Lacunary maximal functions on homogeneous groups

Abstract

We observe that classical arguments of Ricci--Stein can be used to prove $L^p$ bounds for maximal functions associated to lacunary dilates of a fixed measure in the setting of homogenous groups. This recovers some recent results on averages over Korányi spheres and horizontal spherical averages of a type introduced by Nevo--Thangavelu. Moreover, the main theorem applies much more broadly and we explore its consequences through a variety of explicit examples.

20 pages, 1 table, no figures