A Characterization of Some Weighted Norm Inequalities for Maximal Operators

Abstract

It is found class of pairs of exponents $(p(\cdot ),q(\cdot ))$ such that for pairs of Banach function spaces$(L^{p(\cdot )}(R^n),L^{q(\cdot )}(R^n))$ weak Minkowski’s inequality holds. Also some conditions which ensure the boundednness of maximal operator $M_{\phi }:L^{p(\cdot )}(R^n)\to L_w^{q(\cdot )}(R^n)$ are found, when for the pair of Banach function spaces $(L^{p(\cdot )}(R^n),L^{q(\cdot )}(R^n))$ holds weak Minkowski’s inequality.