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Global hypoellipticity and global solvability for a class of operators on compact manifolds. (English) Zbl 0901.58071
Let \(M_1, M_2\) be smooth compact oriented manifolds and let \(P: C^\infty (M_1) \to C^\infty (M_1)\) be a continuous linear operator, admitting continuous extension \({}^t P: D'(M_1) \to D'(M_1)\).
The authors prove that if \(P\) and \(^t P\) are injective, then the following properties are equivalent: (i) \(P\) and \(^t P\) are globally hypoelliptic on \(M_1\times M_2\); (ii) \(P\) and \({}^tP\) are globally solvable on \(M_1\times M_2\); (iii) \(P\) and \({}^tP\) are globally hypoelliptic on \(M_1\); (iv) \(P\) and \({}^tP\) are globally solvable on \(M_1\).
MSC:
58J99 Partial differential equations on manifolds; differential operators
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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