Weak L^p space and inequality concerning Fourier transform

Question

The norm in $L^{p, \infty}$ is defined as $$ \sup_{\alpha > 0} \alpha |\{ x \in \mathbb{R}^n: |f(x)| > \alpha \}|^\frac{1}{p}. $$ Fourier transform and it's reversal is defined by $$ \mathfrak{F}(f)(x) = \hat{f}(x) = \displaystyle\int_{\mathbb{R}^n}^{} e^{-2 \pi i x y}f(y) dy, $$ $$ \mathfrak{F}^{-1}(\hat{f})(x) = \displaystyle\int_{\mathbb{R}^n}^{} e^{2 \pi i x y}\hat{f}(y) dy. $$ Assume that for some $R > 0$ and $f$ we can write $f$ as (At this point don't worry about the existence of such expressions) $$ f = f_{<R} + f_{>R} := \mathfrak{F}^{-1}(\hat{f} \chi_{B(0, R)}) + \mathfrak{F}^{-1}(\hat{f} \chi_{\mathbb{R}^n \setminus B(0, R)}). $$ I would like to know how to prove the following inequality: $$\| f_{<R}\|_{L^{q, \infty}} \leq C\|f \|_{L^{q, \infty}}.$$ It is taken from https://arxiv.org/pdf/1303.6351.pdf, page 11, part of some proof. As I can read, authors say that it follows using (2.4) which in the paper is the fact that $$\| f + g \|_{L^{p, \infty}} \leq 2 \| f \|_{L^{p, \infty}} + 2 \| g \|_{L^{p, \infty}}$$ but i don't see how applying this triangle inequality helps. Is it really hard to prove or am I missing something?