Spherical summation: a problem of E.M. Stein

Cordoba, Antonio; Lopez-Melero, B.

Annales de l'Institut Fourier, Tome 31 (1981) no. 3, pp. 147-152

Writing $(T_R^{\lambda }f)\widehat{}\left(\xi \right)=(1-|\xi {|}^2/R^2{)}_{+}^{\lambda }\widehat{f}\left(\xi \right)$. 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 $\frac{4}{3+2\lambda }<p<\frac{4}{1-2\lambda }$.

Écrivons $(T_R^{\lambda }f)\widehat{}\left(\xi \right)=(1-|\xi {|}^2/R^2{)}_{+}^{\lambda }\widehat{f}\left(\xi \right)$. 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 $\frac{4}{3+2\lambda }<p<\frac{4}{1-2\lambda }$.

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