TY - JOUR
T1 - Hardy inequalities on metric measure spaces, III
T2 - The case q ≤ p ≤ 0 and applications
AU - Kassymov, A.
AU - Ruzhansky, M.
AU - Suragan, D.
N1 - 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 ‘Qualitative analysis for nonlocal and fractional models’ from Nazarbayev University.
Publisher Copyright:
© 2023 The Authors.
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.
KW - metric measure space
KW - reverse Hardy inequality
KW - reverse Hardy-Littlewood-Sobolev inequality
KW - reverse Stein-Weiss inequality
UR - http://www.scopus.com/inward/record.url?scp=85146267628&partnerID=8YFLogxK
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U2 - 10.1098/rspa.2022.0307
DO - 10.1098/rspa.2022.0307
M3 - Article
AN - SCOPUS:85146267628
SN - 1364-5021
VL - 479
JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
IS - 2269
M1 - 20220307
ER -