Existence of principal values of some singular integrals on Cantor sets, and Hausdorff dimension
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 R d of Ha...
| Autores: | , , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:287011 |
| Acceso en línea: | https://ddd.uab.cat/record/287011 https://dx.doi.org/urn:doi:10.2140/pjm.2023.326.285 |
| Access Level: | acceso abierto |
| Palabra clave: | Cauchy singular integral Riesz singular integral Cantor set Hausdorff dimension Martingale |
| Sumario: | 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 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. |
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