Uniqueness theorems for Cauchy integrals
If µ is a finite complex measure in the complex plane C we denote by Cµ its Cauchy integral defined in the sense of principal value. The measure µ is called reflectionless if it is continuous (has no atoms) and Cµ = 0 at µ-almost every point. We show that if µ is reflectionless and its Cauchy maxima...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2008 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:28262 |
| Acceso en línea: | https://ddd.uab.cat/record/28262 https://dx.doi.org/urn:doi:10.5565/PUBLMAT_52208_03 |
| Access Level: | acceso abierto |
| Palabra clave: | Cauchy integral Reflectionless measure |
| Sumario: | If µ is a finite complex measure in the complex plane C we denote by Cµ its Cauchy integral defined in the sense of principal value. The measure µ is called reflectionless if it is continuous (has no atoms) and Cµ = 0 at µ-almost every point. We show that if µ is reflectionless and its Cauchy maximal function Cµ ∗ is summable with respect to then µ is trivial. An example of a reflectionless measure whose maximal function belongs to the "weak" L1 is also constructed, proving that the above result is sharp in its scale. We also give a partial geometric description of the set of reflectionless measures on the line and discuss connections of our results with the notion of sets of finite perimeter in the sense of De Giorgi. |
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