Matrix-Weighted Variable Besov Spaces

Abstract

In this article, using variable matrix ${\mathscr {A}}_{p(\cdot ),\infty }$ weights, we introduce the matrix-weighted variable Besov space $B^{s(\cdot )}_{p(\cdot ),q(\cdot )}(W)$ and the corresponding reducing variable Besov space $B^{s(\cdot )}_{p(\cdot ),q(\cdot )}({\mathbb {A}})$, and prove that they are equivalent. Applying this, we establish the $\varphi $-transform characterization of $B^{s(\cdot )}_{p(\cdot ),q(\cdot )}(W)$. By 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 $b^{s(\cdot )}_{p(\cdot ),q(\cdot )}(W)$ corresponding to $B^{s(\cdot )}_{p(\cdot ),q(\cdot )}(W)$, which, together with the aforementioned $\varphi $-transform characterization, is further used to establish various decomposition characterizations of $B^{s(\cdot )}_{p(\cdot ),q(\cdot )}(W)$, respectively, in terms of molecules, wavelets, and atoms. Applying the wavelet decomposition of $B^{s(\cdot )}_{p(\cdot ),q(\cdot )}(W)$, we obtain the trace theorem and the extension property of $B^{s(\cdot )}_{p(\cdot ),q(\cdot )}(W)$, and applying the molecular characterization, we obtain the boundedness of Calderón–Zygmund operators on $B^{s(\cdot )}_{p(\cdot ),q(\cdot )}(W)$.

Keywords

Variable Besov space / Variable matrix weight / Almost diagonal operator / Atom / Molecule / Wavelet / Trace operator / Calderón–Zygmund operator / 46E35 / 47A56 / 15A15 / 46E40 / 42B35

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Funding

National Natural Science Foundation of China(12371093 and 12431006)

Beijing Natural Science Foundation(1262011)

Fundamental Research Funds for the Central Universities(2253200028)

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