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On the local solution of the tangential Cauchy-Riemann equations. (English) Zbl 0679.32019
G. M. Khenkin has constructed explicit operators P and Q for the tangential Cauchy-Riemann complex on a suitably small domain D in a strictly pseudoconvex real hypersurface M in complex n-space. Under appropriate restrictions on D, Khenkin obtained the representation formula $\phi ={\bar \partial}_ bP\phi +Q {\bar \partial}_ b\phi$ for (0,s)-forms $$\phi$$, $$1\leq s\leq n-3$$, restricted to D. For $$U\subset \subset D$$, the author derives a $$C^ k$$-norm estimate of the form $\| P\phi \|_{U,k}\leq K\| \phi \|_{D,k}.$ The same estimate holds for Q. Unfortunately, the constant K blows up as U increases to D. For the case $$k=0$$, Khenkin has obtained such an estimate, where K remains bounded as U increases to D. Whether this is possible for $$K>0$$ is open. The author applies his estimates elsewhere (see the following review) to the local CR embedding problem.
Reviewer: P.M.Gauthier

##### MSC:
 32V40 Real submanifolds in complex manifolds
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##### References:
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