Bounds for discrete multilinear spherical maximal functions in higher dimensions

Abstract

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 ( Z d ) × l q ( Z d ) → l r ( 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, ℓ-linear operators.