Christ, Michael; Geller, Daryl Counterexamples to analytic hypoellipticity for domains of finite type. (English) Zbl 0758.35024 Ann. Math. (2) 135, No. 3, 551-566 (1992). 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) Cited in 3 ReviewsCited in 19 Documents MSC: 35H10 Hypoelliptic equations 32T99 Pseudoconvex domains 30C40 Kernel functions in one complex variable and applications Keywords:Szegö kernel; orthogonal projection; Lebesgue measure PDFBibTeX XMLCite \textit{M. Christ} and \textit{D. Geller}, Ann. Math. (2) 135, No. 3, 551--566 (1992; Zbl 0758.35024) Full Text: DOI