Lp- Lq estimates for the circular maximal operator on Heisenberg radial functions

Abstract

© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.Lp boundedness of the circular maximal function MH1 on the Heisenberg group H1 has received considerable attentions. While the problem still remains open, Lp boundedness of MH1 on Heisenberg radial functions was recently shown for p> 2 by Beltran et al. (Ann Sc Norm Super Pisa Cl Sci. https://doi.org/10.2422/2036-2145.202001-006, 2021). In this paper we extend their result considering the local maximal operator MH1 which is defined by taking supremum over 1 < t< 2. We prove Lp–Lq estimates for MH1 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.