The Cauchy Integral, Analytic Capacity, and Uniform Rectifiability

Abstract

Several explanations concerning notation, terminology, and background are in order. First notation: by 7Hi we have denoted the one-dimensional Hausdorff measure (i.e. length), and A(z,r) stands for the closed disc with center z and radius r. A curve F is called AD-regular, that is, Ahlfors-David-regular, if it satisfies (1.2) (with E = F). Since the lower bound is automatic for curves, this means that 'Hl (r n A(z, r)) 0. General sets satisfying (1.2) are called AD-regular. It is simplest to define the L2-boundedness of the Cauchy singular integral operator via the truncated integrals: we say that CE is bounded in L2(E) (without really defining the operator CE itself) if there is M < ox such that