Existence of principal values of some singular integrals on Cantor sets, and Hausdorff dimension
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866916085858566144 |
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| author | Cufí, J. Donaire, J. J. Mattila, P. Verdera, J. |
| author_facet | Cufí, J. Donaire, J. J. Mattila, P. Verdera, J. |
| contents | Consider a standard Cantor set in the plane of Hausdorff dimension 1. If the linear density of the associated measure $μ$ vanishes, then the set of points where the principal value of the Cauchy singular integral of $μ$ exists has Hausdorff dimension 1. The result is extended to Cantor sets in $\mathbb{R}^d$ of Hausdorff dimension $α$ and Riesz singular integrals of homogeneity $-α$, 0 < $α$ < d : the set of points where the principal value of the Riesz singular integral of $μ$ exists has Hausdorff dimension $α$. A martingale associated with the singular integral is introduced to support the proof. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_05015 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Existence of principal values of some singular integrals on Cantor sets, and Hausdorff dimension Cufí, J. Donaire, J. J. Mattila, P. Verdera, J. Classical Analysis and ODEs 42B20 (primary), 30E20 (secondary), 60F17 (secondary) Consider a standard Cantor set in the plane of Hausdorff dimension 1. If the linear density of the associated measure $μ$ vanishes, then the set of points where the principal value of the Cauchy singular integral of $μ$ exists has Hausdorff dimension 1. The result is extended to Cantor sets in $\mathbb{R}^d$ of Hausdorff dimension $α$ and Riesz singular integrals of homogeneity $-α$, 0 < $α$ < d : the set of points where the principal value of the Riesz singular integral of $μ$ exists has Hausdorff dimension $α$. A martingale associated with the singular integral is introduced to support the proof. |
| title | Existence of principal values of some singular integrals on Cantor sets, and Hausdorff dimension |
| topic | Classical Analysis and ODEs 42B20 (primary), 30E20 (secondary), 60F17 (secondary) |
| url | https://arxiv.org/abs/2306.05015 |