# The asymptotic property of the hyperparabolic maximal function

The asymptotic property of the hyperparabolic maximal function We establish the strong Lp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document}-inequality for the maximal function over hyper-surface based on Euclidean ball supt>01|Bn|∫Bnf(x-tu,y-t2|u|2)du\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sup _{t>0}\frac{1}{|B^n|}\int _{B^n}f(x-tu,y-t^2|u|^2){\mathrm{d}}u$$\end{document}. We prove the Lp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document}-boundedness of the maximal function for p>1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p>1$$\end{document}. Specially, when p=2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p=2$$\end{document}, we prove the L2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2$$\end{document}-boundedness of the maximal function can be controlled by Cn78\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Cn^{\frac{7}{8}}$$\end{document}, where C is independent of n. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of Functional Analysis Springer Journals

# The asymptotic property of the hyperparabolic maximal function

, Volume 12 (4) – Oct 1, 2021
15 pages

/lp/springer-journals/the-asymptotic-property-of-the-hyperparabolic-maximal-function-DoRp6uB9cW
Publisher
Springer Journals
Copyright © Tusi Mathematical Research Group (TMRG) 2021
ISSN
2639-7390
eISSN
2008-8752
DOI
10.1007/s43034-021-00143-8
Publisher site
See Article on Publisher Site

### Abstract

We establish the strong Lp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document}-inequality for the maximal function over hyper-surface based on Euclidean ball supt>01|Bn|∫Bnf(x-tu,y-t2|u|2)du\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sup _{t>0}\frac{1}{|B^n|}\int _{B^n}f(x-tu,y-t^2|u|^2){\mathrm{d}}u$$\end{document}. We prove the Lp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document}-boundedness of the maximal function for p>1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p>1$$\end{document}. Specially, when p=2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p=2$$\end{document}, we prove the L2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2$$\end{document}-boundedness of the maximal function can be controlled by Cn78\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Cn^{\frac{7}{8}}$$\end{document}, where C is independent of n.

### Journal

Annals of Functional AnalysisSpringer Journals

Published: Oct 1, 2021

Keywords: Maximal function; Dimension-free estimate; Fourier multiplier; 42B20; 42B35