Hypoellipticity without loss of derivatives for Fedii's type operators

TL;DR

This paper establishes hypoellipticity without loss of derivatives for a class of second order linear operators on Euclidean space, extending previous results by relaxing conditions on the operators and the vanishing set.

Contribution

It proves hypoellipticity without loss of derivatives for Fedii's type operators under Morimoto's super-logarithmic estimates and specific vanishing conditions, providing new insights into operator regularity.

Findings

Operators are hypoelliptic without loss of derivatives under given conditions.

Examples demonstrate the necessity of hypotheses for hypoellipticity.

Extension of hypoellipticity results to broader classes of operators.

Abstract

We prove that second order linear operators on $\mathbb{R}^{n+m}$ 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 $\mathbb{R}^n$ (but to any arbitrary order) are hypoelliptic without loss of derivarives. We also show examples in which our hypotheses are necessary for hypoellipticity.