Atomic decomposition of variable Hardy spaces via Littlewood–Paley–Stein theory

Abstract

The purpose of this paper is to give a new atomic decomposition for variable Hardy spaces via the discrete Littlewood–Paley–Stein theory. As an application of this decomposition, we assume that $T$ is a linear operator bounded on $L^q$ and $H^{p(\cdot )}$, and we thus obtain that $T$ can be extended to a bounded operator from $H^{p(\cdot )}$ to $L^{p(\cdot )}$.