Suppose that $E$ is an algebraic set in $\mathbb{R}^n (n\ge3)$ with dimension $\le n-2$, and $u$ is locally Lipschitz continuous on $\mathbb{R}^n$. If $u\in C^\infty(E^c)$ and there is a positive constant $C$ such that \begin{equation*} \Delta u(x) \le C, \quad \forall\, x\in E^c, %x\in E^c \cap \Omega, \end{equation*} then the above inequality also holds in the sense of distribution, that is, \begin{equation*} \int_{\mathbb{R}^n} u \,\Delta\varphi \,dx \le C \int_{\mathbb{R}^n} \varphi \,dx, \quad\forall\, \varphi\in C^\infty_0(\mathbb{R}^n),\,\varphi\ge0. % \int_\Omega u \,\Delta\varphi \,dx \le C \int_\Omega \varphi \,dx, \quad\forall\, \varphi\in C^\infty_0(\Omega),\,\varphi\ge0. \end{equation*}
In my limited experience with geometry and analysis, though I found some similar estimates in geometric analysis (such as the Laplacian comparison theorem), I couldn't adapt those ideas to the current situation. So I am not sure whether this claim is even correct and do not know how to prove it.
Are there any related references? Any comment would be highly appreciated.