EXISTENCE OF PRINCIPAL VALUES OF SOME SINGULAR INTEGRALS ON CANTOR SETS, AND HAUSDORFF DIMENSION

Joan Verdera *, Julia Cufi , Juan Jesus Donaire , Pertti Mattila

*Autor corresponent d’aquest treball

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Resum

Consider a standard Cantor set in the plane of Hausdorff dimension 1. If the linear density of the associated measure µ vanishes, then the set of points where the principal value of the Cauchy singular integral of µ exists has Hausdorff dimension 1. The result is extended to Cantor sets in R d of Hausdorff dimension α and Riesz singular integrals of homogeneity −α, 0 < α < d : the set of points where the principal value of the Riesz singular integral of µ exists has Hausdorff dimension α. A martingale associated with the singular integral is introduced to support the proof.

Idioma originalEnglish
Nombre de pàgines14
RevistaPacific Journal of Mathematics
Estat de la publicacióAcceptat en premsa - 1 de des. 2023

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