Precompact Sets in Matrix Weighted Lebesgue Spaces with Variable Exponent

TL;DR

This paper establishes criteria for precompactness in matrix-weighted Lebesgue and Sobolev spaces with variable exponents using translation, average, and approximate identity operators, advancing the understanding of compactness in these complex function spaces.

Contribution

It introduces new conditions for precompactness in matrix-weighted Lebesgue and Sobolev spaces with variable exponents, utilizing multiple operators.

Findings

Provided sufficient conditions for precompactness via translation operator.

Derived criteria for precompactness using average operator.

Extended precompactness results to matrix-weighted Sobolev spaces.

Abstract

In this paper, we first give a sufficiently condition for precompactness in the matrix-weighted Lebesgue spaces with variable exponent by translation operator. Then we obtain a criterion for precompactness in the matrix-weighted Lebesgue space with variable exponent by average operator. Next, we give a criterion for precompactness in the matrix-weighted Lebesgue space with variable exponent by approximate identity. Finally, precompactness in the matrix-weighted Sobolev space with variable exponent is also considered.