Hardy inequalities on metric measure spaces, III: The case q ≤ p ≤ 0 and applications

A. Kassymov, M. Ruzhansky, D. Suragan

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1 Citation (Scopus)


In this paper, we obtain a reverse version of the integral Hardy inequality on metric measure space with two negative exponents. For applications we show the reverse Hardy-Littlewood-Sobolev and the Stein-Weiss inequalities with two negative exponents on homogeneous Lie groups and with arbitrary quasi-norm, the result of which appears to be new in the Euclidean space. This work further complements the ranges of p and q (namely, q≤p<0) considered in the work of Ruzhansky & Verma (Ruzhansky & Verma 2019 Proc. R. Soc. A 475, 20180310 (doi:10.1098/rspa.2018.0310); Ruzhansky & Verma. 2021 Proc. R. Soc. A 477, 20210136 (doi:10.1098/rspa.2021.0136)), which treated the cases 1<p≤q<∞ and p>q, respectively.

Original languageEnglish
Article number20220307
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2269
Publication statusPublished - Jan 25 2023


  • metric measure space
  • reverse Hardy inequality
  • reverse Hardy-Littlewood-Sobolev inequality
  • reverse Stein-Weiss inequality

ASJC Scopus subject areas

  • General Mathematics
  • General Engineering
  • General Physics and Astronomy


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