We define a discrete version of the bilinear spherical maximal function, and show bilinear \(l^{p}(\mathbb {Z}^d)\times l^{q}(\mathbb {Z}^d) \rightarrow l^{r}(\mathbb {Z}^d)\) 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 \(l^{p}(\mathbb {Z}^d)\times l^{\infty }(\mathbb {Z}^d) \rightarrow l^{p}(\mathbb {Z}^d)\) 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.

我们定义双线性球面最大值函数的离散形式，并显示双线性\（l ^ {p}（\ mathbb {Z} ^ d）\ times l ^ {q}（\ mathbb {Z} ^ d）\ rightarrow l ^ {R}（\ mathbb {Z} ^ d）\）为界限\（d \ GE 3 \），\（\压裂{1} {p} + \压裂{1} {q} \ GE \压裂{ 1} {r} \），\（r> \ frac {d} {d-2} \）和\（p，q \ ge 1 \）。由于插值，关键估计为\（l ^ {p}（\ mathbb {Z} ^ d）\ times l ^ {\ infty}（\ mathbb {Z} ^ d）\ rightarrow l ^ {p}（ \ mathbb {Z} ^ d）\）边界，当\（d \ ge 3 \），\（p> \ frac {d} {d-2} \）时保持。我们论点的一个关键特征是使用了circle方法，与Cook的工作相比，它使我们可以将维与函数数量脱钩。