Polar decomposition of semigroups generated by non-selfadjoint quadratic differential operators and regularizing effects

TL;DR

This paper provides a geometric characterization of the regularizing effects of semigroups generated by non-selfadjoint quadratic differential operators, establishing optimal subelliptic estimates and a new polar decomposition representation.

Contribution

It introduces a novel geometric framework and a new polar decomposition for these semigroups, confirming several conjectures and identifying the selfadjoint part as a specific evolution operator.

Findings

Proved optimal subelliptic estimates for the operators.

Established a new representation of the polar decomposition.

Confirmed conjectures by Hitrik, Pravda-Starov, and Viola.

Abstract

We characterize geometrically the regularizing effects of the semigroups generated by accretive non-selfadjoint quadratic differential operators. As a byproduct, we establish the subelliptic estimates enjoyed by these operators, being expected to be optimal. These results prove conjectures by M. Hitrik, K. Pravda-Starov and J. Viola. The proof relies on a new representation of the polar decomposition of these semigroups. In particular, we identify the selfadjoint part as the evolution operator generated by the Weyl quantization of a time-dependent real-valued nonnegative quadratic form for which we prove a sharp anisotropic lower bound.