An endpoint estimate for the discrete spherical maximal function

Ionescu, Alexandru

doi:10.1090/S0002-9939-03-07207-1

An endpoint estimate for the discrete spherical maximal function
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by Alexandru D. Ionescu PDF

Proc. Amer. Math. Soc. 132 (2004), 1411-1417 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