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Abstract

In this article, applying matrix ${\mathcal A}_{p(\cdot),\infty}$ weights introduced in our previous work, we introduce the matrix-weighted variable Besov space via the matrix weight $W$ or the reducing operators ${\mathbb{A}}$ of order $p(\cdot)$ for $W$, Then we show that, defined either by the matrix weight $W$ or the reducing operators ${\mathbb{A}}$ of order $p(\cdot)$ for $W$, the matrix-weighted variable Besov spaces (respectively, the matrix-weighted variable Besov sequence spaces) are both equal. Next, we establish the $\varphi$-transform theorem for matrix-weighted variable Besov spaces and, using this, find that the definition of matrix-weighted variable Besov spaces is independent of the choice of $\varphi$. After that, for the further discussion of variable Besov spaces, we establish the theorem of almost diagonal operators and then, by using this, we establish the molecular characterization. Then, with applying the molecular characterization, we obtain the wavelet and atomic characterizations of matrix-weighted variable Besov spaces. Finally, as an application, we consider some classical operators. By using the wavelet characterization, we establish the trace operator and obtain the theorem of trace operators. Moreover, with applying the molecular characterization, we establish the theorem of Calder\'on--Zygmund operators on matrix-weighted variable Besov spaces.