Bounds for discrete multilinear spherical maximal functions in higher dimensions

TL;DR

This paper establishes the precise boundedness conditions for the discrete bilinear spherical maximal function in higher dimensions, extending to higher degrees and linear operators, and confirms the sharpness of these bounds.

Contribution

It determines the exact range of boundedness for the discrete bilinear spherical maximal function in dimensions d ≥ 5 and extends results to higher degrees and linear operators.

Findings

Boundedness range for d ≥ 5 is sharp and characterized by specific p, q, r conditions.

Results extend to higher degree k and ℓ-linear operators.

Previous bounds for dimensions 3 and 4 remain the best known.

Abstract

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.