Division of holomorphic functions and growth conditions

TL;DR

This paper establishes necessary and sufficient conditions for representing functions as linear combinations of two holomorphic functions with growth constraints, using integral formulas and residue currents in convex domains.

Contribution

It provides explicit criteria and formulas for such representations in higher dimensions, extending classical results to complex convex domains.

Findings

Necessary conditions for representation when n>1.

Sufficient conditions for n=2 with explicit formulas.

Use of residue currents and integral representations.

Abstract

Let D be a strictly convex domain of C^n, f_1 and f_2 be two holomorphic functions defined on a neighborhood of closure of D and set X_l={z, f_l(z)=0}, l=1,2. Suppose that X_l\cap bD is transverse for l=1 and l=2, and that X_1\cap X_2 is a complete intersection. We give necessary conditions when n>1 and sufficient conditions when n=2 under which a function g to be written as g=g_1f_1+g_2f_2 with g_1 and g_2 in L^q(D), q\in [1,+\infty), or g_1 and g_2 in BMO(D). In order to prove the sufficient condition, we explicitly write down the functions g_1 and g_2 using integral representation formulas and new residue currents.