On multilinear maximal operators along homogeneous curves

Lars Becker and Ben Krause

Vol. 342 (2026), No. 2, 207–216

DOI: 10.2140/pjm.2026.342.207

Abstract

Suppose that

$$\overrightarrow{\gamma }(t):=({\gamma }_1(t),\dots <!--FUNCTION\; APPLICATION-->,{\gamma }_n(t))=(a_1{t}^{d_1},\dots <!--FUNCTION\; APPLICATION-->,a_n{t}^{d_n}),1\le d_1<\cdots <d_n\in \mathbb{Z},a_i\ne 0$$

is a homogeneous polynomial curve. We prove that whenever $p_1,\dots <!--FUNCTION\; APPLICATION-->,p_n>1$ and $1∕p={\sum <!--FUNCTION\; APPLICATION-->}_{j=1}^{n}1∕p_j\le 1$, there exists an absolute constant $0<C=C_{p_1,\dots <!--FUNCTION\; APPLICATION-->,p_n}<\infty$ such that

$$\parallel \underset{r>0}{\sup <!--FUNCTION\; APPLICATION-->}\frac{1}{r}{\int \; <!--nolimits-->}_0^r\prod _{i=1}^n|f_i(x-{\gamma }_i(t))|dt{\parallel }_{L^p(ℝ)}\le C\cdot \prod _{i=1}^n\parallel f_j{\parallel }_{L^{p_j}(ℝ)}.$$

Our main tool is a smoothing estimate, adapted from work of Kosz, Mirek, Peluse, Wan, and Wright.

Keywords

multilinear maximal operators, Littlewood–Paley theory, smoothing inequalities

Mathematical Subject Classification

Primary: 42A99

Milestones

Received: 14 August 2025

Revised: 2 December 2025

Accepted: 8 February 2026

Published: 23 April 2026