norms. If m m p = q, we write Mpm instead of Mp,q and if m = 1 then we write Mp,q and Mp for Mp,q m m d m d d and Mp,p . If m(x, ω) = m(x) then M2 (R ) = L2 (R ). The space M1 (R ) is called Feichtinger’s algebra and is isometrically invariant under translation, modulation and Fourier transform (see Feichtinger [10]). For more about modulation spaces see the book of Gr¨ochenig [18]. m Changing the order of integration in the definition of Mp,q (Rd ) we get the modulation m d space Wp,q (R ) with norm ³Z ³Z ´q/p ´1/q m := kf kWp,q |Vg f (x, ω)|p m(x, ω)p dω dx Rd Rd (see Feichtinger and Gr¨ochenig [12]). Of course, Mpm (Rd ) = Wpm (Rd ). For a suitable m weight function the space Wp,q (Rd ) contains the Fourier transforms of the elements of m d Mp,q (R ). More exactly, since |Vg f (x, ω)| = |Vgˆfˆ(ω, −x)| (x, ω ∈ Rd ), (Gr¨ochenig [18, Lemma 3.1.1]) it is easy to see that ˆkM m˜ m ∼ kf kf kWp,q p,q m m ˜ Wp,q (Rd ) = FMp,q (Rd ), and where m(x, ˜ ω) = m(−ω, x) and ∼ denotes the equivalence of norms. d Choosing ψ ∈ S in the definition of W (FLp , Lm q )(R ), we conclude ³Z ´1/q ³ Z ´1/q q q ∧ q q kf kW (FLp ,Lm ∼ kf · T ψk m(x) dx = k(f · T ψ) k m(x) dx . x x p F Lp q ) Rd Rd Then (6) implies m kf kW (FLp ,Lm ∼ kf kWp,q q ) d m d W (FLp , Lm q )(R ) = Wp,q (R ), and (7) whenever m(x, ω) = m(x) and 1 ≤ p, q ≤ ∞. In other words d m )(R ) with equivalent norms, if m(x, ω) = m(ω). In par(Rd ) = W (FLp , Lm FMp,q q d d d ticular, M1 (R ) = FM1 (R ) = W (FL1 , L1 )(R ). These results were proved also in Feichtinger [11] and Gr¨ochenig [18]. 6 In special case, if p = 2 and m(x, ω) = m(x) then (7) yields that kf kW (L2 ,Lm ∼ q ) m , i.e. kf kW2,q ´q/2 ´1/q ³Z ³Z |Vg f (x, ω)|2 dω m(x)q dx ∼ kf kW (L2 ,Lm , q ) Rd Rd whenever g ∈ M1v (Rd ) \ {0}. This can be extended to all g ∈ W (L∞ , Lv1 )(Rd ), that is a larger space than M1v (Rd ), whenever v(x, ω) = v(x). Indeed, Parseval formula and (6) imply ´q/2 ´1/q ³Z ´1/q ³Z ³Z 2 q k(f · Tx g)∧ kq2 m(x)q dx |Vg f (x, ω)| dω m(x) dx = Rd Rd Rd ³Z ´1/q kf · Tx gkq2 m(x)q dx = , Rd where 1 ≤ q ≤ ∞. Thus, by Theorem 2, ´q/2 ´1/q ³Z ³Z 2 q |Vg f (x, ω)| dω m(x) dx ≤ CkgkW (L∞ ,Lv1 ) kf kW (L2 ,Lm q ) Rd Rd with the obvious modification for q = ∞. Similarly, by Corollary 1, ³Z ³Z ´q/2 ´1/q C 2 q v m kf kW (L2 ,Lq ) ≤ kγkW (L∞ ,L1 ) |Vg f (x, ω)| dω m(x) dx . |hγ, gi| Rd Rd We will generalize these inequalities for 1 < p < ∞ below. First of all note that the Hardy-Littlewood inequality for Fourier transforms holds, namely, for h ∈ Lp (Rd ), ³Z ´1/p p ˆ |h(ω)| dω ≤ Cp khkp (1 < p ≤ 2) (8) 2−p Rd ((|ω1 | + 1) · · · (|ωd | + 1)) and ³Z ´1/p p