Existence and weak-type inequalities for Cauchy integrals of general measures on rectifiable curves and sets

Abstract:

If $\mu$ is a finite complex Borel measure and $\Gamma$ a Lipschitz graph in the complex plane, then for $\lambda > 0$ \[ \left | {\left \{ {z \in \Gamma :\sup \limits _{\varepsilon > 0} \left | {\int _{|\zeta - z| \geqslant \varepsilon } {{{(\zeta - z)}^{ - 1}}} d\mu \zeta } \right | > \lambda } \right \}} \right | \leqslant c(\Gamma ){\lambda ^{ - 1}}||\mu |{|_1}.\] It follows that for any finite Borel measure $\mu$ and any rectifiable curve $\Gamma$ the finite principal value \[ \lim \limits _{\varepsilon \downarrow 0} \int _{|\zeta - z| \geqslant \varepsilon } {{{(\zeta - z)}^{ - 1}}d\mu \zeta } \] exists for almost all (with respect to length) $z \in \Gamma$.