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.

In this paper, we prove the reverse Hardy inequality for Hardy operator in weighted variable Lebesgue spaces with exponent less than one. In particular, we establish necessary and sufficient conditions on weight functions for the validity of the reverse Hardy inequality for Hardy operator in weighted variable Lebesgue spaces with negative exponents. It should be noted that in the case of variable Lebesgue space L p ( x ) ( 0 , ∞ ) for 0 ( x) <1 , the obtained necessary and sufficient conditions on the weight functions are different and coincide for some classes of variable exponents. Also, we prove similar results for the dual Hardy operator. The results are illustrated by an example.