ˆ |h(ω)| khkp ≤ Cp dω (2 ≤ p < ∞) 2−p Rd ((|ω1 | + 1) · · · (|ωd | + 1)) (see Edwards [9], Jawerth and Torchinsky [21] and Weisz [27]). (9) Theorem 3 Assume that m is a v-moderate weight, g ∈ W (L∞ , Lv1 )(Rd ) and f ∈ d W (Lp , Lm q )(R ) for some 1 < p ≤ 2, 1 ≤ q ≤ ∞. Then ´q/p ´1/q ³Z ³Z |Vg f (x, ω)|p q dω m(x) dx ≤ Cp kgkW (L∞ ,Lv1 ) kf kW (Lp ,Lm . q ) 2−p Rd Rd ((|ω1 | + 1) · · · (|ωd | + 1)) d In particular, the preceding inequality holds for all f ∈ Lm p (R ). Proof. By using (8) the proof follows from ³Z ³Z ´q/p ´1/q |Vg f (x, ω)|p q dω m(x) dx 2−p Rd Rd ((|ω1 | + 1) · · · (|ωd | + 1)) ³Z ´1/q q q ≤ Cp kf · Tx gkp m(x) dx Rd and from Theorem 2. 7 Using (5), (9) and Theorem 1 we obtain the next theorem. Theorem 4 Assume that m is a v-moderate weight, g, γ ∈ W (L∞ , Lv1 )(Rd ) such that hγ, gi 6= 0. If ³Z ³Z ´q/p ´1/q |Vg f (x, ω)|p q dω m(x) dx 2−p Rd Rd ((|ω1 | + 1) · · · (|ωd | + 1)) d is finite for some 2 ≤ p < ∞, 1 ≤ q ≤ ∞ then f ∈ W (Lp , Lm q )(R ) and ≤ kf kW (Lp ,Lm q ) Cp kγkW (L∞ ,Lv1 ) |hγ, gi| ´q/p ´1/q ³Z ³Z |Vg f (x, ω)|p q dω m(x) dx . 2−p Rd Rd ((|ω1 | + 1) · · · (|ωd | + 1)) d In particular, the result is true for Lm p (R ). Note that in Theorems 3 and 4 m need not be of polynomial growth. These two theorems can be interpreted as kf kW (m·mp ) ≤ Cp kgkW (L∞ ,Lv1 ) kf kW (Lp ,Lm q ) p,q (1 < p ≤ 2, 1 ≤ q ≤ ∞) and kf kW (Lp ,Lm ≤ q ) Cp kγkW (L∞ ,Lv1 ) kf kW (m·mp ) p,q |hγ, gi| (2 ≤ p < ∞, 1 ≤ q ≤ ∞), where m is depending only on x and mp (ω) := ((|ω1 | + 1) · · · (|ωd | + 1))(p−2)/p . 4 STFT multipliers on modulation spaces If h ∈ Lp (Rd ) for some 1 ≤ p ≤ 2 then Fourier inversion formula Z 2πıω·t ˆ h(t) = h(ω)e dω (t ∈ Rd ) Rd ˆ ∈ L1 (Rd ). For general h ∈ Lp (Rd ) let us define holds if h Z 2πıω·t ˆ h(ω)e dω. ρU h(t) := (10) |ω|≤U It is known that ρU h(t) → h(t) in Lp (Rd ) norm as U → ∞ (11) and kρU hkp ≤ Cp khkp 8 (U > 0), (12) whenever h ∈ Lp (Rd ) and 1 < p ≤ 2 (see e.g. Zygmund [30], Grafakos [17] or Weisz [28]). If p > 2 then we suppose that h ∈ Lp (Rd ) ∩ L2 (Rd ) and then we extend the integral in (10) to every h ∈ Lp (Rd ). The following inversion formula is known for the short-time Fourier transform. If g, γ ∈ L2 (Rd ) and hγ, gi 6= 0 then for all f ∈ L2 (Rd ) Z Z 1 f= Vg f (x, ω)Mω Tx γ dω dx, (13) hγ, gi Rd Rd where the equality is understood in a vector-valued weak sense (see Gr¨ochenig [18, p. 44]). Introducing Z Z 1 ρg,γ,U f := Vg f (x, ω)Mω Tx γ dω dx hγ, gi Rd {|ω|≤U } we proved in [14] that lim ρg,γ,U f = f U →∞ in W (Lp , Lq )(Rd ) norm, whenever f ∈ W (Lp , Lq )(Rd ) (1 < p < ∞, 1 ≤ q < ∞) and g, γ ∈ W (L∞ , L1 )(Rd ) such