Molecular decomposition of the modulation spaces

Abstract

The aim of this paper is to develop a theory of decomposition in the weighted modulation spaces $M_{p,q}^{s,W}$ with $0 < p,q \le \infty$, $s \in \mathbb{R}$ and $W \in A_{\infty}$, where $W$ belongs to the class of $A_{\infty}$ defined by Muckenhoupt. For this purpose we shall define molecules for the modulation spaces. As an application we give a simple proof of the boundedness of the pseudo-differential operators with symbols in $M_{\infty,\min(1,p,q)}^{0}$. We shall deal with dual spaces as well.