Subelliptic Estimates for Quadratic Differential Operators

TL;DR

This paper establishes global subelliptic estimates for quadratic differential operators, linking their spectral properties and subellipticity to the algebraic structure of their Hamilton maps and singular spaces.

Contribution

It proves subelliptic estimates for quadratic operators based on the structure of their singular spaces and Hamilton maps, extending previous spectral analysis results.

Findings

Quadratic operators with zero singular space are subelliptic with derivative loss.

Operators with symplectic singular spaces are subelliptic in orthogonal directions.

The subelliptic estimates depend on algebraic properties of the Hamilton maps.

Abstract

We prove global subelliptic estimates for quadratic differential operators. Quadratic differential operators are operators defined in the Weyl quantization by complex-valued quadratic symbols. In a previous joint work with M. Hitrik, we pointed out the existence of a particular linear subvector space in the phase space intrinsically associated to their Weyl symbols, called singular space, which rules spectral properties of non-elliptic quadratic operators. The purpose of the present paper is to prove that quadratic operators whose singular spaces are reduced to zero, are subelliptic with a loss of "derivatives" depending directly on particular algebraic properties of the Hamilton maps of their Weyl symbols. More generally, when singular spaces are symplectic spaces, we prove that quadratic operators are subelliptic in any direction of the symplectic orthogonal complements of their singular spaces.