Compactness of Localization Operators on Modulation Spaces of ω-Tempered Distributions

Abstract

We give sufficient conditions for compactness of localization operators on modulation spaces ${\mathbf {M}}^{p,q}_{m_{\lambda }}( \mathbb {R}^{d})$ of ω-tempered distributions whose short-time Fourier transform is in the weighted mixed space $L^{p,q}_{m_\lambda }$ for m _λ(x) = e ^λω(x).

In this paper we study some properties of localization operators, which are pseudo-differential operators of time-frequency analysis suitable for applications to the reconstruction of signals, because they allow to recover a filtered version of the original signal. To introduce the problem, let us recall the translation and modulation operators

$$\displaystyle \begin{aligned} \begin{array}{rcl} T_{x}f(y)=f(y-x), \quad M_{\xi}f(y)=e^{i y \cdot \xi}f(y), \qquad x,y \in \mathbb{R}^{d}, \end{array} \end{aligned} $$

and, for a window function $\psi \in L^{2}(\mathbb {R}^{d})$, the short-time Fourier transform (briefly STFT) of a function $f \in L^{2}(\mathbb {R}^{d})$

$$\displaystyle \begin{aligned}V_{\psi} f(z)= \langle f, M_{\xi}T_{x}\psi \rangle =\int_{\mathbb{R}^{d}}f(y) \overline{\psi(y-x)}e^{-iy \cdot \xi} \, dy, \qquad z=(x,\xi) \in \mathbb{R}^{2d}. \end{aligned}$$

With respect to the inversion formula for the STFT (see [13, Cor. 3.2.3])

$$\displaystyle \begin{aligned}f= \frac{1}{(2 \pi)^{d} \langle \gamma, \psi \rangle} \int_{\mathbb{R}^{2d}} V_{\psi}f(x, \xi)M_{\xi}T_{x} \gamma\, dx d \xi, \end{aligned}$$

which gives a reconstruction of the signal f, the localization operator, as defined in (0.2), modifies V _ψ f(x, ξ) by multiplying it by a suitable a(x, ξ) before reconstructing the signal, so that a filtered version of the original signal f is recovered.

Another important operator in time-frequency analysis that we shall need in the following is the cross-Wigner transform defined, for $f,g \in L^{2}(\mathbb {R}^{d})$, by

$$\displaystyle \begin{aligned}\operatorname*{\mathrm{Wig}}(f,g)(x, \xi)= \int_{\mathbb{R}^{d}} f \big(x+ \frac{t}{2} \big) \overline{g \big(x- \frac{t}{2} \big)} e^{-i \xi\cdot t } \, dt \qquad x, \xi \in \mathbb{R}^{d}. \end{aligned}$$

The Wigner transform of f is then defined by $\operatorname *{\mathrm {Wig}} f:= \operatorname *{\mathrm {Wig}}(f,f)$.

The above Fourier integral operators, with standard generalizations to more general spaces of functions or distributions, have been largely investigated in time-frequency analysis. In particular, results about boundedness or compactness related to the subject of this paper can be found, for instance, in [1, 7, 10,11,12, 16, 17].

Inspired by Cordero and Gröchenig [7] and Fernández and Galbis [10], our aim in this paper is to study boundedness of localization operators on modulation spaces in the setting of ω-tempered distributions, for a weight functions ω defined as below:

FormalPara Definition 0.1

A non-quasianalytic subadditive weight function is a continuous increasing function ω : [0, +∞) → [0, +∞) satisfying the following properties:

(α):

ω(t ₁ + t ₂) ≤ ω(t ₁) + ω(t ₂), ∀t ₁, t ₂ ≥ 0;

(β):

$\ \ \int _{1}^{+\infty } \frac {\omega (t)}{t^{2}} \, dt < + \infty ;$

(γ):

$\ \ \exists A \in \mathbb {R}$, B > 0 s.t $\omega (t) \geq A+B \log (1+t), \qquad \forall t \geq 0;$

(δ):

φ _ω(t) := ω(e ^t) is convex.

We then consider ω(ξ) := ω(|ξ|) for $\xi \in \mathbb {C}^{d}$.

FormalPara Definition 0.2

The space $\mathcal {S}_{\omega }(\mathbb {R}^{d})$ is defined as the set of all $u \in L^{1}(\mathbb {R}^{d})$ such that $u, \hat {u} \in C^{\infty }(\mathbb {R}^{d})$ and

• (i) $\ \forall \lambda >0, \alpha \in \mathbb {N}^{d}_{0}$: $\sup _{x \in \mathbb {R}^{d}} e^{\lambda \omega (x)} |D^{\alpha }u(x)| < + \infty ,$

• (ii) $\ \ \forall \lambda >0, \alpha \in \mathbb {N}^{d}_{0}$: $\sup _{\xi \in \mathbb {R}^{d}} e^{\lambda \omega (\xi )} |D^{\alpha }\hat {u}(\xi )| < + \infty ,$

where $\mathbb {N}_{0}:= \mathbb {N} \cup \{ 0 \}$.

Note that for $\omega (t)= \log (1+t)$ we obtain the classical Schwartz class $\mathcal {S}(\mathbb {R}^{d})$, while in general $\mathcal {S}_{\omega }(\mathbb {R}^{d}) \subseteq \mathcal {S}(\mathbb {R}^{d})$. For more details about the spaces $\mathcal {S}_{\omega }(\mathbb {R}^{d})$ we refer to [3,4,5,6]. In particular, we can define on $\mathcal {S}_{\omega }(\mathbb {R}^{d})$ different equivalent systems of seminorms that make $\mathcal {S}_{\omega }(\mathbb {R}^{d})$ a Fréchet nuclear space. It is also an algebra under multiplication and convolution.

