In this paper we consider sums of squares of vector fields in $\R^2$ satisfying H\"ormander's condition and with polynomial, but non-(quasi-)homogeneous, coefficients. We obtain a Gevrey hypoellipticity index which we believe to be sharp. The general operator we consider is % $$ P=X^2+Y^2+\sum_{j=1}^{j=L}Z_j^2, $$ with % $$ X=D_x, \quad Y= a_{0}(x, y) x^{q-1}{D_y}, \quad Z_j= a_{j}(x, y) x^{p_j-1}y^{k_j}\,D_y, $$ % 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 $s\geq \frac{1}{1-T}, T = \max_j \frac{q-p_j}{q k_j}.$
Gevrey hypoellipticity for sums of squares with a non-homogeneous degeneracy / Bove, Antonio; Tartakoff, David S.. - In: PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9939. - STAMPA. - 142:4(2014), pp. 1315-1320. [10.1090/S0002-9939-2014-12247-7]
Gevrey hypoellipticity for sums of squares with a non-homogeneous degeneracy.
BOVE, ANTONIO;
2014
Abstract
In this paper we consider sums of squares of vector fields in $\R^2$ satisfying H\"ormander's condition and with polynomial, but non-(quasi-)homogeneous, coefficients. We obtain a Gevrey hypoellipticity index which we believe to be sharp. The general operator we consider is % $$ P=X^2+Y^2+\sum_{j=1}^{j=L}Z_j^2, $$ with % $$ X=D_x, \quad Y= a_{0}(x, y) x^{q-1}{D_y}, \quad Z_j= a_{j}(x, y) x^{p_j-1}y^{k_j}\,D_y, $$ % 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 $s\geq \frac{1}{1-T}, T = \max_j \frac{q-p_j}{q k_j}.$I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
