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

Abstract

We define the mulati-parameter maximal function $\mathcal{M}$ as $$ \mathcal{M} f(x)=\sup _{0<h_1,h_2,\cdots,h_n<1} \frac{1}{h_1h_2\cdots h_n}\left|\int_0^{h_1}\cdots \int_0^{h_n} f(x-P(t_1,\cdots,t_n)) \mathrm{d}t_1\cdots \mathrm{d} t_n\right| $$ 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 $P(t_1,t_2,\cdots,t_n)$.