I encounter the following claim in one paper:

If $(-Delta)^{frac14}uin L^{2,infty}(mathbb{R})$, then $uin BMO(mathbb{R})$. Equivalently, if $uin mathcal{H}^1(mathbb{R})$, then $(-Delta)^{-frac14}uin L^{2,1}(mathbb{R})$. Here $L^{2,infty}$ and $L^{2,1}$ are Lorentz space and $mathcal{H}$ is the hardy space.

I do not know how to show this fact. My knowledge of Riesz potential tells me if $uin mathcal{H}^1(mathbb{R})$, then $(-Delta)^{-frac14}u=I_{1/2}uin L^2(mathbb{R})$, but why does it lie in the smaller space $L^{2,1}$?

The paper says the first half of the claim is contained in the paper: Adams, D. R. (1975). A note on riesz potentials. Duke Mathematical Journal. I read Adams’ paper and could not figure out why.