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2008 Uniqueness Theorems for Cauchy Integrals
Mark Melnikov, Alexei Poltoratski, Alexander Volberg
Publ. Mat. 52(2): 289-314 (2008).


If $\mu$ is a finite complex measure in the complex plane $\mathbb{C}$ we denote by $C^\mu$ its Cauchy integral defined in the sense of principal value. The measure $\mu$ is called reflectionless if it is continuous (has no atoms) and $C^\mu=0$ at $\mu$-almost every point. We show that if $\mu$ is reflectionless and its Cauchy maximal function $C^\mu_*$ is summable with respect to $|\mu|$ then $\mu$ is trivial. An example of a reflectionless measure whose maximal function belongs to the ``weak" $L^1$ 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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Mark Melnikov. Alexei Poltoratski. Alexander Volberg. "Uniqueness Theorems for Cauchy Integrals." Publ. Mat. 52 (2) 289 - 314, 2008.


Published: 2008
First available in Project Euclid: 5 August 2008

zbMATH: 1172.30013
MathSciNet: MR2436727

Primary: 30E20
Secondary: 31A15, 42B20

Rights: Copyright © 2008 Universitat Autònoma de Barcelona, Departament de Matemàtiques


Vol.52 • No. 2 • 2008
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