A characterization of maximal operators associated with radial fourier multipliers

Kim, Jongchon

doi:10.1090/proc/13445

A characterization of maximal operators associated with radial fourier multipliers
HTML articles powered by AMS MathViewer

by Jongchon Kim PDF

Proc. Amer. Math. Soc. 145 (2017), 1077-1085 Request permission

Abstract:

We give a simple necessary and sufficient condition for maximal operators associated with radial Fourier multipliers to be bounded on $L^p_{rad}$ and $L^p$ for certain $p$ greater than $2$. The range of exponents obtained for the $L^p_{rad}$ characterization is optimal for the given condition. The $L^p$ characterization is derived from an inequality of Heo, Nazarov, and Seeger regarding a characterization of radial Fourier multipliers.

References

Anthony Carbery, José L. Rubio de Francia, and Luis Vega, Almost everywhere summability of Fourier integrals, J. London Math. Soc. (2) 38 (1988), no. 3, 513–524. MR 972135, DOI 10.1112/jlms/s2-38.3.513
Leonardo Colzani, Giancarlo Travaglini, and Marco Vignati, Bochner-Riesz means of functions in weak-$L^p$, Monatsh. Math. 115 (1993), no. 1-2, 35–45. MR 1223243, DOI 10.1007/BF01311209
Gustavo Garrigós and Andreas Seeger, Characterizations of Hankel multipliers, Math. Ann. 342 (2008), no. 1, 31–68. MR 2415314, DOI 10.1007/s00208-008-0221-8
Gustavo Garrigós and Andreas Seeger, A note on maximal operators associated with Hankel multipliers, Rev. Un. Mat. Argentina 50 (2009), no. 2, 137–148. MR 2656531
Yaryong Heo, Fëdor Nazarov, and Andreas Seeger, On radial and conical Fourier multipliers, J. Geom. Anal. 21 (2011), no. 1, 96–117. MR 2755678, DOI 10.1007/s12220-010-9171-y
Yaryong Heo, Fëdor Nazarov, and Andreas Seeger, Radial Fourier multipliers in high dimensions, Acta Math. 206 (2011), no. 1, 55–92. MR 2784663, DOI 10.1007/s11511-011-0059-x
Yūichi Kanjin, Convergence almost everywhere of Bochner-Riesz means for radial functions, Ann. Sci. Kanazawa Univ. 25 (1988), 11–15. MR 964084
Sanghyuk Lee, Keith M. Rogers, and Andreas Seeger, Square functions and maximal operators associated with radial Fourier multipliers, Advances in analysis: the legacy of Elias M. Stein, Princeton Math. Ser., vol. 50, Princeton Univ. Press, Princeton, NJ, 2014, pp. 273–302. MR 3329855
Sanghyuk Lee and Andreas Seeger, On radial Fourier multipliers and almost everywhere convergence, J. Lond. Math. Soc. (2) 91 (2015), no. 1, 105–126. MR 3338611, DOI 10.1112/jlms/jdu066
Richard O’Neil, Convolution operators and $L(p,\,q)$ spaces, Duke Math. J. 30 (1963), 129–142. MR 146673
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
Terence Tao, The weak-type endpoint Bochner-Riesz conjecture and related topics, Indiana Univ. Math. J. 47 (1998), no. 3, 1097–1124. MR 1665753, DOI 10.1512/iumj.1998.47.1544