Fractional integral on Hardy spaces on product domains

By using the vector-valued theory of singular integrals, we prove a Hardy–Littlewood–Sobolev inequality on product Hardy spaces $H_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}}^p$, which is a parallel result of the classical Hardy–Littlewood–Sobolev inequality. The same technique shows the $H_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}}^p$-boundedness of the iterated Hilbert transform. As a byproduct, new proofs of several recently discovered Hardy type inequalities on product Hardy spaces are obtained, which avoid complicated Calderón–Zygmund theory on product domain, rendering them considerably simpler than the original proofs.