The question comes from the paper: B. Simon, [Schrodinger Semigroups](https://projecteuclid.org/euclid.bams/1183549767), Bull. A.M.S., (1982) Vol. 7 (3). 

On the Theorem C.1.2(subsolution estimate) of the paper, it says that: If $Hu=0$, where $H=-\Delta+V$ for some bounded continuous function $V$. Then
$$|u(x)|\leq C\int_{B_r(x)}|u(y)|dy,$$
where $C$ depends on $r$ and norm of $V$ and $B_r(x)$ locates inside the domain of $u$. 

On Page 499, it  writes that: If $Hu=Eu$ for some $E$ in the discrete spectrum of $H$, we have $e^{a|x|}u\in L^2$ for all $|a|<M$(it is ok to understand), then by the **Subsolution Estimate** $e^{a|x|} u\in L^\infty$.  


**Q** How to deduce the result If $e^{a|x|}u\in L^2$ for all $|a|<M$, then  $u\in L^\infty$.  

PS: What I do not understand is that $e^{a|x|}u$ does not satisfy the equation, is there any result  to deal   with this problem?