Li: Fefferman-Stein type inequality in multiparameter settings and applications

Let $u(x,t)$ be a harmonic function in ${\mathbb{R}}^n\times (0,\mathrm{\infty })$. The non-tangential maximal function \(u^*(x)= \sup _{|x-y|

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.