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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

32V40 Real submanifolds in complex manifolds
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