Lacunary $\delta$-Discretised Spherical Maximal Operators

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