For $2<p \leq \infty$, if we have $$\|\hat{f}\|_p \lesssim \| (1+|x|^2)^{s/2} f(x)\|_p$$ for any $f\in \mathscr{S}(\mathbb{R}^d)$, the Schwartz function, what is the of s?
Just by Hausdorff-Young's inequality and Holder's inequality, we know that when $s>d(1-2/p)$, the inequality is true. And I can get $s\geq d(1-2/p)$ by scaling argument. I believe that when $s=d(1-2/p)$, the inequality is not true, because I prove the case when $p=\infty$. But in the other cases, I have not proved it.
Any idea will be helpful. Thanks.