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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.

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  • $\begingroup$ Sobolev embedding + elliptic regularity (which allows you to pass from $D$ to $W^{2,2}$)? $\endgroup$ Dec 26, 2023 at 5:37
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    $\begingroup$ 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 $H^2$ $\endgroup$
    – MeS
    Dec 26, 2023 at 5:57

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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 $p>2$.

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