On a class of hypoelliptic differential operators with double characteristics
type: Item
instance of: scholarly article
title: On a class of hypoelliptic differential operators with double characteristics
zbMATH Open document ID: 0799.35047
published in: Journal of the Mathematical Society of Japan
publication date: 1994-11-14T00:00:00Z
review text: The author studies hypoellipticity for second-order degenerate elliptic and parabolic operators with real coefficients, giving a unified method with which he can generalize previous results of \textit{Wakabayashi} and \textit{Suzuki} (in print) and \textit{T. Hoshiro} [J. Math. Kyoto Univ. 28, No. 4, 615-632 (1988; Zbl 0692.35034) and ibid. 29, No. 3, 497-513 (1989; Zbl 0725.35036)]. Let \[ P(x,D)= 2 {{\partial^ 2} \over {\partial x_ 1}}+ \sum_{i,j=2}^ n a^{ij}(x) {\partial \over {\partial x_ j}}+ \sum_{i=2}^ n b^ i(x) {\partial \over {\partial x_ i}}+ c(x), \] where the \(a^{ij}\) are the \(C^ \infty\) components of a symmetric contravariant tensor on \(\mathbb{R}^ n\) and \[ \sum_{i,j=2}^ n a^{ij} (x)\xi_ i \xi_ i\geq 0 \quad \text{ on }\quad T^* (\mathbb{R}^ n) \] and \(b^ i,c\in C^ \infty (\mathbb{R}^ n)\). Denote \(\varphi:= (\partial/\partial x_ 1) \otimes (\partial/ \partial x_ 1)+ \sum_{i,j=2}^ n a^{ij} (\partial/ \partial x_ i) \otimes_ s (\partial/\partial x_ j)\), where \(\otimes_ s\) stands for the symmetric tensor product, let \(\Psi: \Gamma( \mathbb{R}^ n, T^* (\mathbb{R}^ n))\to \Gamma(\mathbb{R}^ n, T(\mathbb{R}^ n))\) be the map defined by \(\zeta\to \Phi(\zeta,\cdot)\), \(X_ 1:= \{\Psi (\zeta)\), \(\zeta\in\Gamma (\mathbb{R}^ n, T^* (\mathbb{R}^ n))\), and define the drift vector field \(X_ 0\) by \(X_ 0= \sum_{i=2}^ n b^ i(x) \partial/ \partial x_ i\). Suppose (H): The Lie algebra \({\mathcal L}(X)\) over \(\mathbb{R}\) generated by \(X=X_ 1\cup X_ 0\) has rank \(n\) outside a subset \(S\) of the hypersurface \(x_ 1=0\). The author gives sufficient conditions for hypoellipticity for \(P(x,D)\) under condition (H) (which is Hörmander's celebrated condition). For this, let \(\alpha(x,\xi')= \sum_{i,j=2}^ n a^{ij} (x_ 1, x')\xi_ i \xi_ j\), where \(x_ 1\) is considered as a parameter, and assume: \(\text{A}_ 1)\) \((\exists)\) \(a_ 0>0\) such that \(\sum_{i,j=2}^ n | \partial^ 2\alpha/ \partial x_ i \partial x_ j|^ 2 \leq a_ 0\) for \((x_ 1,x',\xi')\…
Mathematics Subject Classification ID: 35H10
zbMATH DE Number: 469069
zbMATH Keywords: degenerate parabolic operators
zbMATH Keywords: degenerate elliptic operators
zbMATH Keywords: conditions for hypoellipticity
author: Kazuaki Taira
reviewed by: Gheorghe Gussi
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04530391
OpenAlex ID: W2060343291
DOI: 10.2969/JMSJ/04530391
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description: scientific article; zbMATH DE number 469069