Abstract
Letf∈L p(ℝn),n≥2, be a radial function and letS Rf be the spherical partial sums operator. We prove that if\(\frac{{2n}}{{n + 1}}< p< \frac{{2n}}{{n - 1}}\) thenS Rf(x)→f(x) a.e. asR→∞. The result is false for\(p< \frac{{2n}}{{n + 1}}\) and\(p > \frac{{2n}}{{n + 1}}\).
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References
Herz, C.: On the mean inversion of the Fourier and Hankel transforms. Proc. Nat. Acad. Sci. U.S.A.40, 996–999 (1954).
Kenig, C., Tomas, P.: The weak behavior of the spherical means. Proc. Amer. Math. Soc.78, 48–50 (1980).
Chanillo, S.: The multiplier for the ball and radial functions. J. Funct. Anal.55, 18–24 (1984).
Fefferman, C.: The multiplier problem for the ball. Ann. Math.94, 330–336 (1971).
Hunt, R., Young, W. S.: A weighted norm inequality for Fourier series. Bull. Amer. Math. Soc.80, 274–277 (1974).
Stein, E. M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton: Univ. Press. 1971.
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Prestini, E. Almost everywhere convergence of the spherical partial sums for radial functions. Monatshefte für Mathematik 105, 207–216 (1988). https://doi.org/10.1007/BF01636929
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DOI: https://doi.org/10.1007/BF01636929