# Embeddings of the maximal domain for the Laplacian

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.

• Sobolev embedding + elliptic regularity (which allows you to pass from $D$ to $W^{2,2}$)? Dec 26, 2023 at 5:37
• Thank you for the comment. I think the elliptic regularity should imply that if $f\in D$ then also $f \in H^2_{loc}$ and on compact subsets we can use the Sobolev embedding. I edited my question to ask about embeddings on $\Omega$. For this I'm not sure the elliptic regularity works as we have no boundary condition and $D$ is a larger space than $H^2$
– MeS
Dec 26, 2023 at 5:57

There is no hope to gain summability without using boundary conditions. For example the function $$\frac{1}{z \log z}$$ is holomorphic, hence harmonic, and in $$L^2$$ in the disc (in the complex plane) centered at $$\frac 14$$ with radius $$\frac 14$$ but it is not in any $$L^p$$ for $$p>2$$.