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Analytic hypoellipticity for sums of squares of vector fields. (English) Zbl 0933.35047

This paper is a synthetic, lucid and up to date review of the status of the theory concerning analytic hypoellipticity of sums of squares operators. First the author discusses the local theory; there are two famous conjectures by François Treves, that spurred quite a number of papers on the subject. A panorama of the most recent papers is given and model equations are proposed to elucidate them. The second part of the paper deals with the global analytic hypoellipticity problem. In this case it is shown that conditions involving the fields brackets are neither necessary nor sufficient and that a deeper spectral (dynamical) behaviour plays a role.
Reviewer: A.Bove (Bologna)

MSC:

35H10 Hypoelliptic equations
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