I am working on the following problem.
Let $f\in L^{2}(\Omega)$,where $\Omega\subset \mathbb{R}^{n}$ is a bounded region with smooth boundary.Show that if u is a weak solution to $$\Delta u=f,\text{in}\,\Omega,\quad f=0,\text{in}\,\partial{\Omega}.$$ Then $u\in H^{2}(\Omega)$ and $$||u||_{H^{2}}\leq C ||f||_{L^{2}}.$$
The book gives the hint:Use the Fourier transform and Plancherel Theorem.
But I cannot see how can I use Fourier transform since their are no function in Schwartz space.
I tried to take a smooth approximating series $u_{n}\to u,\text{in} H^{1}_{0}.$But I cannot use Fourier transform to $u_{n}$ since I don't know too much about this series.
Any help will be thanked.