A Hardy inequality and applications to reverse Hölder inequalities for weights on $\mathbb{R}$

Abstract

We prove a sharp integral inequality valid for non-negative functions defined on $[0,1]$, with given $L^1$ 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.