Let \(u(x, t)\) be a harmonic function in \(\mathbb R^{n}\times (0,\infty )\). The non-tangential maximal function \(u^*(x)= \sup _{|x-y|<t}|u(y,t)|\) and area integral \[S(u)(x)^2=\int _{|x-y|<t}|\nabla u(y,t)|^2 t^{1-n}\,dy\,dt\] are two fundamental tools in the theory of singular integrals and the related function spaces. Fefferman and Stein first showed that \(\|u^*\|_{L^p(\mathbb R^n)}\approx \|S(u)\|_{L^p(\mathbb R^n)}\), \(0<p\leq 1\), when \(u(x,t)\to 0\) as \(t\to \infty \).
The key objects in their proof are the following inequality \[\left |\left \{x\in \mathbb R^n\colon S(u)(x)>\lambda \right \}\right |\lesssim \left |\left \{x\in \mathbb R^n\colon u^*(x)>\lambda \right \}\right |+{1\over \lambda ^2}\int _0^\lambda s|\{x\in \mathbb R^n: u^*(x)>s\}|\,ds\] and the corresponding inequality of the same type but with \(u^*\) and \(S(u)\) interchanged.
We establish such an inequality in certain multiparameter settings, including the Shilov boundaries of tensor product domains in \(\mathbb C^{2n}\), and the Heisenberg groups \(\mathbb H^n\) with flag structure. Our technique bypasses the use of Fourier or the dependence of group structure. Direct applications include the (global) weak type endpoint estimate for multi-parameter Calderón–Zygmund operators and maximal function characterisation of multi-parameter Hardy spaces.
Ji Li, Fefferman–Stein type inequality, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2024.