Weak-type 1-1 bound for a multi-parameter maximal function
Tom 177 / 2024
Streszczenie
We define the multi-parameter maximal function 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).