Division of holomorphic functions and growth conditions

Abstract

Let D be a strictly convex domain of C n , f1 and f2 be two holomorphic functions defined on a neighborhood of D and set X l = {z, f l (z) = 0}, l = 1, 2. Suppose that X l ∩bD is transverse for l = 1 and l = 2, and that X1 ∩X2 is a complete intersection. We give necessary conditions when n ≥ 2 and sufficient conditions when n = 2 under which a function g to be written as g = g1f1 +g2f2 with g1 and g2 in L q (D), q ∈ [1, +∞), or g1 and g2 in BM O(D). In order to prove the sufficient condition, we explicitly write down the functions g1 and g2 using integral representation formulas and new residue currents.