$L^p-L^q$ estimates for the circular maximal operators on Heisenberg radial functions

Abstract

L p boundedness of the circular maximal function M H 1 on the Heisenberg group H 1 has received considerable attentions. While the problem still remains open, L p boundedness of M H 1 on Heisenberg radial functions was recently shown for p ą 2 by Beltran, Guo, Hickman, and Seeger [2].In this paper we extend their result considering the local maximal operator M H 1 which is defined by taking supremum over 1 ă t ă 2. We prove L p -L q estimates for M H 1 on Heisenberg radial functions on the optimal range of p, q modulo the borderline cases. Our argument also provides a simpler proof of the aforementioned result due to Beltran et al.

Spherical maximal operators on Heisenberg groups: Restricted dilation sets

“…Further examples of quasi-Assouad regular sets include convex sequences, self-similar sets with β = γ (such as Cantor sets) and many more; see [23, §6]. Note that we do not cover the case n = 1; indeed it is currently unknown whether the full circular maximal operator on the Heisenberg group H 1 is bounded on any L p for p < ∞ and L p -improving estimates are even more elusive (see [3,14] for results on Heisenberg-radial functions).…”