Analyticity and loss of derivatives

TL;DR

This paper proves that a certain sum of squares of complex vector fields, previously studied by Kohn, is locally real analytically hypoelliptic despite exhibiting a significant loss of derivatives, using L2 methods.

Contribution

It establishes local real analytic hypoellipticity for a class of operators with derivative loss, advancing understanding of hypoelliptic operators with complex vector fields.

Findings

Proves local real analytic hypoellipticity of the operator.

Shows the operator exhibits a loss of many derivatives.

Uses L2 methods to establish hypoellipticity.

Abstract

We prove local real analytic hypoellipticity for a sum of squares of complex vector fields studied by J.J. Kohn in a paper to appear in the Annals of Mathematics entitled "Hypoellipticity and loss of derivatives". The operator exhibits a loss of many derivatives but is nonetheless hypoelliptic, and, using L2 methods, we prove analytic hypoellipticity.