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Proceedings of the American Mathematical Society

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$ L\sp p$ norms of the Borel transform and the decomposition of measures

Author: B. Simon
Journal: Proc. Amer. Math. Soc. 123 (1995), 3749-3755
MSC: Primary 44A15; Secondary 28A10
MathSciNet review: 1277133
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Abstract: We relate the decomposition over [a, b] of a measure $ d\mu $ (on $ \mathbb{R}$) into absolutely continuous, pure point, and singular continuous pieces to the behavior of integrals $ \smallint\limits_a^b {{(\operatorname{Im} F(x + i\epsilon ))}^p}dx$ as $ \epsilon \downarrow 0$. Here F is the Borel transform of $ d\mu $, that is, $ F(z) = \smallint {(x - z)^{ - 1}}d\mu (x)$.

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