Hypoellipticity without loss of derivatives for Fedii's type operators

Timur Akhunov; Lyudmila Korobenko; Cristian Rios

paper

Hypoellipticity without loss of derivatives for Fedii's type operators

arXiv:1711.00161

Abstract

We prove that second order linear operators on $R^{n + m}$ of the form $L (x, y, D_{x}, D_{y}) = L_{1} (x, D_{x}) + g (x) L_{2} (y, D_{y})$ , where $L_{1}$ and $L_{2}$ satisfy Morimoto's super-logarithmic estimates and $g$ is smooth, nonnegative, and vanishes only at the origin in $R^{n}$ (but to any arbitrary order) are hypoelliptic without loss of derivarives. We also show examples in which our hypotheses are necessary for hypoellipticity.

35 pages, 1 figure