The corresponding strong dual space is denoted by $\mathcal {S}^{\prime }_{\omega }(\mathbb {R}^{d})$ and its elements are called ω-tempered distributions. Moreover, $\mathcal {S}' (\mathbb {R}^{d})\subseteq \mathcal {S}^{\prime }_{\omega }(\mathbb {R}^{d})$ and the Fourier transform, the short-time Fourier transform and the Wigner transform are continuous from $\mathcal {S}_{\omega }(\mathbb {R}^{d})$ to $\mathcal {S}_{\omega }(\mathbb {R}^{d})$ and from $\mathcal {S}^{\prime }_{\omega }(\mathbb {R}^{d})$ to $\mathcal {S}^{\prime }_{\omega }(\mathbb {R}^{d})$.

The “right” function spaces in time-frequency analysis to work with the STFT are the so-called modulation spaces, introduced by H. Feichtinger in [9]. In this context, we consider the weight m _λ(z) := e ^λω(z), for $\lambda \in \mathbb R$, and define $L^{p,q}_{m_{\lambda }}(\mathbb {R}^{2d})$ as the space of measurable functions f on $\mathbb R^{2d}$ such that

$$\displaystyle \begin{aligned}\| f \|{}_{L^{p,q}_{m_{\lambda}}}:= \int_{\mathbb{R}^{d}} \bigg( \int_{\mathbb{R}^{d}} |f(x, \xi)|{}^{p} m_{\lambda}(x, \xi)^{p} \, dx \big)^{\frac{q}{p}} \, d \xi \bigg)^{\frac{1}{q}}< + \infty, \end{aligned}$$

for 1 ≤ p, q < +∞, with standard changes if p (or q) is + ∞. We define then, for 1 ≤ p, q ≤ +∞, the modulation space

$$\displaystyle \begin{aligned}{\mathbf{M}}_{m_{\lambda}}^{p,q}(\mathbb{R}^{d}):= \{ f \in \mathcal{S}^{\prime}_{\omega}(\mathbb{R}^{d}): V_{\varphi}f \in L^{p,q}_{m_{\lambda}}(\mathbb{R}^{2d}) \}, \end{aligned}$$

which is independent of the window function $\varphi \in \mathcal {S}_{\omega }(\mathbb {R}^{d}) \setminus \{ 0 \}$ and is a Banach space with norm $\| f \|{ }_{{\mathbf {M}}_{m_{\lambda }}^{\,p,q}}:= \| V_{\varphi } f \|{ }_{L^{p,q}_{m_{\lambda }}}$ (see [4]). Moreover, for 1 ≤ p, q < +∞, the space $\mathcal {S}_{\omega }(\mathbb {R}^{d})$ is a dense subspace of ${\mathbf {M}}_{m_{\lambda }}^{p,q}$ by Boiti et al. [4, Prop. 3.9]. We shall denote ${\mathbf {M}}_{m_{\lambda }}^{p}(\mathbb {R}^{d})= {\mathbf {M}}_{m_{\lambda }}^{p,p}(\mathbb {R}^{d})$ and ${\mathbf {M}}^{p,q}(\mathbb {R}^{d})={\mathbf {M}}_{m_0}^{p,q}(\mathbb {R}^{d})$.

As in [13, Thm. 12.2.2] if p ₁ ≤ p ₂, q ₁ ≤ q ₂, and λ ≤ μ then ${\mathbf {M}}^{\, p_{1}, q_{1}}_{m_{\mu }} \subseteq {\mathbf {M}}^{\, p_{2}, q_{2}}_{m_{\lambda }}$ with continuous inclusion (see [8, Lemma 2.3.16]). Set

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle m_{\lambda, 1}(x):= m_{\lambda}(x,0), \quad m_{\lambda, 2}(\xi):= m_{\lambda}(0,\xi),\\ & &\displaystyle v_{\lambda}(z)=e^{| \lambda| \omega(z)}, \quad v_{\lambda, 1}(x):= v_{\lambda}(x,0), \quad v_{\lambda, 2}(\xi):= v_{\lambda}(0,\xi), \end{array} \end{aligned} $$

and prove the following generalization of [7, Prop. 2.4]:

FormalPara Proposition 0.3

Let 1 ≤ p, q, r, t, t′≤ +∞ such that $\frac {1}{p}+ \frac {1}{q}-1= \frac {1}{r}$ and $\frac {1}{t}+ \frac {1}{t'}=1$ . Then, for all $\lambda , \mu \in \mathbb {R}$ and 1 ≤ s ≤ +∞,

$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathbf{M}}^{ \, p,st}_{m_{\lambda, 1} \otimes m_{\mu, 2}}( \mathbb{R}^{d}) * {\mathbf{M}}^{\, q, st'}_{v_{\lambda,1} \otimes v_{\lambda,2} m_{-\mu,2}}(\mathbb{R}^{d}) \hookrightarrow {\mathbf{M}}_{m_{\lambda}}^{ \, r,s}(\mathbb{R}^{d}) \end{array} \end{aligned} $$

