Lebesgue bounds for multilinear spherical and lacunary maximal averages

TL;DR

This paper establishes Lebesgue space bounds for multilinear spherical averaging operators and lacunary maximal averages in higher dimensions, expanding understanding of their boundedness properties.

Contribution

It provides new Lebesgue space bounds for multilinear spherical averages and lacunary maximal averages, including endpoint estimates in higher dimensions.

Findings

Boundedness of $\\mathcal A^n$ from $L^{p_1} imes \\cdots imes L^{p_n}$ to $L^r$

Mapping of $L^1 imes \\cdots imes L^1$ to $L^1$ for spherical averages

Optimal bounds for lacunary maximal spherical averages

Abstract

We establish $L^{p_1}(\mathbb R^d) \times \cdots \times L^{p_n}(\mathbb R^d) \rightarrow L^r(\mathbb R^d)$ bounds for spherical averaging operators $\mathcal A^n$ in dimensions $d \geq 2$ for indices $1\le p_1,\dots , p_n\le \infty$ and $\frac{1}{p_1}+\cdots +\frac{1}{p_n}=\frac{1}{r}$. We obtain this result by first showing that $\mathcal A^n$ maps $L^1 \times \cdots \times L^1 \rightarrow L^1$. We also obtain similar estimates for lacunary maximal spherical averages in the largest possible open region of indices.