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

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 $$|a|(it is ok to understand), then by the Subsolution Estimate $$e^{a|x|} u\in L^\infty$$.

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

1 Answer

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