On Multi-linear Maximal Operators Along Homogeneous Curves

TL;DR

This paper establishes boundedness of multi-linear maximal operators along homogeneous polynomial curves, extending harmonic analysis tools to a broader class of geometric configurations.

Contribution

It proves new boundedness results for multi-linear maximal operators along homogeneous curves, utilizing a novel smoothing estimate adapted from recent harmonic analysis research.

Findings

Boundedness of multi-linear maximal operators along homogeneous polynomial curves.

Existence of a uniform constant controlling the operator norm.

Extension of harmonic analysis techniques to polynomial curve settings.

Abstract

Suppose that \[ \vec{\gamma}(t) := (\gamma_1(t),\dots,\gamma_n(t)) = (a_1 t^{d_1},\dots,a_n t^{d_n}), \; \; \; 1\leq d_1 < \dots < d_n, \ a_i \neq 0\] is a homogeneous polynomial curve. We prove that whenever $p_1,\dots,p_n > 1$ and $\frac{1}{p} = \sum_{j=1}^n \frac{1}{p_j} \leq 1$, there exists an absolute constant $0 < C = C_{p_1,\dots,p_n;\vec{\gamma}} < \infty$ so that \[ \| \sup_{r > 0} \ \frac{1}{r} \int_{0}^r \prod_{i=1}^n |f_i(x-\gamma_i(t))| \ dt \|_{L^p(\mathbb{R})} \leq C \cdot \prod_{i=1}^n \| f_j \|_{L^{p_j}(\mathbb{R})}. \] Our main tool is a smoothing estimate, adapted from work of Kosz-Mirek-Peluse-Wright.