Estimates of the modulus of continuity of the logarithmic double layer potential in the closure of domain

TL;DR

This paper provides precise estimates for the modulus of continuity of the real part of the Cauchy-type integral in domains bounded by Ahlfors-regular curves, improving upon classical results.

Contribution

It introduces more accurate estimates for the modulus of continuity of the Cauchy-type integral's real part, with proofs of their sharpness through explicit examples.

Findings

New estimates are more exact than Zygmund's classical estimate.

Constructed examples demonstrate the sharpness of the estimates.

Results apply to domains bounded by Ahlfors-regular curves.

Abstract

We obtain estimates of the modulus of continuity for the real part of the Cauchy-type integral in the closure of domain bounded by an Ahlfors-regular integration curve. These estimates are more exact than the well-known Zygmund estimate for the modulus of continuity of the Cauchy-type integral. The accuracy of estimates is proved by constructing an example of a curve and an integral density for which the specified estimates are exact with respect to the order of smallness.