A remark on the maximal operator for radial measures
HTML articles powered by AMS MathViewer
- by Adrián Infante PDF
- Proc. Amer. Math. Soc. 139 (2011), 2899-2902 Request permission
Abstract:
The purpose of this paper is to prove that there exist measures $d\mu (x)=\gamma (x)dx$, with $\gamma (x)=\gamma _{0}(|x|)$ and $\gamma _{0}$ being a decreasing and positive function, such that the Hardy-Littlewood maximal operator, $\mathcal {M}_{\mu }$, associated to the measure $\mu$ does not map $L^{p}_{\mu }(\mathbb {R}^{n})$ into weak $L^{p}_{\mu }(\mathbb {R}^{n})$, for every $p<\infty$. This result answers an open question of P. Sjögren and F. Soria.References
- Liliana Forzani, Roberto Scotto, Peter Sjögren, and Wilfredo Urbina, On the $L^p$ boundedness of the non-centered Gaussian Hardy-Littlewood maximal function, Proc. Amer. Math. Soc. 130 (2002), no. 1, 73–79. MR 1855622, DOI 10.1090/S0002-9939-01-06156-1
- Peter Sjögren, A remark on the maximal function for measures in $\textbf {R}^{n}$, Amer. J. Math. 105 (1983), no. 5, 1231–1233. MR 714775, DOI 10.2307/2374340
- Peter Sjögren and Fernando Soria, Sharp estimates for the non-centered maximal operator associated to Gaussian and other radial measures, Adv. Math. 181 (2004), no. 2, 251–275. MR 2026859, DOI 10.1016/S0001-8708(03)00050-1
- Ana M. Vargas, On the maximal function for rotation invariant measures in $\textbf {R}^n$, Studia Math. 110 (1994), no. 1, 9–17. MR 1279371, DOI 10.4064/sm-110-1-9-17
Additional Information
- Adrián Infante
- Affiliation: Department of Mathematics, Universidad Simón Bolívar, Caracas, Venezuela
- Email: ainfante@usb.ve
- Received by editor(s): April 6, 2010
- Received by editor(s) in revised form: August 4, 2010
- Published electronically: January 14, 2011
- Communicated by: Michael T. Lacey
- © Copyright 2011 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 139 (2011), 2899-2902
- MSC (2000): Primary 42B25
- DOI: https://doi.org/10.1090/S0002-9939-2011-10727-5
- MathSciNet review: 2801630