In this paper, the authors assert that
...the domain of realization of the Laplacian in $L^{1}(\mathbb{R}^{n})$ is not contained in $W^{2,1}(\mathbb{R}^{n})$ if $n>1$. However, it is continuously embedded in $W^{1+\beta,1}(\mathbb{R}^{n})$ for each $\beta\in(0,1)$. Therefore, if $u$ and $\Delta u$ are in $L^{1}(\mathbb{R}^{n})$, then any first order derivative $D_{i}u$, $i=1,\ldots,n$, belongs to $W^{\beta,1}(\mathbb{R}^{n})$for each $\beta$, and by Sobolev embedding it belongs to $L^{p}(\mathbb{R}^{n})$ for each $p<n/n-1$, with
$$\left\|D_{i}u\right\|_{L^{p}}\leq C(p)\left(\left\|u\right\|_{L^{1}}+\left\|\Delta u\right\|_{L^{1}}\right)$$
Question. Could someone help me with a sketch for proving the continuous embedding into $W^{1+\beta,1}(\mathbb{R}^{n})$?
Here's what I know. If $u\in L^{1}(\mathbb{R}^{n})$ and $\Delta u=f\in L^{1}(\mathbb{R}^{n})$, then in the sense of (tempered) distributions
$$(-2\pi i)^{2}\left|\xi\right|^{2}\widehat{u}=\widehat{\Delta u}=\widehat{f}$$
Since $u,f\in L^{1}(\mathbb{R}^{n})$, $\widehat{u}$, $\widehat{f}$ belong to $C_{0}(\mathbb{R}^{n})$ and away from the origin $\widehat{u}(\xi)=-4\pi^{2}\widehat{f}(\xi)\left|\xi\right|^{-2}$.
I wanted to arrive at $u=f\ast K$, where $K$ is the fundamental solution of the Laplacian, and try to go from there; however, I hit the wall at this point.