Global (and local) analyticity for second order operators constructed from rigid vector fields on products of tori

Abstract:

We prove global analytic hypoellipticity on a product of tori for partial differential operators which are constructed as rigid (variable coefficient) quadratic polynomials in real vector fields satisfying the Hörmander condition and where $P$ satisfies a “maximal” estimate. We also prove an analyticity result that is local in some variables and global in others for operators whose prototype is \[ P=\left ( \frac \partial {\partial x_1}\right ) ^2+\left ( \frac \partial { \partial x_2}\right ) ^2+\left ( a(x_1,x_2)\frac \partial {\partial t}\right )^2 \] (with analytic $a(x),a(0)=0$, naturally, but not identically zero). The results, because of the flexibility of the methods, generalize recent work of Cordaro and Himonas in [Global analytic hypoellipticity of a class of degenerate elliptic operators on the torus, Math. Res. Lett. 1 (1994), 501–510] and Himonas in [On degenerate elliptic operators of infinite type, Math. Z. (to appear)] which showed that certain operators known not to be locally analytic hypoelliptic (those of Baouendi and Goulaouic [Analyticity for degenerate elliptic equations and applications, Proc. Sympos. Pure Math., vol. 23, Amer. Math. Soc., Providence, RI, 1971, pp. 79–84], Hanges and Himonas [Singular solutions for sums of squares of vector fields, Comm. Partial Differential Equations 16 (1991), 1503–1511], and Christ [Certain sums of squares of vector fields fail to be analytic hypoelliptic, Comm. Partial Differential Equations 10 (1991), 1695–1707]) were globally analytic hypoelliptic on products of tori.