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

Writing $\left({T}_{R}^{\lambda }f\right)\stackrel{^}{}\left(\xi \right)=\left(1-|\xi {|}^{2}/{R}^{2}{\right)}_{+}^{\lambda }\stackrel{^}{f}\left(\xi \right)$. E. Stein conjectured

 $\parallel {\left(\sum _{j}|{T}_{{R}_{j}}^{\lambda }{f}_{i}{|}^{2}\right)}^{1/2}{\parallel }_{p}\le C\parallel {\left(\sum _{j}|{f}_{j}{|}^{2}\right)}^{1/2}{\parallel }_{p}$

for $\lambda >0$, $\frac{4}{3}\le p\le 4$ and $C={C}_{\lambda ,p}$. We prove this conjecture. We prove also $f\left(x\right)={lim}_{j\to \infty }{T}_{{2}^{j}}^{\lambda }f\left(x\right)$ a.e. We only assume $\frac{4}{3+2\lambda }.

Écrivons $\left({T}_{R}^{\lambda }f\right)\stackrel{^}{}\left(\xi \right)=\left(1-|\xi {|}^{2}/{R}^{2}{\right)}_{+}^{\lambda }\stackrel{^}{f}\left(\xi \right)$. E. Stein a supposé que

 $\parallel {\left(\sum _{j}|{T}_{{R}_{j}}^{\lambda }{f}_{i}{|}^{2}\right)}^{1/2}{\parallel }_{p}\le C\parallel {\left(\sum _{j}|{f}_{j}{|}^{2}\right)}^{1/2}{\parallel }_{p}$

pour $\lambda >0$, $\frac{4}{3}\le p\le 4$ et $C={C}_{\lambda ,p}$. Nous démontrons cette conjecture. Nous démontrons aussi $f\left(x\right)={lim}_{j\to \infty }{T}_{{2}^{j}}^{\lambda }f\left(x\right)$ presque partout. Nous supposons seulement $\frac{4}{3+2\lambda }.

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author = {Cordoba, Antonio and Lopez-Melero, B.},
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, Volume 31 (1981) no. 3, pp. 147-152. doi : 10.5802/aif.842. http://archive.numdam.org/articles/10.5802/aif.842/

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[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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