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 Calderón–Zygmund theory on product domain, rendering them considerably simpler than the original proofs.

💡 Research Summary

The paper addresses the problem of extending the classical Hardy–Littlewood–Sobolev (HLS) inequality to product Hardy spaces (H^{p}{\mathrm{prod}}(\mathbb{R}^{d})). In the one‑parameter setting the HLS inequality is well understood and has been transferred to the Hardy space framework (Krantz, 1982) by means of atomic decompositions and a geometric property of Euclidean balls. The product setting, however, introduces a multi‑parameter structure: the definition of (H^{p}{\mathrm{prod}}) involves a separate scale (\delta_{i}>0) for each coordinate, and the atoms are no longer simple balls but more intricate dyadic rectangles with mixed cancellation conditions. Because the geometric lemma used in Krantz’s proof (if (x\in B) and (y\in 2B) then (|x-y|\lesssim|y|)) fails for general open sets, a direct adaptation of the classical argument is impossible. Moreover, the Calderón–Zygmund theory on product domains is substantially more delicate (Journé’s counter‑example shows that the classical theory does not carry over verbatim).

To overcome these obstacles the author adopts a vector‑valued viewpoint. For a Banach space (X) (in practice a Hilbert or UMD space) one defines the vector‑valued Hardy space (H^{p}(\mathbb{R}^{d};X)) and uses the Lusin‑area (or (S)) function to characterize its norm. The key observation is that the product fractional integral \