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \hspace{-20mm}\mathit{\mbox{and}}\qquad \qquad \| f * g \|{}_{{\mathbf{M}}_{m_{\lambda}}^{r,s}} \leq \| f \|{}_{{\mathbf{M}}^{\, p,st}_{m_{\lambda, 1} \otimes m_{\mu, 2}}} \| g \|{}_{{\mathbf{M}}^{\, q, st'}_{v_{\lambda,1} \otimes v_{\lambda,2} m_{-\mu,2}}}. \end{array} \end{aligned} $$

(0.1)

FormalPara Proof

For the Gaussian function $g_{0}(x)=e^{- \pi |x|{ }^{2}} \in \mathcal {S}_{\omega }(\mathbb {R}^{d})$ consider on ${\mathbf {M}}^{r,s}_{m_{\lambda }}$ the modulation norm with respect to the window function $g(x):=g_{0}* g_{0}(x)= 2^{-d/2} e^{- \frac {\pi }{2} | x |{ }^{2}} \in \mathcal {S}_{\omega }(\mathbb {R}^{d})$. Since m _λ(x, ξ) ≤ m _λ(x, 0)v _λ(0, ξ) and $\overline {g_{0}(-x)}=g_{0}(x)$, by Gröchenig [13, Lemma 3.1.1], Young and Hölder inequalities:

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle \| f *h \|{}_{{\mathbf{M}}^{r,s}_{m_{\lambda}}} = \| V_{g}(f*h) \|{}_{L^{r,s}_{m_{\lambda}}}= \bigg( \int_{\mathbb{R}^{d}} \bigg( \int_{\mathbb{R}^{d}} | V_{g}(f*h)|{}^{r} m_{\lambda}^{r} (x, \xi) \, dx \bigg)^{\frac{s}{r}} d \xi \bigg)^{\frac{1}{s}} \\ & &\displaystyle \leq \bigg( \int_{\mathbb{R}^{d}} \bigg( \int_{\mathbb{R}^{d}} | (f*M_{\xi}g_{0})*(h*M_{\xi}g_{0})(x)|{}^{r} m_{\lambda}(x,0)^{r} \, dx \bigg)^{\frac{s}{r}} v_{\lambda}^{s}(0, \xi) \, d \xi \bigg)^{\frac{1}{s}}\\ & &\displaystyle = \bigg( \int_{\mathbb{R}^{d}} \| (f*M_{\xi}g_{0})*(h*M_{\xi}g_{0}) \|{}^{s}_{L^{r}_{m_{\lambda,1}}} v_{\lambda}^{s}(0, \xi) \, d \xi \bigg)^{\frac{1}{s}}\\ & &\displaystyle \leq \bigg( \int_{\mathbb{R}^{d}} \| f * M_{\xi}g_{0} \|{}_{L^{p}_{m_{\lambda,1}}}^{s} \| h * M_{\xi}g_{0} \|{}_{L^{q}_{v_{\lambda,1}}}^{s} v_{\lambda}^{s}(0, \xi) \, d \xi \bigg)^{\frac{1}{s}}\\ & &\displaystyle = \bigg( \int_{\mathbb{R}^{d}} \| V_{g_{0}}f \|{}_{L^{p}_{m_{\lambda,1}}}^{s} m_{\mu}^{s}(0, \xi) \| V_{g_{0}}h \|{}_{L^{q}_{v_{\lambda,1}}}^{s} m_{- \mu}^{s}(0, \xi) v_{\lambda}^{s}(0, \xi) \, d \xi \bigg)^{\frac{1}{s}}\\ & &\displaystyle \leq \| f \|{}_{{\mathbf{M}}^{p,st}_{m_{\lambda,1}\otimes m_{\mu,2}}} \| h \|{}_{{\mathbf{M}}^{q, st'}_{v_{\lambda,1} \otimes v_{\lambda,2} m_{- \mu,2}}}. \end{array} \end{aligned} $$

□

Given two window functions $\psi , \gamma \in \mathcal {S}_{\omega }(\mathbb {R}^{d}) \setminus \{ 0 \}$ and a symbol $a \in \mathcal {S}^{\prime }_{\omega }(\mathbb {R}^{2d})$, the corresponding localization operator $L^{a}_{\psi , \gamma }$ is defined, for $f \in \mathcal {S}_{\omega }(\mathbb {R}^{d})$, by

$$\displaystyle \begin{aligned} L^{a}_{\psi, \gamma}f=V^{*}_{\gamma}(a \cdot V_{\psi} f)= \int_{\mathbb{R}^{2d}} a(x, \xi) V_{\psi} f(x, \xi) M_{\xi}T_{x} \gamma \,dx d \xi, \end{aligned} $$

(0.2)

where $V^{*}_{\gamma }$ is the adjoint of V _γ. As in [2, Lemma 2.4] we have that $L^{a}_{\psi , \gamma }$ is a Weyl operator $L^{a^{w}}$ with symbol a ^w = a ∗Wig(γ, ψ):

$$\displaystyle \begin{aligned} L^{a^{w}}f:= \frac{1}{(2 \pi)^{d}} \int_{\mathbb{R}^{2d}} \hat{a}^{w}(\xi, u) e^{-i \xi \cdot u} T_{-u}M_{\xi}f \,du d \xi. \end{aligned} $$

(0.3)

