Bounds for discrete multilinear spherical maximal functions in higher dimensions

We find the sharp range for boundedness of the discrete bilinear spherical maximal function for dimensions $d \geq 5$. That is, we show that this operator is bounded on $l^{p}(\mathbb{Z}^d)\times l^{q}(\mathbb{Z}^d) \to l^{r}(\mathbb{Z}^d)$ for $\frac{1}{p} + \frac{1}{q} \geq \frac{1}{r}$ and $r>\frac{d}{2d-2}$ and we show this range is sharp. Our approach mirrors that used by Jeong and Lee in the continuous setting. For dimensions $d=3,4$, our previous work, which used different techniques, still gives the best known bounds. We also prove analogous results for higher degree $k$, $\ell$-linear operators.