The Multilinear Spherical Maximal Function in one dimension

Abstract

boundedness for the multilinear spherical maximal function in the largest possible open set of indices and we provide counterexamples that indicate the optimality of our results.

Sparse bounds for the bilinear spherical maximal function

“…In particular, in 𝑑 = 1, the bilinear circular maximal function behaves very differently from all its higher dimensional analogues. For instance, the boundedness result for  obtained in [28] only applies in 𝑑 ⩾ 2, and such Lebesgue space bounds in 𝑑 = 1 were only very recently obtained, independently, in [13,20] via somewhat similar approaches involving parametrizing pieces of the circle. For the single scale operators, the study of the 𝐿 𝑝 improving estimates for  𝑡 in 𝑑 = 1 was initiated by Oberlin [36] several decades ago but one still only has limited knowledge on the optimal range for such estimates (see [3,42]).…”