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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ L\sp p$ function decomposition on $ C\sp \infty$ totally real submanifolds of $ {\bf C}\sp n$


Author: William S. Calbeck
Journal: Proc. Amer. Math. Soc. 117 (1993), 187-194
MSC: Primary 32A40; Secondary 32D99
MathSciNet review: 1116253
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Abstract: For $ 1 < p < \infty $ we show that $ {L^p}$ functions defined on a $ {C^\infty }$ totally real submanifold of $ {\mathbb{C}^n}$ can be locally decomposed into the sum of boundary values of holomorphic functions in wedges such that the boundary values are in $ {L^p}$.

The general case of a $ {C^\infty }$ totally real submanifold is reduced to the flat case of $ {\mathbb{R}^n}$ in $ {\mathbb{C}^n}$ by an almost analytic change of variables. $ {L^p}$ results in the flat case are then obtained using Fourier multipliers. In transporting these results back to the manifold we lose analyticity, so it is necessary to solve a $ \overline \partial $ problem in an appropriate domain. This gives holomorphy in the wedges but produces a $ {C^\infty }$ error on the edge. This $ {C^\infty }$ function is then holomorphically decomposed using the FBI transform with a careful analysis to check that the functions are $ {C^\infty }$ up to the edge and do not destroy the $ {L^p}$ behavior.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1116253-7
PII: S 0002-9939(1993)1116253-7
Article copyright: © Copyright 1993 American Mathematical Society