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

A. Kassymov; M. Ruzhansky; D. Suragan

doi:10.1098/rspa.2022.0307

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

A. Kassymov
, M. Ruzhansky
, D. Suragan

Department of Mathematics

Ghent University
Al Farabi Kazakh National University
Queen Mary University of London
Nazarbayev University

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

Abstract

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 language	English
Article number	20220307
Journal	Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume	479
Issue number	2269
DOIs	https://doi.org/10.1098/rspa.2022.0307
Publication status	Published - Jan 25 2023

Funding

The first and second authors were supported in parts by the FWO Odysseus 1 grant no. G.0H94.18N: Analysis and Partial Differential Equations, by the Methusalem programme of the Ghent University Special Research Fund (BOF) (grant no. 01M01021) and by the EPSRC (grant no. EP/R003025/2). Also, this research has been funded by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (grant no. AP19676031) and partially supported by the collaborative research programme ‘Qualitative analysis for nonlocal and fractional models’ from Nazarbayev University.

Keywords

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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10.1098/rspa.2022.0307

Cite this

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

abstract = "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 1q, respectively.",

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author = "A. Kassymov and M. Ruzhansky and D. Suragan",

note = "Funding Information: The first and second authors were supported in parts by the FWO Odysseus 1 grant no. G.0H94.18N: Analysis and Partial Differential Equations, by the Methusalem programme of the Ghent University Special Research Fund (BOF) (grant no. 01M01021) and by the EPSRC (grant no. EP/R003025/2). Also, this research has been funded by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (grant no. AP19676031) and partially supported by the collaborative research programme {\textquoteleft}Qualitative analysis for nonlocal and fractional models{\textquoteright} from Nazarbayev University. Publisher Copyright: {\textcopyright} 2023 The Authors.",

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PY - 2023/1/25

Y1 - 2023/1/25

N2 - 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 1q, respectively.

AB - 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 1q, respectively.

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

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