𝐿^{𝑝} norms of the Borel transform and the decomposition of measures

Simon, B.

doi:10.1090/S0002-9939-1995-1277133-1

$L^ p$ norms of the Borel transform and the decomposition of measures
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Proc. Amer. Math. Soc. 123 (1995), 3749-3755 Request permission

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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Operators with singular continuous spectrum

Hausdorff dimension, rank one perturbations, and localizations

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