# real analysis – Weighted Hausdorff-Young’s inequality

For $$2, if we have $$|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.