An endpoint estimate for the discrete spherical maximal function
HTML articles powered by AMS MathViewer
- by Alexandru D. Ionescu
- Proc. Amer. Math. Soc. 132 (2004), 1411-1417
- DOI: https://doi.org/10.1090/S0002-9939-03-07207-1
- Published electronically: August 20, 2003
- PDF | Request permission
Abstract:
We prove that the discrete spherical maximal function extends to a bounded operator from $L^{d/(d-2),1}(\mathbb {Z}^d)$ to $L^{d/(d-2),\infty }(\mathbb {Z}^d)$ in dimensions $d\geq 5$. This is an endpoint estimate for a recent theorem of Magyar, Stein and Wainger.References
- Jean Bourgain, Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 10, 499–502 (French, with English summary). MR 812567
- J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Analyse Math. 47 (1986), 69–85. MR 874045, DOI 10.1007/BF02792533
- Anthony Carbery, Andreas Seeger, Stephen Wainger, and James Wright, Classes of singular integral operators along variable lines, J. Geom. Anal. 9 (1999), no. 4, 583–605. MR 1757580, DOI 10.1007/BF02921974
- Akos Magyar, $L^p$-bounds for spherical maximal operators on $\mathbf Z^n$, Rev. Mat. Iberoamericana 13 (1997), no. 2, 307–317. MR 1617657, DOI 10.4171/RMI/222
- A. Magyar, E. M. Stein, and S. Wainger, Discrete analogues in harmonic analysis: spherical averages, Ann. of Math. (2) 155 (2002), no. 1, 189–208. MR 1888798, DOI 10.2307/3062154
- A. Seeger, T. Tao and J. Wright, Endpoint mapping properties of spherical maximal operators, J. Inst. Math. Jussieu 2 (2003), 109–144.
- Elias M. Stein, Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2174–2175. MR 420116, DOI 10.1073/pnas.73.7.2174
Bibliographic Information
- Alexandru D. Ionescu
- Affiliation: Department of Mathematics, University of Wisconsin at Madison, Madison, Wisconsin 53706
- MR Author ID: 660963
- Email: ionescu@math.wisc.edu
- Received by editor(s): November 11, 2002
- Received by editor(s) in revised form: December 31, 2002
- Published electronically: August 20, 2003
- Additional Notes: The author was supported in part by the National Science Foundation under NSF Grant No. 0100021
- Communicated by: Andreas Seeger
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 1411-1417
- MSC (2000): Primary 42B25
- DOI: https://doi.org/10.1090/S0002-9939-03-07207-1
- MathSciNet review: 2053347