Bounds for discrete multilinear spherical maximal functions in higher dimensions

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