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

TL;DR

The paper addresses when principal values of singular integrals exist on Cantor-type measures by linking PV existence to a density sequence $a_n$ associated with a nested cube construction. It introduces a martingale framework $S_n$ tied to truncated integrals and develops a stopping-time argument, complemented by Hungerford’s dimension lemma, to obtain lower bounds on the Hausdorff dimension of PV-existence sets. The main results show PV exists on a set of dimension at least $1$ in the plane when $a_n\to0$, and extend the theory to $\alpha$-Riesz transforms in $\mathbb{R}^d$ with dimension $\alpha$, with Appendix 1 giving a dimension lemma and Appendix 2 outlining higher-dimensional adaptations. The methods connect density decay, martingale convergence, and capacity arguments to characterize where principal values exist, contributing to the understanding of singular integrals on fractal supports and their dimensional properties.

Abstract

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 $\mathbb{R}^d$ 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.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Remark 2.1
• Theorem 2.2
• Lemma 2.3
• proof : Proof of Lemma \ref{['inc']}
• Lemma 2.4
• proof
• Lemma 3.1
• Lemma 3.2