Atomic Characterization and Its Applications of Matrix-Weighted Variable Hardy Spaces

Abstract

In this article, by means of the matrix-weighted grand maximal function we first introduce the variable Hardy space $H_W^{p(\cdot )}$ on ${\mathbb{R}}^n$ with the ${\mathcal{A}}_{p(\cdot ),\infty }$ matrix weight $W$ and with the variable exponent $p(\cdot )$ having globally log-Hölder continuity, and then via using several different convex body valued maximal functions we establish its various maximal function equivalent characterizations. Combining a refined Whitney decomposition with both the convex body valued maximal function and its corresponding convex-body reducing operator, we obtain the atomic characterization of $H_W^{p(\cdot )}$. As applications, we give its dual space and establish the boundedness of Calderón--Zygmund operators from $H_W^{p(\cdot )}$ to the matrix-weighted variable Lebesgue space $L_W^{p(\cdot )}$ and to itself. This approach to establishing atomic characterization differs from all previous ones.