Year: 2025
Type: preprint
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 $φ$-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 $B^{s(\cdot)}_{p(\cdot),q(\cdot)}(W)$.
Source: arXiv (Cornell University)
Authors: Yang, Dachun, Yuan, Wen, Zeng, Zongze
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416068101
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