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

Abstract

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