that hγ, gi 6= 0. This motivates the investigation of multiplier operators. For a given multiplier λ ∈ L∞ (Rd ) the multiplier operator is defined for Fourier transforms by Z 2πıω·t ˆ Mλ h(t) := λ(ω)h(ω)e dω (h ∈ S(Rd )). (14) Rd Multiplier operators acting on various spaces are studied very intensively in the literature (see e.g. Larsen [22], Stein [26], Zygmund [30] and Grafakos [17]). For the short-time Fourier transform and multiplier λ ∈ S 0 (R2d ) the ST F T multiplier operator is defined formally by Z Z 1 Mg,γ,λ f := Mλ f := λ(x, ω)Vg f (x, ω)Mω Tx γ dω dx, (15) hγ, gi Rd Rd where hγ, gi 6= 0. Sometimes it is more convenient to interpret the definition of Mλ in a weak sense: hMλ f, hi = hλVg f, Vγ hi = hλ, Vg f Vγ hi, f, h ∈ S(Rd ). (16) 2 If g(t) = γ(t) = e−πt , then Mλ is the classical Anti-Wick operator. Mλ was investigated in many papers, such as Berezin [1], Shubin [25], Wong [29], Feichtinger and Nowak [13], Boggiato, Cordero, Gr¨ochenig, Tabacco [2, 4, 5, 8]. If f, g ∈ S(Rd ) then Vf g ∈ S(Rd ) and if f, g ∈ L2 (Rd ) then Vf g ∈ L2 (Rd ) and kVf gk2 = kf k2 kgk2 (see Gr¨ochenig [18]). From this it follows that |hMλ f, hi| ≤ kλk∞ kVg f Vγ hk1 ≤ kλk∞ kVg f k2 kVγ hk2 ≤ kλk∞ kgk2 kγk2 kf k2 khk2 (17) and 9 Theorem 5 If λ ∈ L∞ (R2d ) and g, γ ∈ L2 (Rd ) then Mλ is bounded on L2 (Rd ) and kMλ f k2 ≤ kλk∞ kgk2 kγk2 kf k2 . Note that this theorem was proved also in Wong [29]. If we choose g and γ from a smaller class then we get a larger class of multipliers. First of all, since the dual of 1/m m Mp,q (Rd ) is Mp0 ,q0 (Rd ), m khk kVg f Vγ hk1 ≤ Ckf kMp,q 1/m , M p0 ,q 0 if g, γ ∈ M1v (Rd ). This together with (17) yield that Mλ is bounded on the modulation m (Rd ) (1 ≤ p, q ≤ ∞), whenever λ ∈ L∞ (R2d ). However, Cordero and space Mp,q Gr¨ochenig [5] proved a more general theorem: Theorem 6 Suppose that m is a v-moderate weight of polynomial growth. If λ ∈ m (Rd ) for 1 ≤ p, q ≤ ∞ and M∞ (R2d ) and g, γ ∈ M1v (Rd ) then Mλ is bounded on Mp,q m ≤ kλkM kgkM v kγkM v kf kM m . kMλ f kMp,q ∞ p,q 1 1 In particular, Mλ is bounded on L2 (Rd ). The proof follows from (16) and from the inequality (see [5]) m khk kVg f Vγ hkM1 ≤ kgkM1v kγkM1v kf kMp,q 1/m . M p0 ,q 0 Note that M∞ (R2d ) ⊃ W (L1 , L∞ )(R2d ). The converse of Theorem 6 is not true and it seems hopeless to find a characterization of the multipliers λ (see Cordero and m Gr¨ochenig [5]). We can verify Theorem 6 for the spaces Wp,q (Rd ) in the same way. Theorem 7 Suppose that m is a v-moderate weight of polynomial growth. If λ ∈ m M∞ (R2d ) and g, γ ∈ M1v (Rd ) then Mλ is bounded on Wp,q (Rd ) for 1 ≤ p, q ≤ ∞ and m ≤ kλkM kgkM v kγkM v kf kW m . kMλ f kWp,q ∞ p,q 1 1 d In particular, Mλ is bounded on W (L2 , Lm q )(R ), whenever m(x, ω) = m(x) and 1 ≤ q ≤ ∞. 