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arXiv · 2306.05015

Existence of principal values of some singular integrals on Cantor sets, and Hausdorff dimension

J. Cufí·J. J. Donaire·P. Mattila·J. Verdera
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Abstract

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

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J. Cufí, J. J. Donaire, P. Mattila, J. Verdera. 2023-06-08. Existence of principal values of some singular integrals on Cantor sets, and Hausdorff dimension. https://doi.org/10.2140/pjm.2023.326.285

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