Norm estimate of the Cauchy transform on \(L^p(\Omega)\) (Q814358)

Let \(\Omega \) be a bounded domain in the complex plane. The Cauchy integral operator is defined by \[ Cf(z)=-\frac{1}{\pi}\int_\Omega \frac{f(\xi)}{\xi -z } dx dy, \quad \xi=x+iy. \] Two-sided estimates of the norm of \(Cf\) are given in the case when \(\Omega \) is the unit disc or the bounded simply connected domain with piecewise \(C^1\) boundary. A consequence is a better constant in Poincaré inequalities.