This problem is from Tao's lecture note on Harmonic Analysis,
(Stein-Weiss inequality) Let $f_1,\ldots,f_N$ be functions with $N\geq 2.$ Show that $$\Vert f_1+\cdots f_N\Vert_{L^{1,\infty}} \lesssim \log N\left(\Vert f_1\Vert_{L^{1,\infty}}+\cdots+\Vert f_N\Vert_{L^{1,\infty}}\right).$$
Also show by example that $\log N$ in the right hand side cannot be replaced by a smaller quantity.
The hint in the exercise says that I can reduce the inequality to showing that $$\mu(\{f_1+\cdots+f_N\geq 1\})\lesssim \log N\left(\Vert f_1\Vert_{L^{1,\infty}}+\cdots+\Vert f_N\Vert_{L^{1,\infty}}\right)$$ with $f_n$'s satisfy $\frac{1}{2N}\leq f_n \leq 1$ for all $n$.
I already understood this reduction process, but I have trouble proceeding further. I attempted as follows:$$\mu(\{f_1+\cdots+f_N\geq 1\})\leq \sum_{n=1}^N \mu(\{f_n\geq 1/N\})\leq N\sum_{n=1}^N\Vert f_n\Vert_{L^{1,\infty}},$$ but I merely ended up with the factor $N$, which is worse than I needed.
How should I do?
Thanks in advance!