# ap.analysis of pdes – Boundedness of Riesz potential on hardy space

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.