Let $\Omega\subset\mathbb{R}^d$ be open,and $P(D)=\operatorname{\sum_{|\alpha|\le N}}f_{\alpha}D^{\alpha}$ be an elliptic differential operator. Rudin proves in Functional Analysis Part II the Regularity Theorem for Elliptic Operators that if $f_{\alpha}$ are smooth and $f_{\alpha}$ are constants for $|\alpha|=N$, then $P(D)u$ is locally in $H_s(\Omega)$ if and only if $u$ is locally in $H_{s+N}(\Omega)$, where \begin{equation} H_s=\{u\in D'(\mathbb{R}^d):(1+|y|^2)^{s/2}\hat{u}\in\mathcal{L}^2\}. \end{equation}
I know almost nothing about partial differential equations, and am ignorant of special examples especially. I need some examples showing the conclusion is false for other types of differential operators. That is, something like $Lu=v$ where $v$ is good but $u$ is bad when $L$ is not elliptic.
Thanks!