Analytic regularity for an operator with Trèves curves
type: Item
instance of: scholarly article
title: Analytic regularity for an operator with Trèves curves
zbMATH Open document ID: 1053.35046
author: Nicholas Hanges
published in: Journal of Functional Analysis
publication date: 2004-08-16T00:00:00Z
review text: The author studies a partial differential operator \(\mathcal{H}\) with analytic coefficients in \(\mathbb{R}^{3},\) which is of the form \[ \mathcal{H}=-\frac{\partial^2}{\partial t^2}-t^2\left( \frac {\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2 }\right) -\left( x_1\frac{\partial}{\partial x_2}-x_2\frac{\partial }{\partial x_1}\right) ^2.\tag{1} \] A classical result of Hörmander guarantees the hypoellipticity of \(\mathcal{H}\) on any open set \(\Omega\subset\mathbb{R}^{3}.\) The main results of the paper deal with the analytic hypoellipticity of the operator \(\mathcal{H}\) . The essential theorems proved by the author are: Theorem 1. Let \(\Omega\subset\mathbb{R}^{3}\) be open. If \(\Omega\cap\{t=0\}\neq \emptyset,\) then \(\mathcal{H}\) is not analytic hypoelliptic on \(\Omega.\) Theorem 2. The operator \(\mathcal{H}\) is globally analytic hypoelliptic on \(\Omega_{p,q}\), for every \(p,q>0,\) where \[ \Omega_{p,q}=\{(x_1,x_2,t):x_1^2+x_2^2<p^2,t^2<q^2\}. \] Theorem 1 is consistent with the \textit{F. Trèves}' conjectures [Proc. Symp. Pure Math. 65, 201--219 (1999; Zbl 0938.35038)], while as a corollary of Theorem 2 the operator \(\mathcal{H}\) is analytic hypoelliptic at the origin, i. e., if \(u\) is a distribution near the origin and \(\mathcal{H}u\) is analytic near the origin, then \(u\) is analytic near the origin.
Mathematics Subject Classification ID: 35H10
Mathematics Subject Classification ID: 32T27
zbMATH DE Number: 2091721
zbMATH Keywords: analytic hypoellipticity
zbMATH Keywords: sum of squares
zbMATH Keywords: bicharacteristic
reviewed by: Chikh Bouzar
MaRDI profile type: Publication
full work available at URL: j.jfa.2003.12.006
OpenAlex ID: W2023599428
cites work: Nonanalytic-hypoellipticity for some degenerate elliptic operators
cites work: A property of the functions and distributions annihilated by a locally integrable system of complex vector fields
cites work: Remarks about global analytic hypoellipticity
cites work: Global analytic hypoellipticity of the \({\bar \partial}\)-Neumann problem on circular domains
cites work: Real analytic regularity of the Szegö projection on circular domains
cites work: The Szegö projection need not preserve global analyticity
cites work: Q3982258
cites work: Global analytic hypoellipticity in the presence of symmetry
cites work: Counterexamples to analytic hypoellipticity for domains of finite type
cites work: Global analytic hypoellipticity of a class of degenerate elliptic operators on the torus
cites work: Global analytic regularity for sums of squares of vector fields
cites work: Treves curves and the Szego kernel
cites work: Analytic singularities of the Bergman kernel for tubes
cites work: Q4448986
cites work: Hypoelliptic second order differential equations
cites work: Q3671563
cites work: Q3903365
cites work: Q3937740
cites work: Q4041983
cites work: Q3037865
cites work: Q3142876
cites work: The local real analyticity of solutions to d'Alembert-Operator(b) and the (partial d)--Neumann problem
cites work: Global (and local) analyticity for second order operators constructed from rigid vector fields on products of tori
cites work: Sur l'hypoellipticite analytique microlocale des operateurs de type principal
cites work: Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the -neumann problem
cites work: Q4249508
cites work: Q3844537
DOI: 10.1016/J.JFA.2003.12.006
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description: scientific article; zbMATH DE number 2091721