There is a quite interesting problem for me to deal with it :
Let $f_1,f_2,...,f_N$ be in $L^{p,\infty}(X,\mu)$ which is the space of weak $L^p(X,\mu)$
Prove that for $0< p< 1$ we have $$\bigg\lVert\sum_{k=1}^{N} f_k\bigg\rVert_{L^{p,\infty}}\leq N^{\frac{1}{p}}\,\sum_{k=1}^{N}\lVert f_k\rVert_{L^{p,\infty}}$$
proof :
Use the notation $\omega_{|f_k\,|}(\alpha)=\bigg|\{x\in X:|f_k(x)|>\alpha\}\bigg|$.
For each $\alpha>0$ , $\bigg\{x\in X :|f_{1}(x)+\cdots+f_{N}(x)|>\alpha\bigg\}\subseteq\displaystyle\bigcup_{k=1}^{N}\bigg\{x\in X:|f_{k}(x)|>\dfrac{\alpha}{N}\bigg\}$
Then,one has
\begin{align*} \omega_{|f_{1}+\cdots+f_{N}|}(\alpha)&=\bigg|\bigg\{x\in X:|f_{1}(x)+\cdots+f_{N}(x)|>\alpha\bigg\}\bigg|\\ &\leq\sum_{k=1}^{N}\left|\left\{x\in X:|f_{k}(x)|>\frac{\alpha}{N}\right\}\right|\\ &=\sum_{k=1}^{N}\omega_{|f_{k}|}\bigg(\frac{\alpha}{N}\bigg) \end{align*}
Whence,for every $\alpha>0$ \begin{align*} \alpha^p\omega_{|f_{1}+\cdots+f_{N}|}(\alpha)&\leq N^p\dfrac{\alpha^p}{N^p}\sum_{k=1}^{N}\omega_{|f_{k}|}\bigg(\frac{\alpha}{N}\bigg)\\ &=N^p\sum_{k=1}^{N}\frac{\alpha^{p}}{N^{p}}\omega_{|f_{k}|}\bigg(\frac{\alpha}{N}\bigg)\\ &\leq N^p\sum_{k=1}^{N}\sup\bigg\{\frac{\alpha^{p}}{N^{p}}\omega_{|f_{k}|}\bigg(\frac{\alpha}{N}\bigg):\alpha>0\bigg\}\\ &\leq N^p\sum_{k=1}^{N}\lVert f_k\rVert_{L^{p,\infty}}^{p} \end{align*}
Now when $\delta\in [1,\infty)$ use the face that $\bigg(\sum_{k=1}^{N} a_k\bigg)^\delta\leq N^{\delta-1}\sum_{k=1}^{N} a^\delta_k\,$ for each $a_k\in\mathbb{R_{>0}}$
Thus, \begin{align*} \alpha\bigg(\omega_{|f_{1}+\cdots+f_{N}|}(\alpha)\bigg)^{\frac{1}{p}} &\leq N\bigg(\sum_{k=1}^{N}\lVert f_k\rVert_{L^{p,\infty}}^{p}\bigg)^{\frac{1}{p}}\\ &\leq N\cdot N^{(\frac{1}{p}-1)}\sum_{k=1}^{N}\lVert f_k\rVert_{L^{p,\infty}}\\ &=N^{\frac{1}{p}}\sum_{k=1}^{N}\lVert f_k\rVert_{L^{p,\infty}} \end{align*} Then,
$$\bigg\lVert\sum_{k=1}^{N} f_k\bigg\rVert_{L^{p,\infty}}=\sup\bigg\{\alpha\bigg(\omega_{|f_{1}+\cdots+f_{N}|}(\alpha)\bigg)^{\frac{1}{p}}:\alpha>0\bigg\}\leq N^{\frac{1}{p}}\,\sum_{k=1}^{N}\lVert f_k\rVert_{L^{p,\infty}}$$
If you have the time,please checking my working for validity with me and any advice will appreciate.Thanks for patiently reading.
The proof is overall good. Perhaps one line can be sharpen like the following:
\begin{align*} \dfrac{\alpha^{p}}{N^{p}}\omega_{|f_{k}|}\left(\dfrac{\alpha}{N}\right)\leq\sup\{\alpha^{p}\omega_{|f_{k}|}(\alpha): \alpha>0\}=\|f_{k}\|_{L^{p,\infty}}^{p}. \end{align*}