Characterization of relatively compact sets in weighted Lebesgue spaces by weighted modulus of continuity and application

In this paper we consider a weighted modulus of continuity in weighted Lebesgue spaces. We proved that the modulus of continuity is well defined in weighted Lebesgue spaces and has all the properties of usual modulus of continuity. Also, we study relatively compact sets in weighted Lebesgue spaces. The full characterization of relatively compact sets is given in the case of weighted Lebesgue space by weighted modulus of continuity. In particular, we give an analog of the Kolmogorov-Riesz compactness theorem in weighted Sobolev spaces by weighted modulus of continuity. As an application, we give an approximation theorem for a family of averaging operators in weighted Lebesgue spaces by a weighted modulus of continuity.