Osaka Journal of Mathematics

Subelliptic estimates for overdetermined systems of quadratic differential operators

Karel Pravda-Starov

Full-text: Open access

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.

Article information

Source
Osaka J. Math., Volume 49, Number 3 (2012), 563-611.

Dates
First available in Project Euclid: 15 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1350306588

Mathematical Reviews number (MathSciNet)
MR2993058

Zentralblatt MATH identifier
1270.35206

Subjects
Primary: 35B65: Smoothness and regularity of solutions
Secondary: 35N10: Overdetermined systems with variable coefficients

Citation

Pravda-Starov, Karel. Subelliptic estimates for overdetermined systems of quadratic differential operators. Osaka J. Math. 49 (2012), no. 3, 563--611. https://projecteuclid.org/euclid.ojm/1350306588


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