Parenti, Cesare; Parmeggiani, Alberto On the hypoellipticity with a big loss of derivatives. (English) Zbl 1076.35138 Kyushu J. Math. 59, No. 1, 155-230 (2005). 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. Reviewer: Luigi Rodino (Torino) Cited in 2 ReviewsCited in 19 Documents MSC: 35S05 Pseudodifferential operators as generalizations of partial differential operators 35H10 Hypoelliptic equations Keywords:operators with multiple characteristics Citations:Zbl 0344.32009 PDFBibTeX XMLCite \textit{C. Parenti} and \textit{A. Parmeggiani}, Kyushu J. Math. 59, No. 1, 155--230 (2005; Zbl 1076.35138) Full Text: DOI