Moreover, if $f,g \in \mathcal {S}_{\omega }(\mathbb {R}^{d})$ then by definition of adjoint operator we can write

$$\displaystyle \begin{aligned} \langle L^{a}_{\psi, \gamma}f,g \rangle = \langle a \cdot V_{\psi}f, V_{\gamma}g \rangle =\langle a , \overline{ V_{\psi}f} V_{\gamma}g \rangle, \end{aligned}$$

and, similarly as in [13, Thm. 14.5.2] (see also [8, Teo. 2.3.21]), we have, for $a^{w} \in {\mathbf {M}}^{\infty ,1}_{m_{\mu }}( \mathbb {R}^{2d})$ with μ ≥ 0,

$$\displaystyle \begin{aligned} \| L^{a^{w}} f \|{}_{{\mathbf{M}}^{p,q}_{m_{\lambda}}}= \| L^{a}_{\psi, \gamma} f \|{}_{{\mathbf{M}}^{p,q}_{m_{\lambda}}} \leq \| a^{w} \|{}_{{\mathbf{M}}^{\infty,1}_{m_{\mu}}} \| f \|{}_{{\mathbf{M}}^{p,q}_{m_{\lambda}}}, \end{aligned} $$

(0.4)

for all $f \in {\mathbf {M}}^{p,q}_{m_{\lambda }}$ and $\lambda \in \mathbb {R}$.

FormalPara Theorem 0.4

Let $\psi , \gamma \in \mathcal {S}_{\omega }(\mathbb {R}^{d}) \setminus \{0\}$ and $a \in {\mathbf {M}}^{\infty }_{m_{\lambda }}( \mathbb {R}^{2d})$ for some λ ≥ 0. Then $L^{a}_{\psi , \gamma }$ is bounded from ${\mathbf {M}}^{p,q}_{m_{\lambda }}(\mathbb {R}^{d})$ to ${\mathbf {M}}^{p,q}_{m_{\lambda }}(\mathbb {R}^{d})$ , for 1 ≤ p, q < +∞, and

$$\displaystyle \begin{aligned} \begin{array}{rcl} \| L^{a}_{\psi, \gamma} \|{}_{op} \leq \| a \|{}_{{\mathbf{M}}^{\infty}_{m_{- \lambda, 2}}} \| \psi \|{}_{{\mathbf{M}}^{1}_{v_{\lambda}}} \| \gamma \|{}_{{\mathbf{M}}^{p}_{m_{\lambda}}}. \end{array} \end{aligned} $$

FormalPara Proof

By definition $V_{\psi }: {\mathbf {M}}^{p,q}_{m_{\lambda }} \to L^{p,q}_{m_{\lambda }}( \mathbb {R}^{2d})$ and, by Boiti et al. [4, Prop. 3.7], $V_{\gamma }^{*}: L^{p,q}_{m_{\lambda }}( \mathbb {R}^{2d}) \to {\mathbf {M}}^{p,q}_{m_{\lambda }} (\mathbb {R}^{d})$. Let $f \in {\mathbf {M}}^{p,q}_{m_{\lambda }}( \mathbb {R}^{d})$. To prove that $L^{a}_{\psi , \gamma }f=V^{*}_{\gamma }(a \cdot V_{\psi } f) \in {\mathbf {M}}^{p,q}_{m_{\lambda }}$, it is then enough to show that $a \cdot V_{\psi } f \in L^{p,q}_{m_{\lambda }}(\mathbb {R}^{2d})$. By the inversion formula [4, Prop. 3.7], given two window functions $\Phi , \Psi \in \mathcal {S}_{\omega }(\mathbb {R}^{2d})$ with 〈 Φ,  Ψ〉≠ 0, we have, for $z=(z_1,z_2)\in \mathbb R^{2d}\times \mathbb R^{2d}$,

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle \bigg( \int_{\mathbb{R}^{d}} \bigg( \int_{\mathbb{R}^{d}} |a(x, \xi)|{}^{p} |V_{\psi}f(x, \xi)|{}^{p} e^{p \lambda \omega(x, \xi)} \, dx \bigg)^{\frac{q}{p}} \, d \xi \bigg)^{\frac{1}{q}}\\ & \leq&\displaystyle \frac{1}{(2 \pi)^{d}} \frac{1}{|\langle \Phi, \Psi \rangle| } \bigg(\int_{\mathbb{R}^{d}} \bigg( \int_{\mathbb{R}^{d}} \bigg( \int_{\mathbb{R}^{4d}} |V_{\Psi}a(z)|{}^{p}| M_{z_2}T_{z_1} \Phi(x, \xi)|{}^{p} dz \bigg)\\ & &\displaystyle \cdot |V_{\psi}f(x, \xi)|{}^{p} e^{p \lambda \omega(x, \xi)} dx \bigg)^{\frac{q}{p}} d \xi \bigg)^{\frac{1}{q}}\\ & \leq &\displaystyle \frac{1}{(2 \pi)^{d}} \frac{1}{|\langle \Phi, \Psi \rangle| } \bigg(\int_{\mathbb{R}^{d}} \! \bigg( \! \int_{\mathbb{R}^{d}} \bigg(\! \int_{\mathbb{R}^{4d}} \big( |V_{\Psi}a(z)| e^{\lambda \omega(z)}\big)^{p}| M_{z_2}T_{z_1} \Phi(x, \xi)|{}^{p} dz \bigg)\\ & &\displaystyle \cdot |V_{\psi}f(x, \xi)|{}^{p} e^{p \lambda \omega(x, \xi)} dx \bigg)^{\frac{q}{p}} \! d \xi \bigg)^{\frac{1}{q}} \\ & \leq &\displaystyle C \| V_{\Psi}a \|{}_{L^{\infty}_{m_{\lambda}}} \cdot \| V_{\psi} f \|{}_{L^{p,q}_{m_{\lambda}}}=C \| a \|{}_{{\mathbf{M}}^{\infty}_{m_{\lambda}}} \cdot \| f \|{}_{{\mathbf{M}}^{p,q}_{m_{\lambda}}}, \end{array} \end{aligned} $$

for some C > 0. Therefore $a \cdot V_{\psi } f \in L^{p,q}_{m_{\lambda }}(\mathbb {R}^{2d})$ and $L^{a}_{\psi ,\gamma } f \in {\mathbf {M}}^{p,q}_{m_{\lambda }}(\mathbb {R}^{d})$.

