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

Julià Cufí, Juan Jesús Donaire, Pertti Mattila and Joan Verdera

Vol. 326 (2023), No. 2, 285–300

DOI: 10.2140/pjm.2023.326.285

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 ${ℝ}^d$ 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.

Keywords

Cauchy singular integral, Riesz singular integral, Cantor set, Hausdorff dimension, martingale

Mathematical Subject Classification

Primary: 42B20

Secondary: 30E20

Milestones

Received: 8 June 2023

Revised: 26 September 2023

Accepted: 16 October 2023

Published: 9 January 2024