want to prove the duality relation:


$$||f||\_{L^{p,1}} =C\_{p} \cdot \sup\{\int\_X fg d\mu: \text{ for any } ||g||\_{L^{p',\infty}}\le 1 \} $$


where $\frac{1}{p}+\frac{1}{p'}=1, p>1, \mu$ is $\sigma$-finite meaure on $X$, $C\_{p}\in (0, \infty)$ and 
$$||g||\_{L^{p,q}}:=(p \cdot \int\_{0}^{\infty} \lambda^{q-1}\mu(|f|>\lambda)^{\frac{q}{p}}d\lambda)^{\frac{1}{q}}, p>1, q \in (0,\infty].$$


There are several versions of proofs online, but they are all wrong:


<https://www.math.ucla.edu/~tao/247a.1.06f/notes1.pdf> or


<https://www.math.ucla.edu/~killip/247a/Lorentz_space_notes.pdf>



Duality relation of Lorentz space $L^{p,1}$
You may consult the book by Grafakos, Classical Fourier Analysis. A detailed study of Lorentz dual spaces is in Theorem 1.4.17, p.52


