Gabor multipliers for weighted Banach spaces on locally compact abelian groups

Abstract

We use a projective groups representation $\rho$ of the unimodular group $\mathcal{G} \times \hat{\mathcal{G}}$ on $L^2(\mathcal{G}$) to define Gabor wavelet transform of a function $f$ with respect to a window function $g$, where $\mathcal{G}$ is a locally compact abelian group and $\hat{\mathcal{G}}$ its dual group. Using these transforms, we define a weighted Banach $\mathcal{H}^{1, \rho}_w(\mathcal{G})$ and its antidual space $\mathcal{H}^{{1}^{\sim}, \rho}_w(\mathcal{G})$, $w$ being a moderate weight function on $\mathcal{G} \times \hat{\mathcal{G}}$. These spaces reduce to the well known Feichtinger algebra $S_0(\mathcal{G})$ and Banach space of Feichtinger distribution $S'_0(\mathcal{G})$ respectively for $w\equiv 1$. We obtain an atomic decomposition of $\mathcal{H}^{1, \rho}_w(\mathcal{G})$ and study some properties of Gabor multipliers on the spaces $L^2(\mathcal{G}), \mathcal{H}^{1, \rho}_w(\mathcal{G})$ and $\mathcal{H}^{{1}^{\sim}, \rho}_w(\mathcal{G})$. Finally, we prove a theorem on the compactness of Gabor multiplier operators on $L^2(\mathcal{G})$ and $\mathcal{H}^{1, \rho}_w(\mathcal{G})$, which reduces to an earlier result of Feichtinger [Fei 02, Theorem 5.15 (iv)] for $w=1$ and $\mathcal{G}=R^d$.