Variable Matrix-Weighted Besov Spaces

TL;DR

The article advances the theory of Besov spaces by introducing matrix-weighted variable Besov spaces $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$ and their averaging variants, using variable matrix $\mathscr{A}_{p(\cdot),\infty}$ weights. It establishes fundamental equivalences with associated sequence spaces, provides a robust $\varphi$-transform characterization, and develops a full decomposition theory via molecules, wavelets, and atoms. The work then leverages these structures to prove trace, extension, and Calderón–Zygmund operator boundedness results in this non-scalar, variable-exponent setting. Altogether, the paper delivers a coherent, operator-rich framework for matrix-weighted Besov spaces in the variable-exponent context, with potential applications to harmonic analysis and PDEs under nonuniform media. Throughout, all results are formulated in terms of $p(\cdot)$, $q(\cdot)$, $s(\cdot)$, and matrix weights $W$ within ${\mathscr{A}}_{p(\cdot),\infty}$.

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 averaging 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 this and via first establishing the boundedness of $α$-convexification $η$-type operators on variable Lebesgue spaces, we obtain the boundedness of almost diagonal operators on the sequence space $b^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$ related to $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$, which 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 properties of $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$, and, applying the molecular characterization, we obtain the boundedness of Calderón--Zygmund operators on $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$.

Theorems & Definitions (149)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 2.8
• Lemma 2.9
• Definition 2.10