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

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 Rd 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. © 2023 The Authors, under license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). Open Access made possible by subscribing institutions via Subscribe to Open.