$u\in H^1(\Omega)$ implies $\Delta u\in H^{-1}(\Omega)$? Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary.
Is it true that if $u  \in H^1(\Omega)$, then $\Delta u \in H^{-1}(\Omega)$?
Thanks

Hint: First check that the Laplacian operator on $u$ corresponds to multiplication of the Fourier transform $\hat u$ by ${\xi_1}^2+\dots+{\xi_n}^2$ (up to a sign that depends on the convention in the definition of $\Delta$). Then compare that with the definition of the Sobolev spaces in terms of Fourier transforms.
