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 $u\in \mathcal{S}(\Bbb{R}^{n})$?


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