Spherical summation: a problem of E.M. Stein

Cordoba, Antonio; Lopez-Melero, B.

Cordoba, Antonio ; Lopez-Melero, B.

Annales de l'Institut Fourier, Tome 31 (1981), p. 147-152 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé
Abstract

Écrivons $(T_{R}^{λ} f) \hat{} (ξ) = (1 - | ξ |^{2} / R^{2})_{+}^{λ} \hat{f} (ξ)$ . E. Stein a supposé que

∥ {(\sum_{j} | T_{R_{j}}^{λ} f_{i} |^{2})}^{1 / 2} ∥_{p} \leq C ∥ {(\sum_{j} | f_{j} |^{2})}^{1 / 2} ∥_{p}

pour $λ > 0$ , $\frac{4}{3} \leq p \leq 4$ et $C = C_{λ, p}$ . Nous démontrons cette conjecture. Nous démontrons aussi $f (x) = {lim}_{j \to \infty} T_{2^{j}}^{λ} f (x)$ presque partout. Nous supposons seulement $\frac{4}{3 + 2 λ} < p < \frac{4}{1 - 2 λ}$ .

Writing $(T_{R}^{λ} f) \hat{} (ξ) = (1 - | ξ |^{2} / R^{2})_{+}^{λ} \hat{f} (ξ)$ . E. Stein conjectured

∥ {(\sum_{j} | T_{R_{j}}^{λ} f_{i} |^{2})}^{1 / 2} ∥_{p} \leq C ∥ {(\sum_{j} | f_{j} |^{2})}^{1 / 2} ∥_{p}

for $λ > 0$ , $\frac{4}{3} \leq p \leq 4$ and $C = C_{λ, p}$ . We prove this conjecture. We prove also $f (x) = {lim}_{j \to \infty} T_{2^{j}}^{λ} f (x)$ a.e. We only assume $\frac{4}{3 + 2 λ} < p < \frac{4}{1 - 2 λ}$ .

Publié le : 1981-01-01

MR 83g:42008 | Zbl 0464.42006

DOI : https://doi.org/10.5802/aif.842

@article{AIF_1981__31_3_147_0,
     author = {Cordoba, Antonio and Lopez-Melero, B.},
     title = {Spherical summation: a problem of E.M. Stein},
     journal = {Annales de l'Institut Fourier},
     volume = {31},
     year = {1981},
     pages = {147-152},
     doi = {10.5802/aif.842},
     mrnumber = {83g:42008},
     zbl = {0464.42006},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1981__31_3_147_0}
}

Cordoba, Antonio; Lopez-Melero, B. Spherical summation: a problem of E.M. Stein. Annales de l'Institut Fourier, Tome 31 (1981) pp. 147-152. doi : 10.5802/aif.842. http://gdmltest.u-ga.fr/item/AIF_1981__31_3_147_0/

Bibliographie

[1] E.M. Stein, Some problems in harmonic analysis, Proc. Sym. Pure Math., Volume XXXV, Part. 1, (1979). | MR 80m:42027 | Zbl 0445.42006

[2] L. Carleson and P. Sjölin, Oscillatory integrals and a multiplier problem for the disc, Studia Math., 44 (1972), 288-299. | MR 50 #14052 | Zbl 0215.18303

[3] A. Cordoba, The Kakeya maximal function and the spherical summation multipliers, Am. J. of Math., Vol. 99, n° 1, (1977), 1-22. | MR 56 #6259 | Zbl 0384.42008

[4] A. Cordoba, Some remarks on the Littlewood-Paley theory, To appear, Rendiconti di Circolo Mat. di Palermo. | Zbl 0506.42022

[5] C. Fefferman, The multiplier problem for the ball, Annals of Math., 94 (1971), 330-336. | MR 45 #5661 | Zbl 0234.42009