This kind of question has always confused me, I need to confirm if I am right..
Let $S^1$ be the circle and consider $u \in L^p(S^1)$ for $1<p<\infty$.
If $\Delta u$ is a well-defined measurable function via the formula \begin{equation} \int_{S^1} \phi \Delta u = \int_{S^1} u \Delta \phi \text{ for all } \phi \in C^\infty(S^1) \end{equation} and in fact $\Delta u \in L^p(S^1)$, then is it necessarily true that \begin{equation} u \in W^{2,p}(S^1) \end{equation} holds?
I am quite sure this is the case, but I cannot provide correctly. I know that $\Delta$ is essentially self-adjoint, but do not see how this fact is used here.
Could anyone please help me?
In the $\mathbb S^1$ case this is rather obvious. Let $\theta$ be the angular coordinate, so that $\Delta=\partial_\theta^2$. Therefore the assumptions read $u\in L^p$ and $\partial_\theta^2 u\in L^p$. Which, by definition, means that $u\in W^{2, p}$.
This kind of questions become harder when you have more than one coordinate, i.e. on higher dimensional domains. There, it is not obvious that having a control on the Laplacian $\Delta= \partial_{x_1}^2+\ldots+\partial_{x_n}^2$ implies a control on the single derivatives $\partial_{x_1}^2, \partial_{x_2}^2, \ldots, \partial_{x_n}^2$.