Convolution operators in matrix weighted, variable lebesgue spaces

TL;DR: This paper extends matrix weighted theory to variable Lebesgue spaces, proving boundedness of convolution operators and convergence of approximate identities, with applications to matrix weighted, variable exponent Sobolev spaces and the Calderón-Zygmund theorem.

Abstract: We extend the theory of matrix weights to the variable Lebesgue spaces. The theory of matrix [Formula: see text] 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 [Formula: see text] condition to the variable exponent setting. We prove boundedness of the convolution operator [Formula: see text] for [Formula: see text], and show that the approximate identity defined using [Formula: see text] converges in matrix weighted, variable Lebesgue spaces [Formula: see text] for [Formula: see text] in matrix [Formula: see text]. 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 [Formula: see text] theorem for matrix weighted, variable exponent Sobolev spaces.