Two-weight dyadic Hardy's inequalities
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866911794782535680 |
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| author | Arcozzi, Nicola Chalmoukis, Nikolaos Levi, Matteo Mozolyako, Pavel |
| author_facet | Arcozzi, Nicola Chalmoukis, Nikolaos Levi, Matteo Mozolyako, Pavel |
| contents | We present various results concerning the two-weight Hardy's inequality on infinite trees. Our main scope is to survey known characterizations (and proofs) for trace measures, as well as to provide some new ones. Also for some of the known characterizations we provide here new proofs. In particular, we obtain a new characterization based on a new reverse Hölder inequality for trace measures, and one based on the well known Muckenhoupt-Wheeden-Wolff inequality, of which we here give a new probabilistic proof. We provide a new direct proof for the so called isocapacitary characterization and a new simple proof, based on a monotonicity argument, for the so called mass-energy characterization. Furthermore, we introduce a conformally invariant version of the two-weight Hardy's inequality, we characterize the compactness of the Hardy operator, we provide a list of open problems and suggest some possible lines of future research. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_05450 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Two-weight dyadic Hardy's inequalities Arcozzi, Nicola Chalmoukis, Nikolaos Levi, Matteo Mozolyako, Pavel Classical Analysis and ODEs Complex Variables 05C05, 05C63, 31E05, 42B25, 42B35, 31A15, 30C85 We present various results concerning the two-weight Hardy's inequality on infinite trees. Our main scope is to survey known characterizations (and proofs) for trace measures, as well as to provide some new ones. Also for some of the known characterizations we provide here new proofs. In particular, we obtain a new characterization based on a new reverse Hölder inequality for trace measures, and one based on the well known Muckenhoupt-Wheeden-Wolff inequality, of which we here give a new probabilistic proof. We provide a new direct proof for the so called isocapacitary characterization and a new simple proof, based on a monotonicity argument, for the so called mass-energy characterization. Furthermore, we introduce a conformally invariant version of the two-weight Hardy's inequality, we characterize the compactness of the Hardy operator, we provide a list of open problems and suggest some possible lines of future research. |
| title | Two-weight dyadic Hardy's inequalities |
| topic | Classical Analysis and ODEs Complex Variables 05C05, 05C63, 31E05, 42B25, 42B35, 31A15, 30C85 |
| url | https://arxiv.org/abs/2110.05450 |