5 STFT multipliers on Wiener amalgam spaces Given a Banach space B let us define the space M (B) of multipliers λ for which the multiplier operator Mλ is bounded on B and equip with the norm kλkM (B) := kMλ kB→B . The space M(B) = Mg,γ (B) of STFT multipliers is defined analogously. For simd plicity we assume that λ is real. It is known that the dual of W (Lp , Lm q )(R ) is 1/m 1/m d d W (Lp0 , Lq0 )(Rd ) and so M (W (Lp , Lm q )(R )) = M (W (Lp0 , Lq 0 )(R )). Since |hMg,γ,λ f, hi| = |hλVg f, Vγ hi| = |hλVγ h, Vg f i| = |hMγ,g,λ h, f i|, 10 we obtain 1/m d d Mg,γ (W (Lp , Lm q )(R )) = Mγ,g (W (Lp0 , Lq 0 )(R )) (1 ≤ p, q ≤ ∞) with equal norms. Here and in the future it is not assumed anymore that m is of polynomial weight. It follows by interpolation that the normed spaces Mg,g (W (Lp , Lq )(Rd )) are nested, that is, for 1 ≤ p ≤ q ≤ 2 we have Mg,g (Lp (Rd )) ⊂ Mg,g (Lq (Rd )) ⊂ Mg,g (L2 (Rd )), Mg,g (W (Lp , L2 )(Rd )) ⊂ Mg,g (W (Lq , L2 )(Rd )) ⊂ Mg,g (L2 (Rd )), Mg,g (W (L2 , Lp )(Rd )) ⊂ Mg,g (W (L2 , Lq )(Rd )) ⊂ Mg,g (L2 (Rd )). The analogous results hold also for modulation spaces and for the spaces d M (W (Lp , Lm q )(R )). d Now we extend Theorems 5 and 7 to the Wiener amalgam spaces W (Lp , Lm q )(R ). v d We choose the windows from another space than M1 (R ). First we assume g, γ ∈ W (C, Lv1 )(Rd ), that is larger than M1v (Rd ) if v(x, ω) = v(x), and then we choose g, γ ∈ W (L∞ , Lv1 )(Rd ). We will show that if λ(x, ·) is a multiplier for Lp (Rd ) uniformly in d x then it is also an STFT multiplier for W (Lp , Lm q )(R ), independent of q. Other, incomparable multiplier theorems for Lp (Rd ) spaces can be found in Boggiatto and Wong [3, 4]. d m d The closed subspace W (Lp , Lm ∞,0 )(R ) of W (Lp , L∞ )(R ) consists of all functions for which kf · Tx 1[0,1)d kp m(x) is bounded in x and has limit zero if |x| → ∞. Theorem 8 Assume that m is a v-moderate weight and g, γ ∈ W (C, Lv1 )(Rd ) such that d hγ, gi 6= 0. If λ(x, ·) ∈ M (Lp (Rd )) uniformly in x then λ ∈ Mg,γ (W (Lp , Lm q )(R )), i.e., kMg,γ,λ f kW (Lp ,Lm ≤ q ) Cp sup kλ(x, ·)kM (Lp ) kγkW (C,Lv1 ) kgkW (C,Lv1 ) kf kW (Lp ,Lm , q ) |hγ, gi| x∈Rd whenever 1 ≤ p, q < ∞ or 1 < p, q ≤ ∞. The same holds for the STFT multipliers of d m d m d the spaces W (L1 , Lm ∞,0 )(R ), W (C, L1 )(R ) and Lp (R ). Proof. Since M (Lp (Rd )) ⊂ M (L2 (Rd )) = L∞ (Rd ), we have λ ∈ L∞ (R2d ). First suppose that 1 ≤ p, q < ∞ and f, g, γ ∈ S(Rd ). Then the multiplier