Singular solutions for sums of squares of vector fields

Abstract

Recently J. J. Kohn (2005) proved C ∞ hypoellipticity for P k = LL + L|z| 2k L = −L * L − (z k L) * z k L with L = ∂ ∂z + iz ∂ ∂t , (the negative of) a singular sum of squares of complex vector fields on the complex Heisenberg group, an operator which exhibits a loss of k − 1 derivatives. Subsequently, M. Derridj and D. S. Tartakoff proved analytic hypoellipticity for this operator using rather different methods going back to earlier methods of Tartakoff. Those methods also provide an alternate proof of the hypoellip-ticity given by Kohn. In this paper, we consider the equation P m,k = L m L m + L m |z| 2k L m with L m = ∂ ∂z + iz|z| 2m ∂ ∂t , for which the underlying manifold is only of finite type, and prove analytic hypoellipticity using methods of Derridj and Tartakoff. This operator is also subelliptic with large loss of derivatives, but the exact loss plays no role for analytic hypoellipticity. Nonetheless, these methods give a proof of C ∞ hy-poellipticity with precise loss as well, which is to appear in a forthcoming paper by A. Bove, M. Derridj, J. J. Kohn and the author.

Gevrey Hypoellipticity for Grushin Operators

“…When fe = l, we have 6=2 which is known the optimal index by the result of Baouendi-Goulaouic, [2] and also we refer to [15], p.310. For general integer k, it turns out that the index 0=1+ fe is also optimal by the result of [11] and [31] as was mentioned in the introduction.…”

“…Finally we note that for the example M given above, Gevrey index 0=l + k is optimal by the result of [11], where is constructed a function u-u (x, y, z) satisfying the equation Mu -0 in a neighborhood of (0, 0, 0) €E U 3 …”

“…After then there have been investigated the problem of analytic and non-analytic hypoellipticity of the Grushin operators, (cf. [19], [2], [9], [27], [28], [30], [31], [11]). …”

Global regularity in ultradifferentiable classes

“…But this condition is not sufficient for the local analytic hypoellipticity of P as it was first observed by Baouendi and Goulaouic [7]. Other classes of locally hypoelliptic operators which fail to be locally analytic hypoelliptic have been found and there are important results on analytic regularity (see, f.i., [26,41,24,25,12,13] and the references therein). All such operators also fail to be locally hypoelliptic in the setting of ultradifferentiable function spaces (see, Propositions 4.1 and 4.2 for example).…”

Non-isotropic Gevrey Hypoellipticity for Grushin Operators

“…After then, there has been investigated the problem of analytic and non-analytic hypoellipticity of the Grushin operators [1], [2], [13], [28], etc.…”