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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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