Bounds for discrete multilinear spherical maximal functions

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