Estimates of the Modulus of Continuity of the Logarithmic Double Layer Potential

For the real part of the Cauchy-type integral, which is known to be the logarithmic double layer potential, we obtain estimates of the modulus of continuity 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 for the logarithmic double layer potential 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.