$L^p-L^q$ 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.

18 pages

References in corpus (1)

• Lebesgue space estimates for spherical maximal functions on Heisenberg groups