Existence of principal values of some singular integrals on Cantor sets, and Hausdorff dimension
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.
math.CA↗