Global Hypoellipticity for First-Order Operators on Closed Smooth Manifolds

Abstract

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 $L$.

53 pages; updated version (fixed typos and new references, resized to fit on fewer pages) to appear in J. Anal. Math

References in corpus (2)

• Schatten classes on compact manifolds: Kernel conditions
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