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 $H^p_{\rm{prod}}$, which is a parallel result of the classical Hardy--Littlewood--Sobolev inequality. The same technique shows the $H^p_{\rm{prod}}$-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 Calder\'on--Zygmund theory on product domain, rendering them considerably simpler than the original proofs.