Spherical summation : a problem of E.M. Stein

Cordoba, Antonio, and Lopez-Melero, B.. "Spherical summation : a problem of E.M. Stein." Annales de l'institut Fourier 31.3 (1981): 147-152. <http://eudml.org/doc/74501>.

@article{Cordoba1981,
abstract = {Writing $(T^\lambda _R f)\hat\{\}(\xi ) = (1- |\xi |^2/R^2)^\lambda _+ \hat\{f\}(\xi )$. E. Stein conjectured\begin\{\}\Big \Vert \Big (\sum \_j |T^\lambda \_\{R\_j\} f\_i|^2\Big )^\{1/2\}\Big \Vert \_p \le C\Big \Vert \Big (\sum \_j |f\_j|^2\Big )^\{1/2\}\Big \Vert \_p\end\{\}for $\lambda &gt;0$, $\{4\over 3\} \le p \le 4$ and $C= C_\{\lambda ,p\}$. We prove this conjecture. We prove also $f(x) = \lim _\{j\rightarrow \infty \} T^\lambda _\{2^j\} f(x)$ a.e. We only assume $\{4\over 3+2\lambda \} &lt; p &lt; \{4\over 1-2\lambda \}$.},
author = {Cordoba, Antonio, Lopez-Melero, B.},
journal = {Annales de l'institut Fourier},
keywords = {Fourier multiplier operator},
language = {eng},
number = {3},
pages = {147-152},
publisher = {Association des Annales de l'Institut Fourier},
title = {Spherical summation : a problem of E.M. Stein},
url = {http://eudml.org/doc/74501},
volume = {31},
year = {1981},
}

TY - JOUR
AU - Cordoba, Antonio
AU - Lopez-Melero, B.
TI - Spherical summation : a problem of E.M. Stein
JO - Annales de l'institut Fourier
PY - 1981
PB - Association des Annales de l'Institut Fourier
VL - 31
IS - 3
SP - 147
EP - 152
AB - Writing $(T^\lambda _R f)\hat{}(\xi ) = (1- |\xi |^2/R^2)^\lambda _+ \hat{f}(\xi )$. E. Stein conjectured\begin{}\Big \Vert \Big (\sum _j |T^\lambda _{R_j} f_i|^2\Big )^{1/2}\Big \Vert _p \le C\Big \Vert \Big (\sum _j |f_j|^2\Big )^{1/2}\Big \Vert _p\end{}for $\lambda &gt;0$, ${4\over 3} \le p \le 4$ and $C= C_{\lambda ,p}$. We prove this conjecture. We prove also $f(x) = \lim _{j\rightarrow \infty } T^\lambda _{2^j} f(x)$ a.e. We only assume ${4\over 3+2\lambda } &lt; p &lt; {4\over 1-2\lambda }$.
LA - eng
KW - Fourier multiplier operator
UR - http://eudml.org/doc/74501
ER -