×

On the hypoellipticity with a big loss of derivatives. (English) Zbl 1076.35138

The authors consider the class of pseudo-differential operators studied by Grushin, Sjöstrand and others, see in particular L. Boutet de Monvel, A. Grigis and B. Helffer [Astérisque 34–35, 93–121 (1976; Zbl 0344.32009)]. Besides vanishing to order \(k\) on the characteristic manifold \(\Sigma\) and transversal ellipticity, the authors assume that \(\Sigma\) is symplectic and flat, i.e., given by \(\Sigma= \{x=0,\xi=0\}\) in \((\mathbb{R}^n_x\times \mathbb{R}^m_y)\times (\mathbb{R}^m_\xi \times\mathbb{R}^m_\eta/0)\). The results of the above mentioned authors grant that an operator \(P=p(x,y,D_x,D_y)\) of this type is hypoelliptic with the (minimal) loss of \(k/2\) derivatives if and only if for the localization operator in \(\mathbb{R}^n\) with polynomial coefficients \(A_\rho (x,D_x)\), \(\rho\in\Sigma\), we have \(\text{Ker}\,A_\rho=\{0\}\). Here the authors go much further with the analysis, and show that when the condition \(\text{Ker}\,A_\rho= \{0\}\) is not satisfied, one can still obtain hypoellipticity, with a loss of \(r>k/2\) derivatives, under suitable additional assumptions. A striking example is given by \[ P=(D^2_x+ \lambda^2x^2|D_y|^2)^2+\text{lower order terms}, \] that, depending upon the choice of l.o.t., is hypoelliptic with loss of \(2,3,4\) or 5 derivatives.

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
35H10 Hypoelliptic equations

Citations:

Zbl 0344.32009
PDFBibTeX XMLCite
Full Text: DOI