real analysis – Weighted Hausdorff-Young’s inequality

For $2<p leq infty$, if we have $$|hat{f}|_p lesssim | (1+|x|)^{s/2} f(x)|_p$$ for any $fin mathscr{S}(mathbb{R}^d)$, the Schwartz function, what is the restriction 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 $sgeq d(1-2/p)$ 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.