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A division problem for\(\bar \partial _b - CLOSED\) forms

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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 Journé. 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.

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The author was partially supported by the Swedish Natural Research Council.

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Andersson, M. A division problem for\(\bar \partial _b - CLOSED\) forms. J. Anal. Math. 68, 39–58 (1996). https://doi.org/10.1007/BF02790203

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  • DOI: https://doi.org/10.1007/BF02790203

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