On a Hardy Type General Weighted Inequality in Spaces L p(·)

Abstract

A Hardy type two-weighted inequality is investigated for the multidimensional Hardy operator in the norms of generalized Lebesgue spaces Lp(·). Equivalent necessary and sufficient conditions are found for the \({L^{p(\cdot)} \longrightarrow L^{q(\cdot)}}\) boundedness of the Hardy operator when exponents q(0) < p(0), q(∞) < p(∞). It is proved that the condition for such an inequality to hold coincides with the condition for the validity of two-weighted Hardy inequalities with constant exponents if we require of the exponents to be regular near zero and at infinity.