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Transactions of the American Mathematical Society

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

 

 

Kernels for the tangential Cauchy-Riemann equations


Author: Al Boggess
Journal: Trans. Amer. Math. Soc. 262 (1980), 1-49
MSC: Primary 32F20; Secondary 35N15
DOI: https://doi.org/10.1090/S0002-9947-1980-0583846-0
MathSciNet review: 583846
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Abstract: On certain codimension one and codimension two submanifolds in $ {{\textbf{C}}^n}$, we can solve the tangential Cauchy-Riemann equations $ {\bar \partial _b}u\, = \,f$ with an explicit integral formula for the solution.

Let $ M\, = \,\partial D$, where D is a strictly pseudoconvex domain in $ {{\textbf{C}}^n}$. Let $ \omega \, \subset \, \subset \,M$ be defined by $ \omega \, = \,\{ z\, \in \,M;\,\operatorname{Re} \,h(z)\, < \,0\} $, where h is holomorphic near D. Points on the boundary of $ \omega $, $ \partial \omega $, where the tangent space of $ \partial \omega $ becomes complex linear, are called characteristic points.

Theorem 1. Suppose $ \partial \omega $ is admissible (in particular if $ \partial \omega $ has two characteristic points). Suppose $ f\, \in \,{\mathcal{E}}_M^{p,q}(\bar \omega )$, $ 1\, \leqslant \,q\, \leqslant \,n\, - \,3$, is smooth on $ \omega $ and satisfies $ {\bar \partial _M}f\, = \,0$ on $ \omega $; then there exists $ u\, \in \,{\mathcal{E}}_M^{p,q - 1}(\omega )$ which is smooth on $ \omega $ except possibly at the characteristic points on $ \partial \omega $ and which solves the equation $ {\bar \partial _M}u\, = \,f$ on $ \omega $.

Theorem 2. Suppose $ f\, \in \,{\mathcal{E}}_M^{p,q}(\omega )$, $ 2\, \leqslant \,q\, \leqslant \,n\, - \,3$, is smooth on $ \omega $; vanishes near each characteristic point; and $ {\bar \partial _M}f\, = \,0$ on $ \omega $. Then there exists $ u\, \in \,{\mathcal{E}}_M^{p,q - 1}(\omega )$ satisfying $ {\bar \partial _M}u\, = \,f$ on $ \omega $.

Theorem 3. Suppose $ f\, \in \,{\mathcal{D}}_M^{p,q}(\omega )$, $ 2\, \leqslant \,q\, \leqslant \,n - \,3$, is smooth with compact support in $ \omega $, and $ {\bar \partial _M}f\, = \,0$. Then there exists $ u\, \in \,{\mathcal{D}}_M^{p,q - 1}(\omega )$ with compact support in $ \omega $ and which solves $ {\bar \partial _M}u\, = \,f$.

In all three theorems we have an explicit integral formula for the solution.

Now suppose $ S\, = \,\partial \omega $. Let $ {C_s}$ be the set of characteristic points on S. We construct an explicit operator $ E:\,{\mathcal{D}}_S^{p,q}(S\, - \,{C_S})\, \to \,{\mathcal{E}}_S^{p,q - 1}(S\, - \,{C_S})$ with the following properties.

Theorem 4. The operator E maps $ L_{p,\operatorname{comp} }^{\ast}(S\, - \,{C_S})\, \to \,L_{p,\operatorname{loc} }^{\ast}(S\, - {C_S})$ and if $ f\, \in \,{\mathcal{D}}_S^{p,q}(S\, - \,{C_S})$, $ 1\, \leqslant \,q\, \leqslant \,n\, - \,3$, then $ f\, = \,{\bar \partial _S}\{ E(f)\} \, + \,E({\bar \partial _S}f)$.


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DOI: https://doi.org/10.1090/S0002-9947-1980-0583846-0
Article copyright: © Copyright 1980 American Mathematical Society