To prove that $L^{a}_{\psi . \gamma }$ is bounded, consider $g \in \mathcal {S}_{\omega }(\mathbb {R}^{d})$ and set $\Psi = \operatorname *{\mathrm {Wig}}(g,g) \in \mathcal {S}_{\omega }(\mathbb {R}^{2d})$. For $\xi =( \xi _{1}, \xi _{2}) \in \mathbb {R}^{2d}$, we set $\tilde {\xi }=( \xi _{2},- \xi _{1})$. By Cordero and Gröchenig [7, Lemma 2.2]

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle \| \operatorname*{\mathrm{Wig}}( \gamma, \psi) \|{}_{{\mathbf{M}}^{1,p}_{m_{\lambda,2}}}= \| V_{\Psi} \operatorname*{\mathrm{Wig}}( \gamma, \psi) \|{}_{L^{1,p}_{m_{\lambda,2}}}\\ & &\displaystyle = \Big( \int_{\mathbb{R}^{2d}} \Big( \int_{\mathbb{R}^{2d}} \Big| V_{g} \psi \Big(z+ \frac{\tilde{\xi}}{2} \Big) V_{g} \gamma \Big(z- \frac{\tilde{\xi}}{2} \Big) \Big| \, dz \Big)^{p} m_{\lambda,2}^{p} (\xi)\, d \xi \Big)^{\frac{1}{p}}. \end{array} \end{aligned} $$

By the change of variables $z+ \frac {\tilde {\xi }}{2}=\tilde {z}$ and [4, formula (3.12)] we obtain (cf. also [7, Prop. 2.5]):

$$\displaystyle \begin{aligned} \begin{array}{rcl} \| \operatorname*{\mathrm{Wig}}( \gamma, \psi ) \|{}_{{\mathbf{M}}^{1,p}_{m_{\lambda,2}}} & =&\displaystyle \bigg( \int_{\mathbb{R}^{2d}} \bigg( \int_{\mathbb{R}^{2d}} | V_{g} \psi ( \tilde{z})| | V_{g} \gamma (\tilde{z}- \tilde{\xi})| \, d \tilde{z} \bigg)^{p} m_{\lambda,2}^{p} (\xi)\, d \xi \bigg)^{\frac{1}{p}}.\\ & =&\displaystyle \bigg( \int_{\mathbb{R}^{2d}}(|V_{g} \psi(\tilde z)|*|V_{g} \gamma(-\tilde z)|)^{p}(\tilde{\xi}) \,m_{\lambda,2}^{p}( \tilde{\xi}) \, d \tilde{\xi} \bigg)^{\frac{1}{p}}\\ {} & &\displaystyle \leq \| V_{g} \psi\|{}_{L^{1}_{v_{\lambda}}}\| V_{g}\gamma\|{}_{L^{p}_{m_{\lambda}}} = \| \psi \|{}_{{\mathbf{M}}^{1}_{v_{\lambda}}} \| \gamma \|{}_{{\mathbf{M}}^{p}_{m_{\lambda}}}. \end{array} \end{aligned} $$

(0.5)

Therefore $\operatorname *{\mathrm {Wig}}(\gamma ,\psi ) \in {\mathbf {M}}^{1}_{m_{\lambda ,2}}( \mathbb {R}^{2d})$ and hence, from Proposition 0.3 (with p = t = r = +∞, q = s = t′ = 1, λ = 0 and μ = −λ), we have that ${\mathbf {M}}^{\infty }_{m_{- \lambda ,2}}* {\mathbf {M}}^{1}_{m_{\lambda ,2}} \subseteq {\mathbf {M}}^{\infty , 1}$, so that $a^{w}=a* \operatorname *{\mathrm {Wig}}( \gamma , \psi ) \in {\mathbf {M}}^{\infty ,1}$ and by (0.4) with μ = 0

$$\displaystyle \begin{aligned} \| L^{a}_{\psi, \gamma} \|{}_{op} \leq \| a^{w} \|{}_{{\mathbf{M}}^{\infty,1}}. \end{aligned}$$

From (0.1) and (0.5) we finally have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \| L^{a}_{\psi, \gamma} \|{}_{op} & &\displaystyle \leq \| a* \operatorname*{\mathrm{Wig}}( \gamma, \psi ) \|{}_{{\mathbf{M}}^{\infty,1}} \leq \| a \|{}_{{\mathbf{M}}^{\infty}_{m_{- \lambda, 2}}} \| \operatorname*{\mathrm{Wig}}( \gamma, \psi) \|{}_{{\mathbf{M}}^{1}_{m_{\lambda, 2}}} \\ & &\displaystyle \leq \| a \|{}_{{\mathbf{M}}^{\infty}_{m_{- \lambda, 2}}} \| \psi \|{}_{{\mathbf{M}}^{1}_{v_{\lambda}}} \| \gamma \|{}_{{\mathbf{M}}^{p}_{m_{\lambda}}}. \end{array} \end{aligned} $$

□

A boundedness result analogous to that of Theorem 0.4 is proved, with different techniques, in [16] under further restrictions on the symbol a(x, ξ) and without estimates on the norm of $L^a_{\psi ,\gamma }$.

