Growth estimates for cauchy integrals of measures and rectifiability

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Resum

We show that if μ is a finite Borel measure on the complex plane such that C* μ (z) = sup ε > 0 |C ε μ (z)| = sup ε > 0 | ∫ |ξ- z| > ε 1/ξ - z d/μ (ξ)| < ∞ for μ-a.e. z ∈ ℂ, then μ must be the addition of some point masses, plus some measure absolutely continuous with respect to arc length on countably many rectifiable curves, plus another measure with zero linear density. We also prove that the same conclusion holds if instead of the condition C * μ (z) < ∞ μ-a.e. one assumes C ∈μ(z) = o(μ(B(z,ε))/ε) as ε →, 0+ μ-a.e. © Birkhäuser Verlag, Basel 2007.
Idioma originalAnglès
Pàgines (de-a)605-643
RevistaGeometric and Functional Analysis
Volum17
DOIs
Estat de la publicacióPublicada - 1 de juny 2007

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