Norm Convergence of Partial Sum Operator on Schwartz Space

Question

Let's define the partial sum operator $S_R$ as $$ (\widehat{S_R f}) = \chi_{B_R}\widehat{f}. $$

I need to prove that for $2 < p < \infty$ and $f \in \mathcal{S}(\mathbb{R})$, the Schwartz space, then $\|S_R f - f\|_p \rightarrow 0$ as $R \rightarrow \infty$. Based on the line of questions, I think I basically need the two following facts:

1. $\|S_R f\| \leq C_p \|f\|_p$ for all $1 < p < \infty$,

2. Hausdorff-Young Inequality: If $f \in L^p$, $1 < p < 2$, then $\widehat{f} \in L^{p^\prime}$ and $$ \|\widehat{f}\|_{p^\prime} \leq \|f\|_p .$$

My struggle is regarding how to use 2., which was given with the advice to use said inequality to estimate $\|S_Rf - f\|_p$ in terms of its Fourier transform. That confuses me as it seems on the wrong side of the inequality to get my desired result in any case. Any help and/or advice would be a great help. Thank you.

Answer 1

2. says that $\|g\|_{q} \leq \|\hat{g}\|_{q'}$ for $q > 2$, take $q=p,g=S_rf-f$

(from that $\hat{f}\in S$ we get that $|\hat{f}(x)| \le C/(1+ x^2)$ obtaining a bound for $\|\hat{g}\|_{q'}$)