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.
I found this problem is related to theorem 2.3 in the paper.
my question is why estimate (*) is equivalent to the multiplier form problem:
$$\|\mathcal{F}^{-1}\left(\frac{\hat{u}}{-4\pi^2|\xi|^2+z}\right)\|_q \le\|u\|_p$$
I found typically,we need to convert the problem into muliplier form in order to apply Stein's interpolation theorem.