## 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

### Keywords:

Szegö kernel; orthogonal projection; Lebesgue measure
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