arXiv subjects
Publications and source records attributed to Gregorio Chinni.
The global analytic hypoellipticity is proved for a class of second order partial differential equations with non-negative characteristic form globally defined on the torus. The class considered in this work generalizes at some degree the class of sum of squares considered by Bove-Chinni and also by Cordaro-Himonas.
We study the microlocal regularity of the analytic/Gevrey vectors for the following class of second order partial differential equations \begin{align*} P(x,D) = \sum_{\ell,j=1}^{n} a_{\ell,j}(x) D_{\ell} D_{j} + \sum_{\ell=1}^{n} i b_{\ell}(x) D_{\ell} +c(x), \end{align*} where $a_{\ell,j}(x) = a_{j,\ell}(x)$, $b_{\ell}(x)$, $\ell,j \in \lbrace 1,\dots,\, n\rbrace$, are real valued real Gevrey functions of order $s$ and $c(x)$ is a Gevrey function of order $s$, $s \geq 1$, on $Ω$ open neighborhood of the origin in $\mathbb{R}^{n}$. Thus providing a microlocal version of a result due to M. Derridj in "Gevrey regularity of Gevrey vectors of second order partial differential operators with non negative characteristic form", Complex Anal. Synerg. $\mathbf{6}$, 10 (2020), https://doi.org/10.1007/s40627-020-00047-8.
The sharp Gevrey hypoellipticity is provided for the following generalization of the Métivier operator, "Non-hypoellipticité analytique pour $D_{x}^{2}+\left( x^{2} + y^{2}\right)D_{y}^{2}$" by G. Métivier, \begin{align*} D_{x}^{2}+\left(x^{2n+1}D_{y}\right)^{2}+\left(x^{n}y^{m}D_{y}\right)^{2}, \end{align*} in $Ω$ open neighborhood of the origin in $\mathbb{R}^{2}$, where $n$ and $m$ are positive integers.
We consider sums of squares operators globally defined on the torus. We show that if some assumptions are satisfied the operators are globally analytic hypoelliptic. The purpose of the assumptions is to rule out the existence of a Hamilton leaf on the characteristic variety lying along the fiber of the cotangent bundle, i.e. the case of the (global) Métivier operator.
The global and semi-global analytic hypoellipticity on the torus is proved for two classes of sums of squares operators, introduced in "Analytic Hypoellipticity for Sums of Squares and the Treves Conjecture" by P. Albano and A. Bove and M. Mughetti, and in "Analytic Hypoellipticity for Sums of Squares and the Treves Conjecture. II" by A. Bove and M. Mughetti, satisfying the Hörmander condition and which fail to be neither locally nor microlocally analytic hypoelliptic.
We prove via FBI-transform a result concerning the microlocal Gevrey regularity of analytic vectors for operators sums of squares of vector fields with real-valued real analytic coefficients of Hörmander type, thus providing a microlocal version, in the analytic category, of a result due to M. Derridj in "Local estimates for Hörmander's operators of first kind with analytic Gevrey coefficients and application to the regularity of their Gevrey vectors", concerning the problem of the local regularity for the Gevrey vectors for sums of squares of vector fields with real-valued real analytic/Gevrey coefficients. Nous démontrons , en utilisant la transformation de Fourier-Bros-Iagolnitzer, un résultat de régularité Gevrey microlocale , optimale, des vecteurs analytiques d'opérateurs de Hörmander de type "Sommes de carrés de champs de vecteurs" à coefficients analytiques sur un ouvert. Ce résultat est, dans le cadre analytique, la version microlocale du résultat de M.Derridj "Local estimates for Hörmander's operators of first kind with analytic Gevrey coefficients and application to the regularity of their Gevrey vectors", obtenu pour les vecteurs de Gevrey de tels opérateurs à coefficients Gevrey.
The micro-local Gevrey regularity of a class of "sums of squares" with real analytic coefficients is studied in detail. Some partial regularity result is also given.
We prove a couple of results concerning pseudodifferential perturbations of differential operators being sums of squares of vector fields and satisfying Hörmander's condition. The first is on the minimal Gevrey regularity: if a sum of squares with analytic coefficients is perturbed with a pseudodifferential operator of order strictly less than its subelliptic index it still has the Gevrey minimal regularity. We also prove a statement concerning real analytic hypoellipticity for the same type of pseudodifferential perturbations, provided the operator satisfies to some extra conditions (see Theorem 1.2 below) that ensure the analytic hypoellipticity.