Convolution Operators in Matrix Weighted, Variable Lebesgue Spaces

TL;DR

This paper extends the theory of matrix weights to variable Lebesgue spaces, proving boundedness of convolution operators and convergence of approximate identities, and applies these results to a matrix-weighted Sobolev space version of the H=W theorem.

Contribution

It generalizes the matrix p condition to variable exponents and establishes boundedness and convergence results for convolution operators in this setting.

Findings

Proved boundedness of convolution operators in matrix weighted, variable Lebesgue spaces.

Showed convergence of approximate identities in these spaces.

Established a version of the H=W theorem for matrix weighted, variable exponent Sobolev spaces.

Abstract

We extend the theory of matrix weights to the variable Lebesgue spaces. The theory of matrix $\mathcal{A}_p$ weights has attracted considerable attention beginning with the work of Nazarov, Treil, and Volberg in the 1990s. We extend this theory by generalizing the matrix $\mathcal{A}_p$ condition to the variable exponent setting. We prove boundedness of the convolution operator $\mathbf{f}\mapsto \phi\ast \mathbf{F}$ for $\phi \in C_c^\infty(\Omega)$, and show that the approximate identity defined using $\phi$ converges in matrix weighted, variable Lebesgue spaces $L^{p(\cdot)}(W,\Omega)$ for $W$ in matrix $\mathcal{A}_{p(\cdot)}$. Our approach to this problem is through averaging operators; these results are of interest in their own right. As an application of our work, we prove a version of the classical $H=W$ theorem for matrix weighted, variable exponent Sobolev spaces.