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Singular solutions for a class of Grusin type operators. (English) Zbl 0858.35025
The authors construct singular solutions for the following family of Grushin operators, given, for \(k\) an odd integer and \(\lambda\in\mathbb{C}\), by \[ P_\lambda= \partial^2_x+ x^{2k}\partial^2_y+ i(\lambda+k) x^{k-1}\partial_y, \] in \(\mathbb{R}^2\). This is a simple model of PDE with double characteristics and with complex lower order terms. This model was introduced by A. Gilioli and F. Trèves [Am. J. Math. 96, 367-385 (1974; Zbl 0308.35022)]. In continuation of this paper the authors present two methods of construction of singular solutions when \(\lambda=2j(k+1)\) or \(\lambda=2j(k+1)+2\) for some \(j\in\mathbb{N}\). One is by certain Fourier integrals with symbols expressed in terms of the solutions of the operator \[ Q_\lambda:= -\partial^2_x+x^{2k}+ (\lambda+k)x^{k-1}. \] The other is by the method of concatenations of Gilioli and Treves.
Reviewer: B.Helffer (Orsay)

35H10 Hypoelliptic equations
35A20 Analyticity in context of PDEs
Full Text: DOI
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