Lacunary $\hat{I}´$-Discretised Spherical Maximal Operators

Abstract

We study the lacunary analogue of the $\hat{I}´$-discretised spherical maximal operators introduced by Hickman and Jančar, for $\hat{I}´\in (0,1/2)$, and establish the boundedness on $L^p$ for all $1<p<\infty$, along with the endpoint weak-type estimate $H^1\to L^{1,\infty }$. We also prove the corresponding $L^p$ boundedness for the multi-parameter variant. The constants in these bounds are uniform in $\hat{I}´$, and thus, by taking the limit $\hat{I}´\to 0^{+}$, our results recover the classical boundedness of the lacunary spherical maximal function.

Ji Li and Chong-Wei Liang would like to thank Professor Sanghyuk Lee for discussions on the spherical average