Spherical summation: a problem of E.M. Stein
Annales de l'Institut Fourier, Tome 31 (1981) no. 3, pp. 147-152.

Écrivons (T R λ f) ^(ξ)=(1-|ξ| 2 /R 2 ) + λ f ^(ξ). E. Stein a supposé que

j | T R j λ f i | 2 1 / 2 p C j | f j | 2 1 / 2 p

pour λ>0, 4 3p4 et C=C λ,p . Nous démontrons cette conjecture. Nous démontrons aussi f(x)=lim j T 2 j λ f(x) presque partout. Nous supposons seulement 4 3+2λ<p<4 1-2λ.

Writing (T R λ f) ^(ξ)=(1-|ξ| 2 /R 2 ) + λ f ^(ξ). E. Stein conjectured

j | T R j λ f i | 2 1 / 2 p C j | f j | 2 1 / 2 p

for λ>0, 4 3p4 and C=C λ,p . We prove this conjecture. We prove also f(x)=lim j T 2 j λ f(x) a.e. We only assume 4 3+2λ<p<4 1-2λ.

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     title = {Spherical summation: a problem of {E.M.} {Stein}},
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Cordoba, Antonio; Lopez-Melero, B. Spherical summation: a problem of E.M. Stein. Annales de l'Institut Fourier, Tome 31 (1981) no. 3, pp. 147-152. doi : 10.5802/aif.842. http://archive.numdam.org/articles/10.5802/aif.842/

[1] E.M. Stein, Some problems in harmonic analysis, Proc. Sym. Pure Math., Volume XXXV, Part. 1, (1979). | MR | Zbl

[2] L. Carleson and P. Sjölin, Oscillatory integrals and a multiplier problem for the disc, Studia Math., 44 (1972), 288-299. | MR | Zbl

[3] A. Cordoba, The Kakeya maximal function and the spherical summation multipliers, Am. J. of Math., Vol. 99, n° 1, (1977), 1-22. | MR | Zbl

[4] A. Cordoba, Some remarks on the Littlewood-Paley theory, To appear, Rendiconti di Circolo Mat. di Palermo. | Zbl

[5] C. Fefferman, The multiplier problem for the ball, Annals of Math., 94 (1971), 330-336. | MR | Zbl

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