Growth estimates for cauchy integrals of measures and rectifiability

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Abstract

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.
Original languageEnglish
Pages (from-to)605-643
JournalGeometric and Functional Analysis
Volume17
DOIs
Publication statusPublished - 1 Jun 2007

Keywords

  • Cauchy transform
  • Curvature of measures
  • Maximal Cauchy transform
  • Principal values
  • Rectifiability

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