A distortion theorem for quadratic domains for harmonic functions (Q1922988)

The purpose of the paper is to prove a theorem stated in another paper of the author jointly with \textit{H. Sakai} and \textit{H. S. Shapiro}. In the present paper the author proves that any finite connected domain in the plane can be distorted so that it becomes ``graviequivalent'' to a signed measure \(\mu\) with arbitrary small support. The original domain \(D\) is assumed to have an analytic boundary. If \(\Omega\) is the distorted domain we have \[ \int_\Omega h dx dy=\int h d\mu \] for every integrable harmonic \(h\).