Division of holomorphic functions and growth conditions

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 \overline 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\geq 2 and sufficient conditions when n=2 under which a function g can 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.