Hypoellipticity for a class of degenerate elliptic operators of second order
Michiharu Suzuki · Tsukuba Journal of Mathematics · 1992
\S 1. Introduction and results.Fedii [1] studied hypoellipticity for operators of the form $L=D_{1}^{2}+\phi(x_{1})^{2}D_{2}^{2}$ in $R^{2}$ , and proved thathas a zero of infinite order.Compared with higher dimensional cases, the problem in $R^{2}$ becomes much simpler.So one can expect that one investigates hypoellipticity for more general operators in $R^{2}$ .In this paper we shall give sufficient conditions of hypoellipticity for operators of the form $P(x, D)=D_{1}^{2}+$ $\alpha(x)D_{2}^{2}+\beta(x, D)$ in $R^{2}$ , where $x=(x_{1}, x_{2})\in R^{2},$ $\alpha(x)\in C^{\infty}(R^{2})$ is non-negative and $\beta(x, D)$ is a properly supported classical pseudodifferential operator of order 1.In doing so, we need general criteria for hypoellipticity, which are improvements of ones obtained by Morimoto [5] (see Theorem 1.1 below).Let us define the usual symbol classes $S_{1.\dot{0}}^{m1oc}$and $S_{I}^{m_{0}}$ .We say that a symbol $p(x, \xi)$ belongs to $S_{1,\dot{0}\prime}^{moc}$ (resp.$S_{1}^{m_{0}}$ ) if $p(x, \xi)\in C^{\infty}(T^{*}R^{n})$ and if for any compact subset $K$ of $R^{n}$ and for any multi-indices $\alpha$ and $\beta$ (resp.for any multi-indices $\alpha$ and $\beta$ there is $C_{\alpha,\beta}\equiv C_{K.\alpha.\beta}>0$(resp.$C_{\alpha,\beta}>0$ ) such that $|p_{(\beta)}^{(\alpha)}(x, \xi)|\leqq C_{\alpha.\beta}\langle\xi\rangle^{m-|\alpha|}$for $x\in K$ and $\xi\in R^{n}$ (resp.for $(x,$ $\xi)\in T^{*}R^{n}$ ), where)^{1/2}$ and $T^{*}R^{n}$ is identified with $R^{n}\times R^{n}$ .We denote by $L_{1,0}^{m}$ the set of the pseudodifferential operators whose symbols belong to $S_{1\dot{0}}^{m,1oc}$ .Let $P(x, D)\in L_{10}^{m,}$ be a properly supported pseudodifferential operator, and let $z^{0}=(x^{0}, \xi^{0})\in T^{*}R^{n}\backslash 0(\cong R^{n}\times(R^{n}\backslash \{0\}))$ .It is said that $P(x, D)$ is microhypoelliptic at $z^{0}$ if there is a conic neighbourhood $\mathcal{V}$ of $z^{0}$ in $T^{*}R^{n}\backslash 0$ such that $WF(u)\cap \mathcal{V}=WF(Pu)\cap \mathcal{V}$ if $u\in \mathcal{D}^{\prime},$ $(R^{n})$ .We also say that $P(x, D)$ is microhypoelliptic in a conic in a conic set $\mathcal{V}(\subset T^{*}R^{n}\backslash 0)$ (resp. in $\Omega(\subset R^{n})$ if $P(x, D)$ is microhypoelliptic at each $(x, \xi)\in q\mu$ (resp.at