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Gevrey hypoellipticity for sums of squares with a non-homogeneous degeneracy


Authors: Antonio Bove and David S. Tartakoff
Journal: Proc. Amer. Math. Soc. 142 (2014), 1315-1320
MSC (2010): Primary 35H20; Secondary 35H10
Published electronically: January 29, 2014
MathSciNet review: 3162252
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Abstract: In this paper we consider sums of squares of vector fields in $ \mathbb{R}^2$ satisfying Hörmander's condition and with polynomial, but non-(quasi-)homoge-
neous, coefficients. We obtain a Gevrey hypoellipticity index which we believe to be sharp. The general operator we consider is

$\displaystyle P=X^2+Y^2+\sum _{j=1}^{L}Z_j^2, $

with

$\displaystyle X=D_x, \quad Y= a_{0}(x, y) x^{q-1}{D_y}, \quad Z_j= a_{j}(x, y) x^{p_j-1}y^{k_j}\,D_y, $

with $ a_{j}(0, 0) \neq 0 $, $ j = 0, 1, \ldots , L $ and $ q>p_j, \{k_j\}$ arbitrary. The theorem we prove is that $ P$ is Gevrey-s hypoelliptic for $ s\geq \frac {1}{1-T}, T = \max _j \frac {q-p_j}{q k_j}.$

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Additional Information

Antonio Bove
Affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, Bologna, Italy
Email: bove@bo.infn.it

David S. Tartakoff
Affiliation: Department of Mathematics, University of Illinois at Chicago, m/c 249, 851 S. Morgan Street, Chicago, Illinois 60607
Address at time of publication: 1216 N. Kenilworth Avenue, Oak Park, Illinois 60302
Email: dst@uic.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12247-7
Keywords: Sums of squares of vector fields, hypoellipticity, Gevrey hypoellipticity
Received by editor(s): May 15, 2012
Published electronically: January 29, 2014
Communicated by: James E. Colliander
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.