Set now

$$\displaystyle \begin{aligned}{\mathbf{M}}_{m_{\lambda}}^{0,1}(\mathbb{R}^{d})= \{f \in {\mathbf{M}}^{\infty,1}_{m_{\lambda}}(\mathbb{R}^{d}): \lim_{|x| \to \infty} \| V_{g}f(x,.) \|{}_{L^{1}_{m_{\lambda}}} e^{\lambda \omega(x)}=0 \} \end{aligned}$$

and prove the following compactness result (cf. also [1, Prop. 2.3] and [12, Thm. 3.22]):

FormalPara Theorem 0.5

If $a^{w} \in {\mathbf {M}}_{m_{\lambda }}^{0,1}( \mathbb {R}^{2d})$ for some λ ≥ 0, then $L^{a^{w}}$ is a compact mapping of ${\mathbf {M}}^{p,q}_{m_{\lambda }}(\mathbb {R}^{d})$ into itself, for 1 ≤ p, q < +∞.

FormalPara Proof

The operator $L^{a^{w}}$ maps ${\mathbf {M}}^{p,q}_{m_{\lambda }}(\mathbb {R}^{d})$ into itself by (0.4). To prove that $L^{a^{w}}$ is compact we first assume $a^{w} \in \mathcal {S}_{\omega }( \mathbb {R}^{2d})$. From (0.3)

$$\displaystyle \begin{aligned} \begin{array}{rcl} L^{a^{w}}f(y) =& &\displaystyle \frac{1}{(2 \pi)^{d}} \int_{\mathbb{R}^{2d}} \hat{a}^{w}(\xi,u) e^{-i \xi \cdot u} e^{i \xi \cdot (y+u)} f(y+u) \, du \, d \xi \\ =& &\displaystyle \frac{1}{(2 \pi)^{d}} \int_{\mathbb{R}^{2d}} \hat{a}^{w}(\xi,x-y) e^{i \xi \cdot y} f(x) \, dx \, d \xi\\ =& &\displaystyle \int_{\mathbb{R}^{d}} k(x,y) f(x) \, dx,{} \end{array} \end{aligned} $$

(0.6)

with kernel $k(x, y)= \frac {1}{(2 \pi )^{d}}\int _{\mathbb {R}^{d}} \hat {a}^{w}(\xi ,x-y) e^{i \xi \cdot y} d \xi$. Note that $k(x,y) \in \mathcal {S}_{\omega }( \mathbb {R}^{2d})$ because it is the inverse Fourier transform (with respect to the first variable) of the translation (with respect to the second variable) of $\hat {a}^{w} \in \mathcal {S}_{\omega }( \mathbb {R}^{2d})$.

Now, let $\phi \in \mathcal {S}_{\omega }(\mathbb {R}^{d})$ and α ₀, β ₀ > 0 such that $\{ \phi _{jl} \}_{j,l \in \mathbb {Z}^{d}}= \{ M_{\beta _{0}l}T_{\alpha _{0}j} \phi \}_{j,l \in \mathbb {Z}^{d}}$ is a tight Gabor frame for $L^{2}(\mathbb {R}^{d})$ (see [13, Def. 5.1.1] for the definition). Then $\{ \Phi _{jlmn} \}_{j,l,m,n \in \mathbb {Z}^{d}}=\{ \phi _{jl}(x) \phi _{mn}(y) \}_{j,l,m,n \in \mathbb {Z}^{d}}$ is a tight Gabor frame for $L^{2}( \mathbb {R}^{2d})$. Since $k \in \mathcal {S}_{\omega }(\mathbb {R}^{2d})$ we have that 〈k, Φ_jlmn〉 = V _ϕ k(α ₀ j, α ₀ m, β ₀ l, β ₀ n) ∈ ℓ ¹ and (see [4, Lemma 3.15])

$$\displaystyle \begin{aligned}k= \sum_{j,l,m,n \in \mathbb{Z}^{d}} \langle k, \Phi_{jlmn} \rangle \Phi_{jlmn}.\end{aligned}$$

Therefore from (0.6)

$$\displaystyle \begin{aligned}L^{a^{w}}f=\sum_{j,l,m,n \in \mathbb{Z}^{d}} \langle k, \Phi_{jlmn} \rangle \langle \phi_{jl},f \rangle \phi_{mn},\end{aligned}$$

with 〈k, Φ_jlmn〉∈ ℓ ¹, $(\phi _{jl})_{j,l \in \mathbb {Z}^{d}}$ equicontinuous in ${\mathbf {M}}^{p',q'}_{m_{-\lambda }}= ({\mathbf {M}}^{p,q}_{m_{\lambda }})^{*}$ and $(\phi _{mn})_{m,n \in \mathbb {Z}^{d}}$ bounded in $\bigcup _{n \in \mathbb {N}} n \{ f \in {\mathbf {M}}^{\, p,q}_{m_{\lambda }}: \| f \|{ }_{{\mathbf {M}}^{p,q}_{m_{\lambda }}} <1 \},$ so that $L^{a^{w}}$ is a nuclear operator from ${\mathbf {M}}^{p,q}_{m_{\lambda }}$ to ${\mathbf {M}}^{p,q}_{m_{\lambda }}$ (see [15, §17.3]). From [15, §17.3, Cor. 4] we thus have that $L^{a^{w}}$ is compact.

