Variable Matrix-Weighted Besov Spaces

In this article, using variable matrix ${\mathcal{A}}_{p(\cdot ),\mathrm{\infty }}$ weights, we introduce the matrix-weighted variable Besov space $B_{p(\cdot ),q(\cdot )}^{s(\cdot )}(W)$ and the corresponding averaging variable Besov space $B_{p(\cdot ),q(\cdot )}^{s(\cdot )}(\mathbb{A})$ and prove that they are equivalent. Applying this, we establish the $\phi$-transform characterization of $B_{p(\cdot ),q(\cdot )}^{s(\cdot )}(W)$. By this and via first establishing the boundedness of $\alpha$-convexification $\eta$-type operators on variable Lebesgue spaces, we obtain the boundedness of almost diagonal operators on the sequence space $b_{p(\cdot ),q(\cdot )}^{s(\cdot )}(W)$ related to $B_{p(\cdot ),q(\cdot )}^{s(\cdot )}(W)$, which is further used to establish various decomposition characterizations of $B_{p(\cdot ),q(\cdot )}^{s(\cdot )}(W)$, respectively, in terms of molecules, wavelets, and atoms. Applying the wavelet decomposition of $B_{p(\cdot ),q(\cdot )}^{s(\cdot )}(W)$, we obtain the trace theorem and the extension properties of $B_{p(\cdot ),q(\cdot )}^{s(\cdot )}(W)$, and, applying the molecular characterization, we obtain the boundedness of Calderón–Zygmund operators on $B_{p(\cdot ),q(\cdot )}^{s(\cdot )}(W)$.