𝐿^{𝑝} bound on the strong spherical maximal function

Lee, Juyoung; Lee, Sanghyuk; Oh, Sewook

doi:10.1090/proc/17108

$L^p$ bound on the strong spherical maximal function
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by Juyoung Lee, Sanghyuk Lee and Sewook Oh;

Proc. Amer. Math. Soc. 153 (2025), 1155-1167

DOI: https://doi.org/10.1090/proc/17108

Published electronically: December 17, 2024

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Abstract:

In this note we show that the strong spherical maximal function in $\mathbb R^d$ is bounded on $L^p$ if $p>2(d+1)/(d-1)$ for $d\ge 3$.

References

Richard J. Bagby, A note on the strong maximal function, Proc. Amer. Math. Soc. 88 (1983), no. 4, 648–650. MR 702293, DOI 10.1090/S0002-9939-1983-0702293-X
David Beltran, Jonathan Hickman, and Christopher D. Sogge, Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds, Anal. PDE 13 (2020), no. 2, 403–433. MR 4078231, DOI 10.2140/apde.2020.13.403
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
Jean Bourgain and Ciprian Demeter, The proof of the $l^2$ decoupling conjecture, Ann. of Math. (2) 182 (2015), no. 1, 351–389. MR 3374964, DOI 10.4007/annals.2015.182.1.9
M. Chen, S. Guo, and T. Yang, A multi-parameter cinematic curvature, arXiv:2306.01606, 2023.
A. Córdoba and R. Fefferman, On differentiation of integrals, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 6, 2211–2213. MR 476977, DOI 10.1073/pnas.74.6.2211
Norberto Angel Fava, Weak type inequalities for product operators, Studia Math. 42 (1972), 271–288. MR 308364, DOI 10.4064/sm-42-3-271-288
N. A. Fava, E. A. Gatto, and C. Gutiérrez, On the strong maximal function and Zygmund’s class $L(\textrm {log}^{+}L)^{n}$, Studia Math. 69 (1980/81), no. 2, 155–158. MR 604348, DOI 10.4064/sm-69-2-155-158
Shaoming Guo, Changkeun Oh, Ruixiang Zhang, and Pavel Zorin-Kranich, Decoupling inequalities for quadratic forms, Duke Math. J. 172 (2023), no. 2, 387–445. MR 4541334, DOI 10.1215/00127094-2022-0033
Juyoung Lee, Sanghyuk Lee, and Sewook Oh, The elliptic maximal function, J. Funct. Anal. 288 (2025), no. 1, Paper No. 110693, 31. MR 4800578, DOI 10.1016/j.jfa.2024.110693
J. M. Marstrand, Packing circles in the plane, Proc. London Math. Soc. (3) 55 (1987), no. 1, 37–58. MR 887283, DOI 10.1112/plms/s3-55.1.37
A. Nagel, E. M. Stein, and S. Wainger, Differentiation in lacunary directions, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), no. 3, 1060–1062. MR 466470, DOI 10.1073/pnas.75.3.1060
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
Elias M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993.
Terence Tao, Ana Vargas, and Luis Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998), no. 4, 967–1000. MR 1625056, DOI 10.1090/S0894-0347-98-00278-1