TY - JOUR
T1 - Growth estimates for cauchy integrals of measures and rectifiability
AU - Tolsa, Xavier
PY - 2007/6/1
Y1 - 2007/6/1
N2 - 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.
AB - 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.
KW - Cauchy transform
KW - Curvature of measures
KW - Maximal Cauchy transform
KW - Principal values
KW - Rectifiability
U2 - 10.1007/s00039-007-0598-7
DO - 10.1007/s00039-007-0598-7
M3 - Article
SN - 1016-443X
VL - 17
SP - 605
EP - 643
JO - Geometric and Functional Analysis
JF - Geometric and Functional Analysis
ER -