## Journal of the Mathematical Society of Japan

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

Eleftherios N. NIKOLIDAKIS

#### 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.

#### Article information

Source
J. Math. Soc. Japan, Volume 70, Number 1 (2018), 141-152.

Dates
First available in Project Euclid: 26 January 2018

https://projecteuclid.org/euclid.jmsj/1516957222

Digital Object Identifier
doi:10.2969/jmsj/07017323

Mathematical Reviews number (MathSciNet)
MR3750271

Zentralblatt MATH identifier
06859847

Subjects
Primary: 26D15: Inequalities for sums, series and integrals
Secondary: 42B25: Maximal functions, Littlewood-Paley theory

#### Citation

NIKOLIDAKIS, Eleftherios N. A Hardy inequality and applications to reverse Hölder inequalities for weights on $\mathbb{R}$. J. Math. Soc. Japan 70 (2018), no. 1, 141--152. doi:10.2969/jmsj/07017323. https://projecteuclid.org/euclid.jmsj/1516957222

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