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Counterexamples to analytic hypoellipticity for domains of finite type. (English) Zbl 0758.35024

In this paper the authors prove that if you consider in \(\mathbb{C}^ 2\) a hypersurface \({\mathcal S}\) determined by \(\text{Im} z_ 2=P(z_ 1)=|\text{Re} z_ 1|^ m\), where \(m\) is an even integer \(\geq 4\), then the distribution \(K(z,t)\) on \(\mathbb{C}\times\mathbb{R}\), \({\mathcal S}\) associated to the Szegö kernel \(S((z,t);(w,s))\) by \(K(z,t)=S((z,t);(0,0))\) is not real analytic away from 0.
Here we recall that the Szegö kernel is the distribution kernel associated to the operator defined by the orthogonal projection of \(L^ 2(\mathbb{C}\times\mathbb{R})\), with respect to the Lebesgue measure, onto the Kernel of the operator: \(\overline L=\partial/\partial\overline z- i(\partial P/\partial\overline z)\partial/\partial t\).
Other counterexamples are given concerning the hypoanalyticity of the \(\square_ b\) operator on hypersurfaces of the type above with different subharmonic, non harmonic polynomials.
Reviewer: B.Helffer (Paris)

MSC:

35H10 Hypoelliptic equations
32T99 Pseudoconvex domains
30C40 Kernel functions in one complex variable and applications
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