We consider the following equations in the distribution sense, considering $u\in \mathscr{D}^{'}$ and the derivatives are also in the distribution sense.
Question 1:
In $\mathbb{R}^d$, given $f \in L^2$, and $$-\Delta u +u =f,$$ can we deduce $u \in H^2$?
What I am sure of:
1) if $u \in \mathscr{D}^{'}$, i.e. $u$ is a distribution, then $u \in H^{2}_{loc}$.
2) if $u \in \mathcal{S}^{'}$, i.e. $u$ is a tempered distribution, then after applying Fourier transform, we can get the result.
Question 2:
In $\mathbb{R}^d$, given $f \in L^1$, and $$-\Delta u +u =f,$$ (i.e. the only change is that now $f\in L^1$) can we produce a counterexample that solution $u \notin W^{2,1}$?
Taking $u\in\mathcal D'$ is not strong enough, since $(-\Delta+1)\exp(\omega\cdot x)=0$ for any $\omega\in S^{d-1}$. But taking $u\in\mathcal S'$ removes these exponentially growing solutions.