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

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