×

Gevrey hypoellipticity for an interesting variant of Kohn’s operator. (English) Zbl 1201.35092

Ebenfelt, Peter (ed.) et al., Complex analysis. Several complex variables and connections with PDE theory and geometry. Proceedings of the conference in honour of Linda Rothschild, Fribourg, Switzerland, July 7–11, 2008. Basel: Birkhäuser (ISBN 978-3-0346-0008-8/hbk). Trends in Mathematics, 51-73 (2010).
The authors consider the following operator (similar to Kohn’s operator but with a point singularity)
\[ P = BB^\star + B^\star (t^{2l} + x^{2k}) B, \]
where
\[ B= D_x + i x^{2q-1}. \]
The authors prove that for \(k < lq\) the operator is hypoelliptic and Gevrey hypoelliptic with index \(\frac{lq}{lq-k} = 1 + \frac{k}{lq-k}\). The authors announce a paper which suggests that the above result is sharp.
For the entire collection see [Zbl 1188.32003].

MSC:

35H10 Hypoelliptic equations
35A20 Analyticity in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
PDFBibTeX XMLCite