Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain and $n \geq 2$. Consider the subspace of $L^2$-functions whose distributional Laplacian is also an $L^2$-function:

$$D = \left\{ f \in L^2(\Omega): \ \Delta f \in L^2(\Omega)\right\}$$

This space is known as the domain of the maximal operator associated with the Laplacian on $L^2$. I am wondering whether there are any embedding results for the space $D$ into spaces with higher order integrability, e.g. $L^p$ for some $p>2$ over $\Omega$ (or some weighted spaces). The elliptic regularity implies that $D \subset H^2_{loc}(\Omega)$. So the question boils down to whether one can control the functions from $D$ as we approach the boundary $\partial \Omega$. I'll appreciate any reference that discusses properties of functions from this space.