Endpoint estimates for the maximal operator associated to spherical partial sums on radial functions
HTML articles powered by AMS MathViewer
- by Elena Romera and Fernando Soria PDF
- Proc. Amer. Math. Soc. 111 (1991), 1015-1022 Request permission
Abstract:
Let $Tf(x) = {\sup _{R > 0}}\left | {{S_R}f(x)} \right |$ where ${S_R}$ is the spherical partial sum operator. We show that $T$ is bounded from the Lorentz space ${L_{{p_i},1}}({{\mathbf {R}}^n})$ into ${L_{{p_i},\infty }}({{\mathbf {R}}^n}),i = 0,1$ when acting on radial functions and where ${p_0} = \tfrac {{2n}}{{n + 1}},{p_1} = \tfrac {{2n}}{{n - 1}}$.References
- Sagun Chanillo, The multiplier for the ball and radial functions, J. Funct. Anal. 55 (1984), no. 1, 18–24. MR 733030, DOI 10.1016/0022-1236(84)90015-6 L. Colzani, Convergence of expansions in Legendre polynomials, preprint.
- A. Córdoba, The disc multiplier, Duke Math. J. 58 (1989), no. 1, 21–29. MR 1016411, DOI 10.1215/S0012-7094-89-05802-X
- Charles Fefferman, The multiplier problem for the ball, Ann. of Math. (2) 94 (1971), 330–336. MR 296602, DOI 10.2307/1970864
- Carl S. Herz, On the mean inversion of Fourier and Hankel transforms, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 996–999. MR 63477, DOI 10.1073/pnas.40.10.996
- Yūichi Kanjin, Convergence and divergence almost everywhere of spherical means for radial functions, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1063–1069. MR 954984, DOI 10.1090/S0002-9939-1988-0954984-8
- Carlos E. Kenig and Peter A. Tomas, The weak behavior of spherical means, Proc. Amer. Math. Soc. 78 (1980), no. 1, 48–50. MR 548082, DOI 10.1090/S0002-9939-1980-0548082-8
- Gerd Mockenhaupt, On radial weights for the spherical summation operator, J. Funct. Anal. 91 (1990), no. 1, 174–181. MR 1054117, DOI 10.1016/0022-1236(90)90051-L
- Elena Prestini, Almost everywhere convergence of the spherical partial sums for radial functions, Monatsh. Math. 105 (1988), no. 3, 207–216. MR 939943, DOI 10.1007/BF01636929
- 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
- G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 111 (1991), 1015-1022
- MSC: Primary 42B25
- DOI: https://doi.org/10.1090/S0002-9939-1991-1068130-6
- MathSciNet review: 1068130