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