PomeLemma 5.46.1.8F61C8BDD569 Let $U\subset \mathbb R^ d$ be open and let $w\in L^\infty _{\mathrm{loc}}(U)$. If $\Delta w$ is a locally finite signed Radon measure, then $w\in W^{1,2}_{\mathrm{loc}}(U)$.
Lemma 5.46.18F61C8BDD569
Proof. Fix a smooth function $0\leq \psi \leq 1$ with compact support in $U$, and a relatively compact open neighborhood $V$ of its support. Mollify $w$ inside $V$ with a standard mollifier $\rho _\varepsilon $, writing $w_\varepsilon =w*\rho _\varepsilon $ and $\nu _\varepsilon =(\Delta w)*\rho _\varepsilon $. For small $\varepsilon $, integration by parts and Young's inequality give
\[ \begin{split} \int \psi ^2|\nabla w_\varepsilon |^2 & =-\int \psi ^2w_\varepsilon \nu _\varepsilon -2\int \psi w_\varepsilon \nabla \psi \cdot \nabla w_\varepsilon \\ & \leq \| w\| _{L^\infty (V)}|\Delta w|(V) +\tfrac 12\int \psi ^2|\nabla w_\varepsilon |^2 +2\int w_\varepsilon ^2|\nabla \psi |^2. \end{split} \]
Absorbing the middle term on the right, we obtain a bound for $\int \psi ^2|\nabla w_\varepsilon |^2$ that is independent of $\varepsilon $. Taking $\psi =1$ on a prescribed smaller compact set, weak compactness in $W^{1,2}$ and convergence of the mollifications in $L^2$ identify the weak limit with $w$. This proves the assertion. $\square$
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