Fractional integral on Hardy spaces on product domains
Yiyu Tang
TL;DR
The paper addresses extending Hardy–Littlewood–Sobolev inequalities to product Hardy spaces $H^p_{\mathrm{prod}}$ by employing a vector-valued singular integral framework. It defines the product fractional integral $I_{(\alpha,d)}$ with the kernel $|y_1\cdots y_d|^{-(1-\alpha/d)}$ and proves a Hardy–Littlewood–Sobolev bound $\|I_{(\alpha,d)} f\|_{H^q_{\mathrm{prod}}} \lesssim \|f\|_{H^p_{\mathrm{prod}}}$ for $0<p<q<\infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{d}$, using a Lusin area function $S$ and $X$-valued Hardy theory. It also establishes the $H^p_{\mathrm{prod}}$-boundedness of the iterated Hilbert transform and derives shorter proofs of several Hardy-type inequalities on product spaces, avoiding deep Calderón–Zygmund machinery. The approach leverages iterative one-dimensional arguments, the vector-valued HL-S inequality, and Fefferman-type criteria to address multi-parameter difficulties inherent in $H^p_{\mathrm{prod}}$. Overall, the work provides a robust, simpler pathway to multi-parameter Hardy space inequalities with potential broad impact in harmonic analysis on product domains.
Abstract
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.