On Multi-linear Maximal Operators Along Homogeneous Curves

Abstract

Suppose that \[ \vecγ(t) := (γ_1(t),\dots,γ_n(t)) = (a_1 t^{d_1},\dots,a_n t^{d_n}), \; \; \; 1\leq d_1 < \dots < d_n, \ a_i \neq 0\] is a homogeneous polynomial curve. We prove that whenever $p_1,\dots ,p_n>1$ and $\frac{1}{p}={\sum }_{j=1}^n\frac{1}{p_j}\le 1$, there exists an absolute constant $0<C=C_{p_1,\dots ,p_n;{\vec{\hat{I}}}^3}<\infty$ so that \[ \| \sup_{r > 0} \ \frac{1}{r} \int_{0}^r \prod_{i=1}^n |f_i(x-γ_i(t))| \ dt \|_{L^p(\mathbb{R})} \leq C \cdot \prod_{i=1}^n \| f_j \|_{L^{p_j}(\mathbb{R})}. \] Our main tool is a smoothing estimate, adapted from work of Kosz-Mirek-Peluse-Wright.