I was trying to prove the following estimate
$$ \|u\|_{L^q(\Bbb{R}^n)}\le \|(\Delta +z) u\|_{L^p(\Bbb{R}^n)} \tag{*}$$ For all $z\in \Bbb{C}$,all $u\in C^\infty_c(\Bbb{R}^n)$,with $n\ge 3,$ and $\frac{1}{p}-\frac{1}{q} = \frac{2}{n}$ and $\frac{n+1}{2n} <\frac{1}{p}<\frac{n+3}{2n}$.
Our professor told us we can find the solution in Koenig, Ruiz, Sogge 1987 paper here.
Assume for now I can prove the case when $|z|\ge 1$,the paper says the case when $|z|\le 1$ can be deduced from the case when $|z|\ge 1$.(Before theorem 2.3).
I can't figure out how to deduce $|z|\le1$?