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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)

MSC:
 35H10 Hypoelliptic equations 35A20 Analyticity in context of PDEs
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References:
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