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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



An elementary integral solution operator for the Cauchy-Riemann equations on pseudoconvex domains in $ {\bf C}\sp{n}$

Author: R. Michael Range
Journal: Trans. Amer. Math. Soc. 274 (1982), 809-816
MSC: Primary 32F20; Secondary 35N15
MathSciNet review: 675081
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Abstract: An integral representation formula for $ (0,q)$ forms is constructed on a strictly pseudoconvex domain $ D$ in $ \mathbf{C}^n$ by using only the local geometry of the boundary of $ D$. By combining this representation with elementary results about compact operators in Banach spaces, one obtains the solution of the Levi problem and, more importantly, an integral solution operator for $ \bar{\partial}$ on $ D$. The construction does not need any a priori knowledge of the solvability of $ \bar{\partial}$ and thus allows us to establish fundamental global results by a direct and elementary method.

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Keywords: Strictly pseudoconvex domains, integral representations, Levi problem, Hefer's lemma, integral solution operator for $ \bar{\partial}$
Article copyright: © Copyright 1982 American Mathematical Society