Precompact sets in matrix weighted Lebesgue spaces with variable exponent

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