Extension of holomorphic functions defined on singular complex hypersurfaces with growth estimates in strictly pseudoconvex domains of $C^n$

TL;DR

This paper establishes conditions under which holomorphic functions defined on singular hypersurfaces within strictly pseudoconvex domains can be extended to the entire domain with controlled growth, using integral formulas and residue currents.

Contribution

It provides new sufficient conditions for extending holomorphic functions from singular hypersurfaces to the whole domain with growth estimates.

Findings

Extension criteria for holomorphic functions on singular hypersurfaces.

Use of integral representation formulas and residue currents.

Extension results in $L^q$ and BMO spaces.

Abstract

Let $D$ be a strictly pseudoconvex domain and $X$ be a singular analytic set of pure dimension $n-1$ in $C^n$ such that $X\cap D\neq \emptyset$ and $X\cap bD$ is transverse. We give sufficient conditions for a function holomorphic on $D\cap X$ to admit a holomorphic extension which belongs to $L^q(D),$ $q\in [1,+\infty[$, or to $BMO(D)$. The extension is given by mean of integral representation formulas and residue currents.