A Hardy inequality and applications to reverse Hölder inequalities for weights on ℝ

Abstract

We prove a sharp integral inequality valid for non-negative functions defined on [0, 1], with given 𝐿¹ norm. This is in fact a generalization of the well known integral Hardy inequality. We prove it as a consequence of the respective weighted discrete analogue inequality whose proof is presented in this paper. As an application we find the exact best possible range of 𝑝 > 𝑞 such that any non-increasing 𝑔 which satisfies a reverse Hölder inequality with exponent 𝑞 and constant 𝑐 upon the subintervals of (0, 1], should additionally satisfy a reverse Hölder inequality with exponent 𝑝 and in general a different constant 𝑐′. The result has been treated in [1] but here we give an alternative proof based on the above mentioned inequality.