For $f\in L_p(\mathbb{R}^n)$ ($1\leq p\leq 2$) define $S_Rf$ by $$S_Rf(x)=\int_{|\xi|<R}\hat f(\xi)e^{2\pi i\xi.x}d\xi$$ Since for $f\in L_p$ we have $\hat f\in L_{p'}$ (Hausdorff-Young inequality), and $\chi_{B_R}\in L_p$, the above integrals exist. I was wondering whether the fact that $S_Rf\in L_p$ can be proved trivially and I'm just missing it somehow. I can prove $S_R\phi\in L_p$ for each Schwartz space function $\phi$ using the smoothness of $\phi$ to integrate by parts. But for a general $f\in L_p$ I don't know. Any help would be appreciated.
I could apply convolution theorem and Young's convolution inequality, but I wanted to know if there are simpler ways to prove it.