Extending Holomorphic Functions from Subvarieties

Abstract

Let D be a pseudoconvex domain in ℂ n and V a subvariety of D. The purpose of this paper is to survey the extension problem of holomorphic functions from V to D in various function spaces. First, we study extension theorems in bounded strictly pseudoconvex domains. Next, we discuss extension theorems in bounded weakly pseudoconvex domains with support function and analytic polyhedra. The main tools to construct extension functions are integral formulas in subvarieties obtained by Hatziafratis[27] and Berndtsson[15]. Further, we state the L 2 extension theorem of Ohsawa-Takegoshi[36] in bounded pseudoconvex domains. Finally, we give some counterexamples for extension problems.