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

2013/12/06 by Eleftherios N. Nikolidakis, Nikolidakis, Eleftherios N.
Engineering · Mathematics · #Advanced Harmonic Analysis Research #Differential Equations and Boundary Problems #FOS: Mathematics #Fatigue and fracture mechanics #Functional Analysis (math.FA) #math.FA

paper · pdf · doi:10.48550/arxiv.1312.1991

11 pages

openalex publication_date 2013/12/06 · arxiv created 2014/12/08 · arxiv updated 2014/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 which proof is presented in this paper. As an application we find the exact best possible range of p>q such that any non-increasing f 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 a different in general constant c'. The result has been treated in \cite1 but here we give an alternative proof based on the above mentioned inequality.

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