Convolution Operators in Matrix Weighted, Variable Lebesgue 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.