Let us finally consider the general case $a \in {\mathbf {M}}^{0,1}_{m_{\lambda }}(\mathbb {R}^{2d})$. By Boiti et al. [4, Prop. 3.9] there exist $a_{n} \in \mathcal {S}_{\omega }(\mathbb {R}^{2d})$ converging to a in ${\mathbf {M}}^{\infty ,1}_{m_{\lambda }}$ and hence, by (0.4)

$$\displaystyle \begin{aligned}\| L^{a^w}- L^{a^w_{n}} \|{}_{{\mathbf{M}}_{m_{\lambda}}^{\,p,q} \to {\mathbf{M}}_{m_{\lambda}}^{\,p,q}} \leq \| a- a_{n} \|{}_{{\mathbf{M}}^{\infty, 1}_{m_{\lambda}}} \to 0.\end{aligned}$$

Since the set of compact operators is closed we have that $L^{a^w}$ is compact on ${\mathbf {M}}_{m_{\lambda }}^{\,p,q}(\mathbb {R}^{d}).$ □

We have the following generalization of [10, Lemma 3.4] and [11, Prop. 5.2]:

FormalPara Lemma 0.6

Let $g_{0} \in \mathcal {S}_{\omega }(\mathbb {R}^{d})$ and $a \in {\mathbf {M}}^{\infty }_{m_{\lambda }}(\mathbb {R}^{d})$ , with λ ≥ 0, such that

$$\displaystyle \begin{aligned} \lim_{|x| \to + \infty} \sup_{| \xi | \leq R} |V_{g_{0}} a(x, \xi)| e^{\lambda \omega(x,\xi)}=0, \qquad \forall R>0. \end{aligned} $$

(0.7)

Then $a*H \in {\mathbf {M}}_{m_{\lambda }}^{0,1}(\mathbb {R}^{d})$ for any $H \in \mathcal {S}_{\omega }(\mathbb {R}^{d})$.

FormalPara Proof

The case λ = 0 has been proved in [10, Lemma 3.4]. Let λ > 0. Since $g_{0} \in \mathcal {S}_{\omega }(\mathbb {R}^{d})$ and $H \in \mathcal {S}_{\omega }(\mathbb {R}^{d})$, by Gröchenig and Zimmermann [14, Thm. 2.7] we have that $V_{g_{0}}H \in \mathcal {S}_{\omega }( \mathbb {R}^{2d})$ and hence, for a fixed ℓ > 0 (to be chosen later depending on λ), there exists c _λ > 0 such that

$$\displaystyle \begin{aligned} |V_{g_{0}}H(x, \xi)| \leq c_{\lambda} e^{-3 \ell \lambda \omega(x)} e^{-3 \ell \lambda \omega( \xi)}, \qquad \forall x , \xi \in \mathbb{R}^{d}. \end{aligned}$$

Now, as in the proof of Proposition 0.3, for g = g ₀ ∗ g ₀, we have that $| V_{g}(a*H)(\cdot , \xi )|= |V_{g_{0}}a( \cdot , \xi )*V_{g_{0}}H( \cdot , \xi )|$. Since ω is increasing and subadditive we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle | V_{g}(a*H)(x, \xi)| \leq \int_{\mathbb{R}^{d}} | V_{g_{0}}a(x-y, \xi )| | V_{g_{0}} H(y, \xi)| dy \\ & &\displaystyle \leq c_{\lambda} e^{-3 \ell \lambda \omega(\xi )} \! \int_{\mathbb{R}^{d}} \! | V_{g_{0}}a(x-y, \xi )| e^{-3 \ell \lambda \omega(y)} dy \\ & &\displaystyle = c_{\lambda} e^{-3 \ell \lambda \omega(\xi )} \! \int_{\mathbb{R}^{d}} \! | V_{g_{0}}a(x-y, \xi )| e^{-3 \ell \lambda \omega(y)} \, e^{\lambda\omega(x-y,\xi)}e^{-\lambda\omega(x-y,\xi) } dy \\ & &\displaystyle \leq c_{\lambda} e^{-3 \ell\lambda \omega(\xi )} e^{- \lambda \omega(x)} \! \int_{\mathbb{R}^{d}} \!| V_{g_{0}}a(x-y, \xi )| e^{\lambda \omega(x-y, \xi)}e^{-(3 \ell-1) \lambda \omega(y)} dy. \end{array} \end{aligned} $$

Since $a \in {\mathbf {M}}^{\infty }_{m_{\lambda }}(\mathbb {R}^{d})$ we have that

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle e^{\lambda \omega(x)+ 2 \ell\lambda \omega(\xi)} | V_{g}(a*H)(x, \xi)|\\ {} \leq & &\displaystyle c_{\lambda} e^{- \ell \lambda \omega(\xi)} \int_{\mathbb{R}^{d}} | V_{g}a(x-y, \xi )| e^{\lambda \omega(x-y, \xi)}e^{-(3 \ell-1) \lambda \omega(y)} dy \end{array} \end{aligned} $$

