Atomic Decompositions of Localized Hardy Spaces with Variable Exponents and Applications

In this paper, we introduce the localized Hardy spaces with variable exponents \(h^{p(\cdot )}\) and establish a new atomic decomposition theorem for \(h^{p(\cdot )}\) by using the discrete Littlewood–Paley–Stein theory. As an application of atomic decomposition, we investigate molecule decomposition for \(h^{p(\cdot )}\). Moreover, pseudo-differential operators of order zero are shown to be bounded on \(h^{p(\cdot )}\).