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Abstract

We define the multi-parameter maximal function $\mathcal M$ as $$ \mathcal M f(x)=\sup_{0 \lt h_1,\ldots ,h_n \lt 1} \frac{1}{h_1\cdots h_n}\Big|\int _0^{h_1}\cdots \int _0^{h_n} f(x-P(t_1,\ldots ,t_n)) \,\mathrm{d}t_1\cdots \mathrm{d} t_n\Big| $$ where $P(t_1,\ldots ,t_n)$ is a real-valued multi-parameter polynomial of real variables $t_1,\ldots ,t_n$. We prove that $\mathcal M$ is weak-type 1-1 with a bound that depends only on the coefficients of $P(t_1,\ldots ,t_n)$.