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

In this article, by means of the matrix-weighted grand maximal function we first introduce the variable Hardy space $H^{p(\cdot)}_W$ on $\mathbb{R}^n$ with the $\mathscr{A}_{p(\cdot),\infty}$ matrix weight $W$ and with the variable exponent $p(\cdot)$ having globally log-H\"older 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 $H^{p(\cdot)}_W$. As applications, we give its dual space and establish the boundedness of Calder\'on--Zygmund operators from $H^{p(\cdot)}_W$ to the matrix-weighted variable Lebesgue space $L^{p(\cdot)}_W$ and to itself. This approach to establishing atomic characterization differs from all previous ones.