Local Hypoellipticity by Lyapunov Function

Abstract

We treat the local hypoellipticity, in the first degree, for a class of abstract differential operators complexes; the ones are given by the following differential operators: $L_j=\partial /\partial t_j+(\partial \varphi /\partial t_j)(t,A)A$, $j=\mathrm{1,2},\dots ,n$, where $A:D(A)\subset H\to H$ is a self-adjoint linear operator, positive with $\mathrm{0}\in \rho (A)$, in a Hilbert space $H$, and $\varphi =\varphi (t,A)$ is a series of nonnegative powers of $A^{-\mathrm{1}}$ with coefficients in $C^{\mathrm{\infty }}(\mathrm{\Omega })$, $\mathrm{\Omega }$ being an open set of ${\mathbb{R}}^n$, for any $n\in \mathbb{N}$, different from what happens in the work of Hounie (1979) who studies the problem only in the case $n=\mathrm{1}$. We provide sufficient condition to get the local hypoellipticity for that complex in the elliptic region, using a Lyapunov function and the dynamics properties of solutions of the Cauchy problem $t$′$(s)=-\nabla \mathrm{R}\mathrm{e}\mathrm{}{\varphi }_{\mathrm{0}}(t(s))$, $s\ge \mathrm{0}$, $t(\mathrm{0})=t_{\mathrm{0}}\in \mathrm{\Omega },{\varphi }_{\mathrm{0}}:\mathrm{\Omega }\to \mathbb{C}$ being the first coefficient of $\varphi (t,A)$. Besides, to get over the problem out of the elliptic region, that is, in the points $t$^∗ $\in \mathrm{\Omega }$ such that $\nabla \mathrm{R}\mathrm{e}{\varphi }_{\mathrm{0}}(t$^∗$)$ = 0, we will use the techniques developed by Bergamasco et al. (1993) for the particular operator $A=\mathrm{1}-\mathrm{\Delta }:H^{\mathrm{2}}({\mathbb{R}}^N)\subset L^{\mathrm{2}}({\mathbb{R}}^N)\to L^{\mathrm{2}}({\mathbb{R}}^N)$.