I have to prove that there exists $\epsilon>0$ such that for any $\epsilon_1,\epsilon_2\in(0,\epsilon)$ and $f\in S(\mathbb{R}^n)$ $$ ||f||_{L^{\infty}} \leq C(n,\epsilon) \left[ |||\xi|^{\frac n2-\epsilon_1} \hat{f}(\xi)||_{L^2}+|||\xi|^{\frac n2+\epsilon_2} \hat{f}(\xi)||_{L^2}\right]. $$
I have two ideas but not solutions so far:
1) After rewriting LHS in the terms of $\hat{f}$ the problem is equivalent to (in terms of integrals) $$ \forall_{x\in\mathbb{R}^n} \forall_{f\in S(\mathbb{R}^n)} |\int_{\mathbb{R}^n} e^{it\cdot x} f(t)\,dt| \leq C(\epsilon,n)\left[ (\int_{\mathbb{R}^n} |t|^{n-\varepsilon_1}|f(t)|^2\,dt)^{\frac 12} + (\int_{\mathbb{R}^n} |t|^{n+\varepsilon_1}|f(t)|^2\,dt)^{\frac 12} \right]. $$
2) By Sobolev embedding theorem for any $\epsilon>0$ $$ ||f||_{L^{\infty}}\leq C ||f||_{W^{\frac n2+\epsilon,2}}=C||(1+|\xi|^2)^{\frac n4+\frac{\epsilon}2}\hat{f}(\xi)||_{L^2}. $$
Do you have any idea how to finish it?