On equivalent conditions for the general weighted Hardy type inequality in space $L^{p(\cdot )}$

Abstract

We study the Hardy type two-weighted inequality for the multidimensional Hardy operator in the norms of generalized Lebesgue spaces $L^{p(\cdot )}({\mathbb{R}}^n)$. In this way we prove equivalent conditions for $L^{p(\cdot )}\to L^{q(\cdot )}$ boundedness of Hardy operator in the case of exponents $q(0)\ge p(0)$, $q(\infty )\ge p\left(\infty \right)$. We also prove that the condition for such inequality to hold coincides with condition for validity of two weighted Hardy inequalities with constant exponents, if we require the exponents to be regular near zero and at infinity.