Weak type 1-1 bound of multi-parameter maximal function

Abstract

We define the mulati-parameter maximal function $\mathcal{M}$ as $$\mathcal{M}f(x)=\underset{0<h_1,h_2,\cdots ,h_n<1}{\sup }\frac{1}{h_1{h}_2\cdots h_n}\left|{\int }_0^{h_1}\cdots {\int }_0^{h_n}f(x-P(t_1,\cdots ,t_n))\mathrm{d}{t}_1\cdots \mathrm{d}{t}_n\right|$$ where $P(t_1,t_2,\cdots ,t_n)$ is a real-valued multi-parameter polynomial of real variables $t_1,t_2,\cdots ,t_n$. Then, we prove that $\mathcal{M}$ is of weak-type 1-1 with a bound that depends only on the coefficients of $P(t_1,t_2,\cdots ,t_n)$.