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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)
##### Keywords:
global hypoellipticity; global solvability
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##### References:
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