Fefferman-Stein type inequality in multiparameter settings and applications

Abstract

Let $u(x,t)$ be a harmonic function in ${\mathbb{R}}^n\times (0,\mathrm{\infty })$. The non-tangential maximal function $u^{\ast }(x)=\underset{|x-y|<t}{sup}|u(y,t)|$ and area integral $$S(u)(x{)}^2={\int }_{|x-y|<t}|\mathrm{\nabla }u(y,t){|}^2{t}^{1-n}dydt$$ are two fundamental tools in the theory of singular integrals and the related function spaces. Fefferman and Stein first showed that $\parallel u^{\ast }{\parallel }_{L^p({\mathbb{R}}^n)}\approx \parallel S(u){\parallel }_{L^p({\mathbb{R}}^n)}$, $0<p\le 1$, when $u(x,t)\to 0$ as $t\to \mathrm{\infty }$.

The key objects in their proof are the following inequality $$\left|\{x\in {\mathbb{R}}^n:S(u)(x)>\lambda \}\right|\lesssim \left|\{x\in {\mathbb{R}}^n:u^{\ast }(x)>\lambda \}\right|+\frac{1}{{\lambda }^2}{\int }_0^{\lambda }s|\{x\in {\mathbb{R}}^n:u^{\ast }(x)>s\}|ds$$ and the corresponding inequality of the same type but with $u^{\ast }$ 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.

Reference:

• Ji Li, Fefferman–Stein type inequality, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2024.