Inequality for Weak $L^p$ spaces For each measurable function $f: X \to \mathbf{R}$, define it's distribution function $F: [0,\infty] \to [0,\infty]$ by $F(t) = |{x : |f(x)| > t}|$. Then, for each $p \geq 1$, define the weak $L^p$ norm $\| f \|_{p,\infty}$ as the smallest value $A$ such that for each $t \in [0,\infty]$,
$$ F(t) \leq A^p / t^p. $$
It is easy to see that $\| f \|_{p,\infty}$ is a quasinorm, i.e. that
$$ \| f_1 + \dots + f_N \|_{p,\infty} \leq N( \| f_1 \|_{p,\infty} + \dots + \| f_N \|_{p,\infty}). $$
However, for $p > 1$, $\| \cdot \|_{p,\infty}$ is comparable to a norm. In particular, this implies that we should be able to obtain an inequality of the form
$$ \| f_1 + \dots + f_N \|_{p,\infty} \lesssim_p \| f_1 \|_{p,\infty} + \dots + \| f_N \|_{p,\infty}, $$
independantly of $N$. Is there an elementary proof of the fact that
$$ \| f_1 + \dots + f_N \|_{p,\infty} \lesssim_p \| f_1 \|_{p,\infty} + \dots + \| f_N \|_{p,\infty}? $$
 A: The $L^{p,\infty}$ quasinorms are the $q=\infty$ case of the Lorentz quasinorms $L^{p,q}$. The Lorentz quasinorms are equivalent to norms when $1<p<\infty$ and $1\leq q \leq \infty$. The easiest way I know to see this is to prove the duality statement:
$$
\|f\|_{L^{p,q}} \sim_{p,q} \sup_{\|g\|_{L^{p',q'}} = 1} \left| \int f(x)\overline{g}(x)~dx\right|,
$$
where $p'$ and $q'$ are the H\"older conjugates to $p$ and $q$ respectively. The right-hand side clearly satisfies the triangle inequality in $f$, so once you have the above equivalence it does define a norm.
That being said, the proof of this equivalence is not entirely trivial. First, of course, you need to define the Lorentz quasinorms. After that I suppose there may be multiple ways to proceed, but in the proof I know you first prove that Lorentz functions satisfy a certain atomic decomposition, which reduces most statements about Lorentz functions to the case where $f$ has the form
$$
f = \sum_m 2^m 1_{E_m},
$$
where $E_m$ are pairwise disjoint and $1_{E_m}$ is the indicator function of $E_m$. If you want more details, check out Chapter 3 of these notes. Proposition 3.6 is the duality theorem.
