A priori estimates for conformal mappings on complex plane with parallel slits

Abstract

We study the properties of a conformal mapping $z(k)$ from the plane without vertical slits $\G_n=[u_n-ih_n, u_n+ih_n], n\in\Z$ and $h=(h_n)_{n\in\Z}\in \ell^2$, onto the complex plane without horizontal slits $\g_n\ss\R, n\in\Z$, with the asymptotics $z(iv)=iv+ o(1), v\to\iy$. Here $u_{n+1}-u_n\ge 1, n\in \Z$. Introduce the sequences $l=(|\g_n|)_{n\in\Z}$. % where $J_n\ge 0,J_n^2=\int_{\G_n}|\Im z(k,h)||dk|/\pi$. We obtain a priori two-sided estimates for $\|h\|_{p,\o}, \|l\|_{p,\o}$, where %$\|h\|_{\o}^p$ is the norm of the Banach space %the extension of i)-ii) for the case $h\in\ell_{\o}^p$, where %$\ell_{\o}^p,1\le p\le 2$ with % the norm $\|h\|_{p,\o}^p=\sum \o_n|h_n|^p, 1\le p\le 2$ with any weight $\o_n\ge 1, n\in \Z$. Moreover, we determine other estimates.