Lacunary $δ$-Discretised Spherical Maximal Operators

We study the lacunary analogue of the $δ$-discretised spherical maximal operators introduced by Hickman and Jančar, for $δ\in (0, 1/2)$, 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