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In this work we study the global hypoellipticity of a class of operators of type
$$
L = D_t + a(t)Q(x,D) + ib(t)P(x,D),
\quad D_t = i^{-1} \partial_t,$$
where $(t,x) \in \mathbb{T}\times M^n$, $a, b$ are real smooth functions on $\mathbb{T}$, and $P(x,D), Q(x,D)$ are self-adjoint first order pseudodifferential operators, defined on a compact smooth Riemannian manifold $M^n$. Furthermore, we request that discrete spectrum of $P$, $\sigma(P) = \{\lambda_j\} \subset \mathbb{R}$, satisfies $|\lambda_j| \rightarrow + \infty$ when $j \rightarrow+\infty$.
In this talk, under the commutation hypothesis $[P,Q] = 0$, we will present necessary and sufficient conditions on the symbols of the operators $P$ and $Q$ to ensure global hypoellipticity of $L$. |
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