Abstract
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.
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Acknowledgements
T. C. Anderson was supported in part by NSF DMS-1502464 and NSF DMS-195440. E. A. Palsson was supported in part by Simons Foundation Grant #360560.
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Anderson, T.C., Palsson, E.A. Bounds for discrete multilinear spherical maximal functions. Collect. Math. 73, 75–87 (2022). https://doi.org/10.1007/s13348-020-00308-z
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DOI: https://doi.org/10.1007/s13348-020-00308-z