Bounds for discrete multilinear spherical maximal functions

TL;DR

This paper establishes bounds for a discrete bilinear spherical maximal function in higher dimensions, utilizing the circle method to decouple dimension from the number of functions involved.

Contribution

It introduces a discrete bilinear spherical maximal function and proves new bounds using the circle method, extending previous work by Cook.

Findings

Proved $l^{p} imes l^{q} o l^{r}$ bounds for the discrete bilinear spherical maximal function.

Established key estimates via interpolation, especially the $l^{p} imes l^{} o l^{p}$ bound.

Demonstrated the effectiveness of the circle method in decoupling dimension from the number of functions.

Abstract

We define a discrete version of the bilinear spherical maximal function, and show bilinear $l^{p}(\mathbb{Z}^d)\times l^{q}(\mathbb{Z}^d) \to l^{r}(\mathbb{Z}^d)$ bounds for $d \geq 3$, $\frac{1}{p} + \frac{1}{q} \geq \frac{1}{r}$, $r>\frac{d}{d-2}$ and $p,q\geq 1$. Due to interpolation, the key estimate is an $l^{p}(\mathbb{Z}^d)\times l^{\infty}(\mathbb{Z}^d) \to l^{p}(\mathbb{Z}^d)$ bound, which holds when $d \geq 3$, $p>\frac{d}{d-2}$. A key feature of our argument is the use of the circle method which allows us to decouple the dimension from the number of functions compared to the work of Cook.