Global Analytic Hypoellipticity for a Class of Quasilinear Sums of Squares of Vector Fields

Analytic Hypoellipticity at Non-Symplectic Poisson-Treves Strata for Sums of Squares of Vector Fields

We consider an operator $ P $ which is a sum of squares of vector fields with analytic coefficients. The operator has a non-symplectic characteristic manifold, but the rank of the symplectic form $ σ$ is not constant on $ \Char P $. Moreover the Hamilton foliation of the non symplectic stratum of the Poisson-Treves stratification for $ P $ consists of closed curves in a ring-shaped open set around the origin. We prove that then $ P $ is analytic hypoelliptic on that open set. And we note explicitly that the local Gevrey hypoellipticity for $ P $ is $ G^{k+1} $ and that this is sharp.

Analyticity for Singular Sums of Squares of Degenerate Vector Fields

In this paper we prove local analytic hypoellipticity for a degenerate sum of squares of complex vector fields generalizing those of Kohn in "Hypoellipticity and Loss of Derivatives". Kohn's article is to appear in the Annals of Mathematics with an appendix by Derridj and Tartakoff proving local analyticity in that case. Here we consider vector fields corresponding to a CR manifold of finite type so there is a kind of double degeneracy, and prove local analyticity using our balanced localization of high derivatives in the "missing" direction T. In a forthcoming paper, Bove, Derridj, Kohn and T…

紧流形上平方和算子的全局亚椭圆性

Global hypoellipticity of sums of squares on compact manifolds

本文研究了紧Riemann流形乘积$T \ imes G$上平方和型算子的全局亚椭圆性，提出了弱于Hörmander条件且推广环面Diophantine条件的充要条件，并给出了满足该条件的算子实例。

Analytic Hypoellipticity for Sums of Squares and the Treves Conjecture

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 \ref{th: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 counterex…

Singular Sums of Squares of Degenerate Vector Fields

We simplify and give an alternative proof of hypoellipticity for generalizations of the singular sum of squares of complex vector fields studied by Kohn, with an appendix by Derridj and Tartakoff, in the Annals of Mathematics, vol. 162 no. 2, 2005, pp. 943-986. The main generalization is that the complex vector fields now come from domains of finite type. We also prove real analytic hypoellipticity and the optimality of our estimates.

Some remarks on polar sets to sums of squares

We argue that a necessary condition for hypoellipticity is that the polar is not a spiral domain.

Regularity of solutions to quasilinear infinitely degenerate second order equations

The main result of the paper is on the continuity of weak solutions of infinitely degenerate quasilinear second order equations. Namely, we show that every weak solution to a certain class of degenerate quasilinear equations is continuous. More precisely, we show that it is Hölder continuous with respect to a certain metric associated to the operator. One of the essential features of this metric is that the metric balls are non doubling with respect to Lebesgue measure. The proof of the continuity together with a recent result by Rios et al. completes the result on hypoellipticity of a class o…

Local Real Analyticity of Solutions for sums of squares of non-linear vector fields

We show that all smooth solutions of model non-linear sums of squares of vector fields are locally real analytic. A global result for more general operators is presented in a paper by Makhlouf Derridj and the first author under the title "Global Analytic Hypoellipticity for a Class of Quasilinear Sums of Squares of Vector Fields".

A Class of Sums of Squares with a Given Poisson-Treves Stratification

We study a class of sum of squares exhibiting the same Poisson-Treves stratification as the Oleinik-Radkevič operator. We find three types of operators having distinct microlocal structures. For one of these we prove a Gevrey hypoellipticity theorem analogous to our recent result for the corresponding Oleinik-Radkevič operator.

紧李群上一类全局解析亚椭圆算子

A class of globally analytic hypoelliptic operators on compact Lie groups

本文研究紧李群上一类微分算子的全局解析亚椭圆性，在Hörmander有限型条件、闭子群不变性及横截方向椭圆性三个条件下证明了该性质，并指出这为研究主纤维丛上平方和的 regularity 奠定了基础。

On local Gevrey regularity for Gevrey vectors of subelliptic sums of squares -- an elementary proof of a sharp Gevrey Kotake-Narasimhan theorem

We study the regularity of Gevrey vectors for Hörmander operators $$ P = \sum_{j=1}^m X_j^2 + X_0 + c$$ where the $X_j$ are real vector fields and $c(x)$ is a smooth function, all in Gevrey class $G^{s}.$ The principal hypothesis is that $P$ satisfies the subelliptic estimate: for some $\varepsilon >0, \; \exists \,C$ such that $$\|v\|_\varepsilon^2 \leq C\left(|(Pv, v)| + \|v\|_0^2\right) \qquad \forall v\in C_0^\infty.$$ We prove directly (without the now familiar use of adding a variable $t$ and proving suitable hypoellipticity for $Q=-D_t^2-P$ and then, using the hypothesis on the iterates…

Wave equations on graded groups and hypoelliptic Gevrey spaces

The overall goal of this dissertation is to investigate certain classical results from harmonic analysis, replacing the Euclidean setting, the abelian structure and the elliptic Laplace operator with a non-commutative environment and hypoelliptic operators. More specifically, we consider wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent non-negative propagation speeds that are Hölder-regular or even more so. The corresponding Euclidean problem has been extensively studied in the `80s and some additional results have been recently obta…