Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global Hypoellipticity for First-Order Operators on Closed Smooth
  Manifolds   The main goal of this paper is to address global hypoellipticity issues for
the following class of operators: $L = D_t + C(t,x,D_x)$, where $(t,x) \in
\mathbb{T} \times M$, $\mathbb{T}$ is the one-dimensional torus, $M$ is a
closed manifold and $C(t,x,D_x)$ is a first order pseudo-differential operator
on $M$, smoothly depending on the periodic variable $t$. In the case of
separation of variables, namely, $C(t,x,D_x) = a(t)p(x,D_x)+ib(t)q(x,D_x)$, we
give necessary and sufficient conditions for the global hypoellipticity of $L$.
In particular, we show that, under suitable conditions, the famous (P)
condition of Niremberg-Treves is neither necessary nor sufficient to guarantee
the global hypoellipticity of $L$.

19
The categories of this paper are: eess
16
The categories of this paper are: stat
15
The categories of this paper are: nucl-ex
24
The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph
28
The categories of this paper are: alg-geom
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
12
The categories of this paper are: nucl-th
25
The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
7
The categories of this paper are: hep-th
4
The categories of this paper are: cond-mat
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
33
The categories of this paper are: plasm-ph