$L^p-L^q$ 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2021
- DOI:
- 10.48550/arXiv.2107.01089
- arXiv:
- arXiv:2107.01089
- Bibcode:
- 2021arXiv210701089L
- Keywords:
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- Mathematics - Classical Analysis and ODEs;
- 42B25;
- 22E25;
- 35S30
- E-Print:
- 18 pages