A question of the Schrodinger Semigroup --By B. Simon

Question

The question comes from the paper: B. Simon, Schrodinger Semigroups, 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 $B_r(x)$ is located inside the domain of $u$.

On Page 499, it is stated 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?

Answer 1

$$ |u(x)|^2\leqslant C^2\left(\int |u(y)|e^{a|y|}\cdot e^{-a|y|}\right)^2\leqslant C^2 \|u(y)e^{a |y|}\|_{L^2}^2\cdot \int e^{-2a|y|} $$