A division problem for $$\bar \partial _b - CLOSED$$ forms

Abstract

LetG 1,…,Gm be bounded holomorphic functions in a strictly pseudoconvex domainD such that\(\delta ^2 \leqslant \sum {\left| {G_j } \right|^2 \leqslant 1} \). We prove that for each\(\bar \partial _b - closed\) (0,q)-form ϕ inL p(∂D), 1<p<∞, there are\(\bar \partial _b - closed\) formsu 1, …,u m inL p(∂D) such that ΣG juj=ϕ. This generalizes previous results forq=0. The proof consists in delicate estimates of integral representation formulas of solutions and relies on a certainT1 theorem due to Christ and Journe. For (0,n−1)-forms there is a simpler proof that also gives the result forp=∞. Restricted to one variable this is precisely the corona theorem.