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Hardy inequalities on metric measure spaces, IV: The case p=1

  • Michael Ruzhansky , Anjali Shriwastawa EMAIL logo and Bankteshwar Tiwari
From the journal Forum Mathematicum

Abstract

In this paper, we investigate the two-weight Hardy inequalities on metric measure space possessing polar decompositions for the case p = 1 and 1 q < . This result complements the Hardy inequalities obtained in [M. Ruzhansky and D. Verma, Hardy inequalities on metric measure spaces, Proc. Roy. Soc. A. 475 2019, 2223, Article ID 20180310] in the case 1 < p q < . The case p = 1 requires a different argument and does not follow as the limit of known inequalities for p > 1 . As a byproduct, we also obtain the best constant in the established inequality. We give examples obtaining new weighted Hardy inequalities on homogeneous Lie groups, on hyperbolic spaces and on Cartan–Hadamard manifolds for the case p = 1 and 1 q < .

MSC 2020: 26D10; 22E30; 45J05

Communicated by Jan Frahm


Funding statement: Michael Ruzhansky is supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations, the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021) and by EPSRC grant EP/R003025/2. Anjali Shriwastawa is supported by UGC Non-net fellowship from Banaras Hindu University, India.

Acknowledgements

This paper was completed when the second author was visiting Ghent University, Belgium. She is very grateful to Ghent Analysis & PDE centre, Ghent University, Belgium for the financial support and warm hospitality during her research visit.

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Received: 2023-09-04
Published Online: 2024-01-15

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