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Ruzhansky M, Verma D. Hardy inequalities on metric measure spaces. Proc Math Phys Eng Sci 2019; 475:20180310. [DOI: 10.1098/rspa.2018.0310] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/17/2018] [Accepted: 02/07/2019] [Indexed: 11/12/2022] Open
Abstract
In this note, we give several characterizations of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardy's original inequality. We give examples obtaining new weighted Hardy inequalities on
R
n
, on homogeneous groups, on hyperbolic spaces and on Cartan–Hadamard manifolds. We note that doubling conditions are not required for our analysis.
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Affiliation(s)
- Michael Ruzhansky
- Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK
- Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium
- School of Mathematical Sciences, Queen Mary University of London, London, UK
| | - Daulti Verma
- Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK
- Miranda House College, University of Delhi, Delhi 110007, India
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