On the $\overline\partial$-equation in a Banach space
摘要
We define a separable Banach space X and prove the existence of a <9-closed C^-smooth (0, l)-form / on the unit ball B of X, which is not c^-exact on any open subset. Further, we show that the sheaf cohomology groups H q {^l,0) = 0, q > 1, where 0 is the sheaf of germs of holomorphic functions on X, and f2 is any pseudoconvex domain in X, e.g., ^l = B. As the Dolbeault group H-1 (B) ^ 0, the Dolbeault isomorphism theorem does not generalize to arbitrary Banach spaces. Lastly, we construct a C^-smooth integrable almost complex structure on M = B X C such that no open subset of M is biholomorphic to an open subset of a Banach space. Hence the Newlander-Nirenberg theorem does not generalize to arbitrary Banach manifolds.
引用本文（GB/T 7714）
Imre Patyi. On the $\overline\partial$-equation in a Banach space[J]. Bulletin de la Société mathématique de France, 2000.
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