(0.8)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \leq & &\displaystyle c_{\lambda} e^{- \ell \lambda \omega(\xi)} \| a \|{}_{{\mathbf{M}}^{\infty}_{m_{\lambda}}} \int_{\mathbb{R}^{d}} e^{-(3 \ell-1) \lambda \omega(y)} dy< + \infty, \end{array} \end{aligned} $$

(0.9)

if $\ell >\frac {1}{3}+ \frac {d}{3 B \lambda }$, where B is the constant of condition (γ) in Definition 0.1. Since lim_|ξ|→+∞ ω(ξ) = +∞, from (0.9) we have that for all ε > 0 there exists R ₁ > 0 such that

$$\displaystyle \begin{aligned} e^{\lambda \omega(x)+ 2 \ell\lambda \omega(\xi)} | V_{g}(a*H)(x, \xi)| < \varepsilon, \quad \forall x, \xi \in \mathbb{R}^{d}, \quad | \xi| \geq R_{1}. \end{aligned} $$

(0.10)

We now choose δ > 0 small enough so that

$$\displaystyle \begin{aligned} \delta \bigg(1 + c_{\lambda} \int_{\mathbb{R}^{d}} e^{-(3\ell-1) \lambda \omega(y)}\bigg) dy \leq \varepsilon. \end{aligned} $$

(0.11)

From the hypothesis (0.7) we can choose R ₂ > 0 sufficiently large so that

$$\displaystyle \begin{aligned} \sup_{| \xi| \leq R_{1}} | V_{g_{0}}a(x, \xi)| e^{\lambda \omega(x, \xi)}< \delta, \quad |x| \geq R_{2}, \end{aligned} $$

(0.12)

$$\displaystyle \begin{aligned} \int_{|y| > R_{2}} e^{-(3 \ell-1) \lambda \omega(y)} \, dy < \frac{\delta}{c_{\lambda} e^{- \ell \lambda \omega(\xi) }\| a \|{}_{{\mathbf{M}}_{m_{\lambda}}^{\infty}}}, \qquad | \xi| \leq R_{1}. \end{aligned} $$

(0.13)

Therefore for |x|≥ 2R ₂, |y|≤ R ₂ (so that |x − y|≥ R ₂) and |ξ|≤ R ₁, by (0.8), (0.9), (0.13), (0.12) and (0.11):

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle e^{\lambda \omega(x)+ 2 \ell\lambda \omega(\xi)} | V_{g}(a*H)(x, \xi)| \\ \leq & &\displaystyle c_{\lambda} e^{ - \ell \lambda \omega(\xi)} \| a \|{}_{{\mathbf{M}}_{m_{\lambda}}^{\infty}} \int_{|y| > R_{2}} e^{-(3 \ell-1) \lambda \omega(y)} dy \\ & &\displaystyle + c_{\lambda} e^{-\ell \lambda \omega(\xi )} \int_{|y| \leq R_{2}} | V_{g_{0}}a(x-y, \xi )| e^{ \lambda \omega(x-y, \xi)} e^{- (3 \ell-1) \lambda \omega(y)} dy \\ < & &\displaystyle \delta + c_{\lambda} \delta \int_{\mathbb{R}^{d}} e^{-(3 \ell-1) \lambda \omega(y)} dy \leq \varepsilon. \end{array} \end{aligned} $$

The above estimate, together with (0.10), gives

$$\displaystyle \begin{aligned}e^{\lambda \omega(x)} \int_{\mathbb{R}^{d}} | V_{g}(a*H)(x, \xi)| e^{\lambda \omega(\xi)} d \xi \leq \varepsilon \int_{\mathbb{R}^{d}} e^{- (2 \ell-1) \lambda \omega(\xi)} d \xi, \qquad |x| \geq 2 R_{2}. \end{aligned}$$

Choosing now $\ell >\frac {1}{2}+ \frac {d}{2 B \lambda }>\frac {1}{3}+ \frac {d}{3 B \lambda }$ so that $e^{-(2 \ell -1) \lambda \omega (\xi )} \in L^{1}(\mathbb {R}^{d})$, we finally obtain

$$\displaystyle \begin{aligned}\lim_{|x| \to \infty} e^{\lambda \omega(x)} \| V_{g}(a*H)(x,.) \|{}_{L^{1}_{m_{\lambda}}}=0. \end{aligned}$$

□

FormalPara Theorem 0.7

Let $\psi , \gamma \in \mathcal {S}_{\omega }(\mathbb {R}^{d})$, $g_{0} \in \mathcal {S}_{\omega }(\mathbb {R}^{2d})$ and $a \in {\mathbf {M}}^{\infty }_{m_{\lambda }}(\mathbb {R}^{2d})$ satisfying (0.7), for some λ ≥ 0. Then $L^{a}_{\psi , \gamma }: {\mathbf {M}}_{m_{\lambda }}^{p,q}(\mathbb {R}^{d})\to {\mathbf {M}}_{m_{\lambda }}^{p,q}(\mathbb {R}^{d})$ is compact, for 1 ≤ p, q < +∞.

FormalPara Proof

Set $H:= W( \gamma , \psi ) \in \mathcal {S}_{\omega }( \mathbb {R}^{2d})$. Since $a \in {\mathbf {M}}^{\infty }_{m_{\lambda }}(\mathbb {R}^{2d})$, by Lemma 0.6 we have that $a^{w}= a* H \in {\mathbf {M}}_{m_{\lambda }}^{0,1}(\mathbb {R}^{2d})$ and hence $L^{a}_{\psi , \gamma }=L^{a^{w}}$ is compact by Theorem 0.5. □