Local hypoellipticity by Lyapunov function (Q1669263)

Summary: 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: \(L_j=\partial/\partial t_j+(\partial\phi/\partial t_j)(t,A)A\), \(j=1,2,\dots,n\), where \(A\:D(A)\subset H\to H\) is a self-adjoint linear operator, positive with \(0\in\rho(A)\), in a Hilbert space \(H\), and \(\phi=\phi(t,A)\) is a series of nonnegative powers of \(A^{-1}\) with coefficients in \(C^\infty(\Omega)\), \(\Omega\) being an open set of \(\mathbb R^n\), for any \(n\in\mathbb N\), different from what happens in the work of \textit{J. Hounie} [Trans. Am. Math. Soc. 252, 233--248 (1979; Zbl 0424.35030) who studies the problem only in the case \(n=1\). 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 \(t'(s)=-\nabla\mathrm{Re}\,\phi_0(t(s))\), \(s\geq 0\), \(t(0)=t_0\in\Omega\), \(\phi_0\:\Omega\to\mathbb C\) being the first coefficient of \(\phi(t,A)\). Besides, to get over the problem out of the elliptic region, that is, in the points \(t^*\in\Omega\) such that \(\nabla\mathrm{Re}\,\phi_0(t^*)=0\), we will use the techniques developed by \textit{A. P. Bergamasco} et al. [J. Funct. Anal. 114, No. 2, 267--285 (1993; Zbl 0777.58041)] for the particular operator \(A=1-\Delta\:H^2(\mathbb R^N)\subset L^2 (\mathbb R^N)\to L^2(\mathbb R^N)\).