On nonlinear Rudin–Carleson type theorems
Volume 266 / 2022
Abstract
We study nonlinear interpolation problems for interpolation and peak-interpolation sets of function algebras. The subject goes back to the classical Rudin–Carleson interpolation theorem. In particular, we prove the following nonlinear version of that theorem: Let $\bar {\mathbb D}\subset \mathbb C$ be the closed unit disk, $\mathbb T\subset \bar {\mathbb D}$ the unit circle, $S\subset \mathbb T$ a closed subset of Lebesgue measure zero and $M$ a connected complex manifold. Then for every continuous $M$-valued map $f$ on $S$ there exists a continuous $M$-valued map $g$ on $\bar {\mathbb D}$ holomorphic on its interior such that $g|_S=f$. We also consider similar interpolation problems for continuous maps $f: S\rightarrow \bar M$, where $\bar M$ is a complex manifold with boundary $\partial M$ and interior $M$. Assuming that $f(S)\cap \partial M\ne \emptyset $, we are looking for holomorphic extensions $g$ of $f$ such that $g(\bar {\mathbb D}\setminus S)\subset M$.
