Subelliptic Estimates for Overdetermined Systems of Quadratic Differential Operators

TL;DR

This paper establishes global subelliptic estimates for overdetermined systems of quadratic differential operators, extending previous results on single operators by analyzing their interactions and providing explicit criteria for subellipticity.

Contribution

It introduces a simple criterion for subellipticity of systems of quadratic operators, detailing the derivative loss and interaction effects, advancing understanding of non-elliptic quadratic systems.

Findings

Established subelliptic estimates for systems of quadratic operators.

Provided explicit criteria for subellipticity based on operator interactions.

Quantified the derivative loss depending on algebraic properties of Hamilton maps.

Abstract

We prove global subelliptic estimates for systems of quadratic differential operators. Quadratic differential operators are operators defined in the Weyl quantization by complex-valued quadratic symbols. In a previous work, 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 a number of fairly general properties of non-elliptic quadratic operators. About the subelliptic properties of these operators, we established that quadratic operators with zero singular spaces fulfill global subelliptic estimates with a loss of derivatives depending on certain algebraic properties of the Hamilton maps associated to their Weyl symbols. The purpose of the present work is to prove similar global subelliptic estimates for overdetermined systems of quadratic operators. We establish here a simple criterion for the subellipticity of these systems giving an explicit measure of the loss of derivatives and highlighting the non-trivial interactions played by the different operators composing those systems.