Existence and Weak-Type Inequalities for Cauchy Integrals of General Measures on Rectifiable Curves and Sets

Abstract

Abstract.If ft is a finite complex Borel measure and T a Lipschitz graph in the complex plane, then for X > 0 jzer:sup / (C-z)-'^C >4 <c(T)X-l\\fi\U.( e>0 V|C-z|>e J It follows that for any finite Borel measure /i and any rectifiable curve T the finite principal valueexists for almost all (with respect to length) zeT. IntroductionFor any finite complex Borel measure p on the complex plane C the Cauchy transform p(z) = J(z-zyxdpt; exists for almost all z £ C with respect to area. This is a rather immediate consequence of the fact that the kernel z_1 is locally integrable with respect to the Lebesgue measure. In this paper we prove that much more is true provided we interpret the Cauchy integral as a principal value. Namely, for any rectifiable curve r, the limit Cfi(z) = lim/ (Z-zyldp£ no J\r-Z\>E exists and is finite for %?x almost all z £ Y. Here %fx is the one-dimensional Hausdorff (i.e., length) measure on C. This result follows from the following weak-type inequality, which we prove in §2. Define, for e > 0, z e C, c,,E(z)= I (t-zy'dpt, Jc\B(z,e) C*(z) = sup \Cn,s(z)\, £>0

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