and STFT multiplier operators in (14) and (15) are well defined. Using Theorems 1 and 2 we conclude kMg,γ,λ f kW (Lp ,Lm ) Zq ³Z ° ° ´ 1 ° ° = λ(x, ω)(f · Tx g)∧ (ω)e2πıω· dω Tx γ(·) dx° ° |hγ, gi| Rd W (Lp ,Lm q ) Rd Z ´ ³ 1/q C ≤ kγkW (C,Lv1 ) kMλ(x,·) (f · Tx g)kqp m(x)q dx |hγ, gi| d R ³Z ´1/q Cp ≤ sup kλ(x, ·)kM (Lp ) kγkW (C,Lv1 ) kf · Tx gkqp m(x)q dx |hγ, gi| x∈Rd Rd Cp sup kλ(x, ·)kM (Lp ) kγkW (C,Lv1 ) kgkW (C,Lv1 ) kf kW (Lp ,Lm . ≤ q ) |hγ, gi| x∈Rd 11 d Since S(Rd ) is dense in W (C, Lv1 )(Rd ) and in W (Lp , Lm q )(R ), we can prove the theorem for 1 ≤ p, q < ∞ with a usual density argument. For 1 < p, q ≤ ∞ we get the result by d m d duality. The theorem is proved for the spaces W (L1 , Lm ∞,0 )(R ) and W (C, L1 )(R ) as d well, because S(R ) is dense in these spaces, too. Finally, the result for the last space d m d follows from W (Lp , Lm p )(R ) = Lp (R ). The converse of Theorem 8 is not true even if p = 2 and λ(x, ω) = λ(ω). More exactly, given a function λ(x, ω) = λ(ω) and a weight v(x, ω) = v(x), then λ ∈ d v d Mg,γ (W (L2 , Lm q )(R )) for all 1 ≤ q ≤ ∞ and all g, γ ∈ M1 (R ) do not imply that λ ∈ M (L2 (Rd )) = L∞ (Rd ). Indeed, if λ0 is not bounded but λ0 ∈ W (L1 , L∞ )(Rd ) and λ(x, ω) := λ0 (ω) then we have λ ∈ W (L1 , L∞ )(R2d ) ⊂ M∞ (R2d ). Thus, by Theorem 7, d λ ∈ Mg,γ (W (L2 , Lm q )(R )) but λ is not bounded. 6 0, we To be able to extend this theorem to g, γ ∈ W (L∞ , Lv1 )(Rd ) with hγ, gi = define first the STFT multiplier operator for this case by Z Z 1 Mg,γ,λ f := lim λ(x, ω)Vg f (x, ω)Mω Tx γ dω dx (18) U →∞ hγ, gi Rd {|ω|≤U } d m d in W (Lp , Lm q )(R ) norm, where f ∈ W (Lp , Lq )(R ). The integral in (18) is well defined pointwise, because |Vg f (x, ω)| = |(f · Tx g)∧ (ω)| ≤ kf · Tx gk1 kgkW (L∞ ,Lv1 ) ≤ kf kW (L1 ,Lm kTx gkW (L∞ ,L1/m ) ≤ Cv(x)kf kW (Lp ,Lm q ) ∞) 1 (for the last inequality see Heil [19] and (2) and (3)). Moreover, λ ∈ L∞ (R2d ) and Z v(x)|Tx γ(t)| dx ≤ v(t)kγkLv1 ≤ v(t)kγkW (L∞ ,Lv1 ) < ∞ Rd for all fixed t ∈ Rd . As the following theorem shows, the limit in (18) does exist. Theorem 9 Assume that m is a v-moderate weight and g, γ ∈ W (L∞ , Lv1 )(Rd ) such that hγ, gi 6= 0. If 1 < p < ∞ and λ(x, ·) ∈ M (Lp (Rd )) uniformly in x then the integral d ∗ in (18) converges in W (Lp , Lm q )(R ) norm if 1 ≤ q < ∞ and in the w topology of m d m d W (Lp , L∞ )(R ) if q = ∞. Moreover, λ ∈ Mg,γ (W (Lp , Lq )(R )), i.e., kMg,γ,λ f kW (Lp ,Lm ≤ q ) Cp sup kλ(x, ·)kM (Lp ) kγkW (L∞ ,Lv1 ) kgkW (L∞ ,Lv1 ) kf kW (Lp ,Lm , q ) |hγ, gi| x∈Rd d whenever 1 < p < ∞ and 1 ≤ q ≤ ∞. In particular, the theorem holds for Lm p (R ). Proof. We conclude by (11) that Z λ(x, ω)(f · Tx g)∧ (ω)e2πıω·t dω Mλ(x,·) (f · Tx g)(t) := lim U →∞ {|ω|≤U } 12 (19) in Lp (Rd ) norm for all fixed x ∈ Rd , if (f · Tx g) ∈ Lp (Rd ) (1 < p < ∞). Theorem 1 implies for U < T that ° 1 Z Z ° ° ° λ(x, ω)Vg f (x, ω)Mω Tx γ dω dx° (20) ° hγ, gi Rd {|ω|≤T }\{|ω|≤U } W (Lp ,Lm q ) Z ³Z ° ´ 1 ° ° ° ∧ 2πıω· g) (ω)e dω T γ(·) dx = λ(x, ω)(f · T ° ° x x |hγ, gi| Rd W (Lp ,Lm q ) {|ω|≤T }\{|ω|≤U } Z °Z ° ´1/q ³ q C ° ° q ∧ 2πıω· . kγkW (L∞ ,Lv1 ) λ(x, ω)(f · Tx g) (ω)e dω ° m(x) dx ≤ ° |hγ, gi| p Rd {|ω|≤T }\{|ω|≤U } If 1 ≤ q < ∞ then by (19) the integrand tends to 0 as T, U → ∞ and it can be estimated by ³ ´q sup kλ(x, ·)kM (Lp ) kf · Tx gkqp m(x)q x∈Rd (see (12)) and this is integrable in x by Theorem 2. Lebesgue’s dominated convergence theorem yields that the expression in (20) tends to 0 as T, U → ∞. Assuming q = ∞ we get similarly to (4) that Z ³Z ´ E¯ 1 ¯¯D ¯ (21) λ(x, ω)(f · Tx g)∧ (ω)e2πıω· dω Tx γ(·) dx, h(·) ¯ ¯ |hγ, gi| Rd {|ω|≤T }\{|ω|≤U } Z Z ° °Z ° ° ≤ C kγTu 1[0,1)d k∞ v(u)° λ(x, ω)(f · Tx g)∧ (ω)e2πıω· dω ° m(x) p Rd Rd {|ω|≤T }\{|ω|≤U } 0 ³Z ´ 1/p 1 0 dx du, |h(t)Tx+u 1[0,1)d (t)|p dt m(x + u) Rd 1/m where h ∈ W (Lp0 , L1 )(Rd ). The p-norm in the integrand converges to 0 as above and °Z ° ° ° ∧ 2πıω· λ(x, ω)(f · Tx g) (ω)e dω ° m(x) ° p {|ω|≤T }\{|ω|≤U } ³ ´³ ´ ≤ sup kλ(x, ·)kM (Lp ) sup kf · Tx gkp m(x) , x∈Rd x∈Rd d which is finite if f ∈ W (Lp , Lm ∞ )(R ) because of Theorem 2. We can see by Lebesgue’s dominated convergence theorem that (21) tends to 0. This shows that the expression in (18) is convergent. The inequality in the theorem can be proved similarly. Unfortunately, a characterization of the spaces M (Lp (Rd )) is known only in the cases p = 1, 2, ∞. As we mentioned above, M (L2 (Rd )) = L∞ (Rd ). Moreover, M (L1 (Rd )) = M (L∞ (Rd )) = F(M (Rd )), where M (Rd ) denotes the space of finite Borel measures (see Larsen [22], Zygmund [30] or Grafakos [17]). However, there are several sufficient conditions for a multiplier to be in M (Lp (Rd )). One of the most known condition is the Marcinkiewicz multiplier theorem. Let Ii := (−2i+1 , −2i ) ∪ (2i , 2i+1 ), (i ∈ Z), 13 Ij := Ij1 × . . . × Ijd , (j ∈ Zd ) and λ ∈ L∞ (Rd ) d-times continuously differentiable on each region Ij (j ∈ Zd ). If for all k ∈ {1, . . . , d}, all j1 , . . . , jk ∈ {1, . . . , d}, all lj1 , . . . , ljk ∈ Z and all tn ∈ Jln for n 6∈ {j1 , . . . , jk }, 1 ≤ n ≤ d, we have Z Z ... |∂j1 . . . ∂jk λ(t1 , . . . , td )| dtj1 . . . dtjk ≤ C, Ilj Ilj 1 k then λ ∈ M (Lp (R)) for all 1 < p < ∞. 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