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Failure of analytic hypoellipticity for some operators of $$X^ 2+Y^ 2$$ type. (English) Zbl 0846.35034
We show that the operator ${\partial^2\over \partial x^2}+ \Biggl(x^k {\partial\over \partial y}- x^l {\partial\over \partial t}\Biggr)^2$ in $$\mathbb{R}^3$$ (with nonnegative integers $$k$$, $$l$$ satisfying $$k< l$$) is not analytically hypoelliptic, if either of the following assumptions is satisfied:
(i) $$(l+ 1)/(l- k)$$ is not a positive integer.
(ii) Both $$l-k$$ and $$(l+ 1)/(l- k)$$ are odd integers.

##### MSC:
 35H10 Hypoelliptic equations
##### Keywords:
analytic hypoellipticity
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