We are concerned with the problem of real analytic regularity of the solutions of sums of squares with real analytic coefficients. Treves conjecture states that an operator of this type is analytic hypoelliptic if and only if all the strata in the Poisson-Treves stratification are symplectic.We produce a model operator, P 1 , having a single symplectic stratum and prove that it is Gevrey s 0 hypoelliptic and not better. See Theorem 2.1 for a definition of s 0 . We also show that this phenomenon has a microlocal character.We point out explicitly that this is a counterexample to the sufficient part of Treves conjecture and not to the necessary part, which is still an open problem.
We are concerned with the problem of real analytic regularity of the solutions of sums of squares with real analytic coefficients. The Treves conjecture defines a stratification and states that an operator of this type is analytic hypoelliptic if and only if all the strata in the stratification are symplectic manifolds.Albano, Bove, and Mughetti (2016) produced an example where the operator has a single symplectic stratum, according to the conjecture, but is not analytic hypoelliptic.If the characteristic manifold has codimension 2 and if it consists of a single symplectic stratum, defined again according to the conjecture, it has been shown that the operator is analytic hypoelliptic.We show here that the above assertion is true only if the stratum is single, by producing an example with two symplectic strata which is not analytic hypoelliptic.
We prove Fefferman's SAK Principle for a class of hypoelliptic operators on R 2 whose nonnegative symbol vanishes anisotropically on the characteristic manifold.
In this paper we consider a model sum of squares of complex vector fields in the plane, close to Kohn's operator but with a point singularity,The characteristic variety of P is the symplectic real analytic manifold x D D 0. We show that this operator is C 1 -hypoelliptic and Gevrey hypoelliptic in G s , the Gevrey space of index s, provided k < lq, for every s lq=.lq k/ D 1 C k=.lq k/. We show that in the Gevrey spaces below this index, the operator is not hypoelliptic. Moreover, if k lq, the operator is not even hypoelliptic in C 1 . This fact leads to a general negative statement on the hypoellipticity properties of sums of squares of complex vector fields, even when the complex Hörmander condition is satisfied.was introduced and shown to be hypoelliptic, yet to lose 2 C .k 1/=m derivatives in L 2 Sobolev norms. Christ [2005] showed that the addition of one more variable destroys hypoellipticity altogether. In those seminal works, m D 1, but Kohn, A. Bove, M. Derridj, and D. S. Tartakoff generalized the results to higher m in [Bove et al. 2006] and elsewhere. MSC2010: primary 35H10, 35H20; secondary 35B65. See inside back cover or msp.org/apde for submission instructions.
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