Bounds for discrete multilinear spherical maximal 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 $d\ge 3$, $\frac{1}{p}+\frac{1}{q}\ge \frac{1}{r}$, $r>\frac{d}{d-2}$ and $p,q\ge 1$. 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 $d\ge 3$, $p>\frac{d}{d-2}$. 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.

typo corrected