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.
Categories: math.CA math.NT