Moser iteration is a technique that gives supremum estimates on a second order elliptic equation using an energy bound. The typical Moser iteration looks like

$$ \| u\|^2_{L^\infty(B_R)} \le C (R^{-2} + \sup |V|)^{n/2} \| u\|^2_{L^2(B_{2R})} $$ when $u$ solves

$$ (\Delta + V)u = 0. $$

To prove the Moser iteration, we first prove the Agmon identity. Let $\eta$ be some cutoff function to be chosen later $b$ any number, and consider \begin{align*} (-\Delta u, \eta^2 u^{2b} u) &= (\nabla u, u^b \eta \nabla (\eta u^{b+1}) + \eta u^{b+1} \nabla (\eta u^b)) \\ &= (u^b \eta \nabla u, \nabla (\eta u^{b+1})) + (\eta u^b \nabla u, u \nabla (\eta u^b)) \\ &= \|\nabla (\eta u^{b+1})\|^2 - (u \nabla (\eta u^b), \nabla (\eta u^{b+1})) + (\eta u^b \nabla u, u \nabla (\eta u^b)) \\ &= \|\nabla (\eta u^{b+1})\|^2 + (\eta u^b \nabla u- \nabla (\eta u^{b+1}), u \nabla (\eta u^b)) \\ &= \|\nabla (\eta u^{b+1})\|^2 - \| u \nabla (\eta u^b) \|^2. \end{align*}

Therefore, \begin{align*} 0 &= \| \nabla (\eta u^{b+1}) \|^2 - \| u \nabla (\eta u^b)\|^2 + (V \eta u^{b+1}, \eta u^{b+1}). \end{align*}