Convolution operators in matrix weighted, variable Lebesgue spaces

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.