I am reading a preprint and trying to understand the proof of Lemma 3.5. On Pg. 19 above eqn (3.49) the authors claim that $\Delta W \leq −2(n − 4)V$ where the functions $W$ and $V$ are defined below, \begin{align} V(x)=\sum_{i=1}^{\nu}\left(\frac{\lambda_{i}^{\frac{n+2}{2}} R^{2-n}}{\left\langle y_{i}\right\rangle^{4}} \chi_{\left\{\left|y_{i}\right| \leq R\right\}}+\frac{\lambda_{i}^{\frac{n+2}{2}} R^{-4}}{\left\langle y_{i}\right\rangle^{n-2}} \chi_{\left\{\left|y_{i}\right| \geq R / 2\right\}}\right) \label{deng-V-func-defn}\\ W(x)=\sum_{i=1}^{\nu}\left(\frac{\lambda_{i}^{\frac{n-2}{2}} R^{2-n}}{\left\langle y_{i}\right\rangle^{2}} \chi_{\left\{\left|y_{i}\right| \leq R\right\}}+\frac{\lambda_{i}^{\frac{n-2}{2}} R^{-4}}{\left\langle y_{i}\right\rangle^{n-4}} \chi_{\left\{\left|y_{i}\right| \geq R / 2\right\}}\right)\label{deng-W-func-defn}. \end{align} where $y_i = \lambda_i(x-z_i)$, $\nu\geq 1$, $n\geq 6$, $z_i\in \mathbb{R}^n$, $\lambda_i>0$, $R\gg 1$ and $\langle y_i\rangle = \sqrt{1+y_i^2}.$
I am not sure if it is possible to take the Laplacian of indicator functions, since the result will be a distribution and thus having an upper bound on it seems weird. But maybe I am missing something here.
Any comments explaining why the estimate might or might not be true will be much appreciated.