On~a~class of hypoelliptic operators

V. V. Grushin

Sbornik. Mathematics, Tome 12 (1970) no. 3, pp. 458-476

Résumé

Let the variables in $R^{k+n}$ be broken up into two groups $x=(x',y)$, where $x'\in R^k$ and $y\in R^n$. We consider differential operators $p(x,D)$ with polynomial symbols of the form $$ p(x,D)=\sum_{|\alpha|+|\beta|\leqslant m,\,|\gamma|\leqslant m\delta}a_{\alphay\beta\gamma}y^\gamma D_{x'}^\beta D_y^\alpha,\qquad(\xi,\eta)\in R^k\times R^n, $$ where $\delta>0$. We assume that the symbol $p(x,\xi,\eta)$ is quasihomogeneous: $$ p\biggl(\frac y\lambda;\lambda^{1+\delta}\xi,\lambda\eta\biggr)=\lambda^mp(y;\xi,\eta)\qquad\forall\,\lambda>0 $$ and that $p(x,D)$ is elliptic for $y\ne0$. We have found a necessary and sufficient condition for operators of this class to be hypoelliptic: namely, that the equation $p(y;\xi,D_y)v(y)=\nobreak0$, $\xi\ne0$, have no nontrivial solutions in $S(R_y^n)$. Thus for example, the operator $\Delta_y^l+|y|^{2r}\Delta_{x'}^l$ is hypoelliptic for any integers $l>0$ and $r>0$, and the operator $\Delta^2_y+|y|^4\Delta_{x'}^2+\lambda\Delta_{x'}$ is hypoelliptic if and only if $\lambda$ is not an eigenvalue of the operator $\Delta^2_y+|y|^4$ in $L_2(R_y^n)$. These results are partially extended to operators with variable coefficients and to pseudodifferential operators. Bibliography: 22 titles.