# eigenfunctions of an globally hypoelliptic operator

An operator $$L$$ is said globally hypoelliptic in the Schwartz space $$\mathcal{S}(\Bbb{R}^{n})$$ if $$u\in \mathcal{S}'(\Bbb{R}^{n}), Lu\in \mathcal{S}(\Bbb{R}^{n})\Rightarrow u\in \mathcal{S}(\Bbb{R}^{n})$$

where $$\mathcal{S}'(\Bbb{R}^{n})$$ is the space of all tempered distributions.

Let $$u\in L^p(\Bbb{R}^{n})$$ an eigenfunction of $$L$$ i,e., $$Lu=au$$ with $$a\in \Bbb{R}$$.

Can we say that $$u\in \mathcal{S}(\Bbb{R}^{n})$$?