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Uniqueness theorems for Cauchy integrals
Melnikov, Mark (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Poltoratski, Alexei (Texas A&M University. Department of Mathematics)
Volberg, Alexander (Michigan State University. Department of Mathematics)

Data: 2008
Resum: 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.
Drets: Tots els drets reservats.
Llengua: Anglès.
Document: article ; recerca ; publishedVersion
Matèria: Cauchy integral ; Reflectionless measure
Publicat a: Publicacions Matemàtiques, V. 52 n. 2 (2008) p. 289-314, ISSN 0214-1493

DOI: 10.5565/PUBLMAT_52208_03

26 p, 251.5 KB

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