Hypoellipticity and Non Hypoellipticity for Sums of Squares of Complex Vector Fields

Authors

  • Antonio Bove Università di Bologna

DOI:

https://doi.org/10.6092/issn.2240-2829/2665

Abstract

In this talk we give a report on a paper where we consider a model sum of squares of planar complex vector fields, being close to Kohn's operator but with a point singularity. The characteristic variety of the operator is the same symplectic real analytic manifold as Kohn's. We show that this operator is hypoelliptic and Gevrey hypoelliptic provided certain conditions are satisfied. We show that in the Gevrey spaces below a certain index the operator is not hypoelliptic. Moreover there are cases in which the operator is not even hypoelliptic. This fact leads to some general negative statement on the hypoellipticity properties of sums of squares of complex vector fields, even when the complex H\"ormander condition is satisfied.

References

L. Boutet de Monvel, F. Treves, On a class of pseudodierential operators with double characteristics, Inv. Math. 24(1974), 1-34.

L. Boutet de Monvel, Hypoelliptic operators with double characteristics and related pseudo differential operators, Comm. Pure Appl. Math. 27(1974), 585-639.

A. Bove, M. Mughetti and D. S. Tartako, Hypoellipticity and Non Hypoellipticity for Sums of Squares of Complex Vector Fields, preprint, 2011.

A. Bove and D. S. Tartako, Optimal non-isotropic Gevrey exponents for sums of squares of vector fields, Comm. Partial Dierential Equations 22 (1997), no. 7-8, 1263-1282.

A. Bove and D. S. Tartako, Gevrey Hypoellipticity for Non-subelliptic Operators, preprint, 2008.

A. Bove, M. Derridj, J. J. Kohn and D. S. Tartako, Sums of Squares of Complex Vector Fields and (Analytic-) Hypoellipticity, Math. Res. Lett. 13(2006), no. 5-6, 683-701.

M. Christ, A remark on sums of squares of complex vector elds, arXiv:math.CV/ 0503506.

W. Feller, An Introduction to Probability Theory and Its Applications, John Wiley and Sons, New York, London, Sidney, 1967.

B. Heler, Sur l'hypoellipticite des operateurs a caracteristiques multiples (perte de 3/2 derivees), Memoires de la S. M. F., 51-52(1977), 13-61.

L. Hormander, The Analysis of Partial Dierential Operators, III, Springer Verlag, 1985.

J.J. Kohn, Hypoellipticity and loss of derivatives, Ann. of Math. (2) 162 (2005) 943-982.

J. Sjostrand and M. Zworski, Elementary linear algebra for advanced spectral problems, Festival Yves Colin de Verdiere. Ann. Inst. Fourier (Grenoble) 57(2007), 2095-2141.

Downloads

Published

2011-12-31

How to Cite

Bove, A. (2011). Hypoellipticity and Non Hypoellipticity for Sums of Squares of Complex Vector Fields. Bruno Pini Mathematical Analysis Seminar, 2(1). https://doi.org/10.6092/issn.2240-2829/2665

Issue

Section

Articles