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 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.