We find the sharp range for boundedness of the discrete bilinear spherical maximal function for dimensions $d⩾5$. 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 $1/p+1/q⩾1/r$ and $r>d/(d-2)$ 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.

中文翻译：

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高维离散多线性球面极大函数的界限

我们找到了离散双线性球面极大函数的有界的尖锐范围 $d⩾5$. 也就是说，我们证明这个算子是有界的${升}^{磷}({\mathbb{Z}}^d)\times {升}^q({\mathbb{Z}}^d)\to {升}^r({\mathbb{Z}}^d)$ 为了 $1/磷+1/q⩾1/r$ 和 $r>d/(d-2)$我们表明这个范围是尖锐的。我们的方法反映了 Jeong 和 Lee 在连续设置中使用的方法。对于尺寸$d=3,4$，我们之前的工作使用了不同的技术，仍然给出了最知名的界限。我们还证明了更高学位的类似结果$克$, $\ell$-线性运算符。