The space $\Omega^p_m(R^d)$ and some properties

Abstract

Let $m$ be a $v$-moderate function defined on $R^d$ and let $g \in L^2(R^d)$. In this work, we define $\Omega ^p_m(R^d)$ to be the vector space of $f \in L^2_n(R^d)$ such that the Gabor transform $V_gf$ belongs to $L^p(R^{2d})$, where $1 \leq p < \infty$. We endowe it with a norm and show that it is a Banach space with this norm. We also study some preliminary properties of $\Omega ^p_m(R^d)$. Later we discuss inclusion properties and obtain the dual space of $\Omega ^p_m(R^d)$. At the end of this work, we study multipliers from $L_w^1 (R^d)$ into $\Omega ^p_w(R^d)$ and from $\Omega ^p_w(R^d)$ into $L^{\infty}_{w^{-1}}(R^d)$, where $w$ is Beurling's weight function.