$L^pL^q$ 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 $L^pL^q$ 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.01089
 Bibcode:
 2021arXiv210701089L
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 42B25;
 22E25;
 35S30
 EPrint:
 18 pages