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

Abstract

$L^p$ boundedness of the circular maximal function $\mathcal M_{\mathbb{H}^1}$ on the Heisenberg group $\mathbb{H}^1$ has received considerable attentions. While the problem still remains open, $L^p$ boundedness of $\mathcal M_{\mathbb{H}^1}$ on Heisenberg radial functions was recently shown for $p>2$ by Beltran, Guo, Hickman, and Seeger [2]. In this paper we extend their result considering the local maximal operator $M_{\mathbb{H}^1}$ which is defined by taking supremum over $1<t<2$. We prove $L^p-L^q$ estimates for $M_{\mathbb{H}^1}$ 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.