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On global hypoellipticity on the torus. (English) Zbl 0886.35048
Our main purpose is to study global hypoellipticity for a class of pseudodifferential operators on the $$n$$-torus, $$T^n$$, $$n\geq 2$$, of the form $P=p(D_1^2)+ e^{imx_1}+ ae^{-imx_1},$ where $$a=\pm1$$, $$m\in \mathbb{N}$$, $$D_1= (1/i)(\partial/\partial x_1)$$, and $$p$$ is a classical symbol satisfying the additional conditions $p(0)=0; \quad |p(1)|\geq 1;\quad |p(t^2)|>2, \quad t\in\mathbb{N}, \quad t\geq 2.$ We recall that an operator $$P$$ is said to be globally hypoelliptic (GH) on $$T^n$$ if the properties $$u\in{\mathcal D}'(T^n)$$ and $$Pu\in C^\infty(T^n)$$ imply $$u\in C^\infty (T^n)$$. We present a necessary and sufficient condition for the operator to be (GH). Our examples show, in particular, that in the case when $$p(t)=\lambda t^2$$, $$1<\lambda< 2$$, the situation $$m>1$$ is different from the case $$m=1$$, namely, when $$m>1$$, the operator may fail to be (GH).
##### MSC:
 35H10 Hypoelliptic equations 35S05 Pseudodifferential operators as generalizations of partial differential operators
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