The class of convolution operators on the Marcinkiewicz spaces

Lau, Ka-Sing. "The class of convolution operators on the Marcinkiewicz spaces." Annales de l'institut Fourier 31.3 (1981): 225-243. <http://eudml.org/doc/74505>.

@article{Lau1981,
abstract = {Let $\{\cal T\}_X$ denote the operator-norm closure of the class of convolution operators $\Phi _\mu : X\rightarrow X$ where $X$ is a suitable function space on $R$. Let $\{\cal M\}^p_r$ be the closed subspace of regular functions in the Marinkiewicz space $\{\cal M\}^p$, $1\le p&lt; \infty $. We show that the space $\{\cal T\}_\{\{\cal M\}^p_r\}$ is isometrically isomorphic to $\{\cal T\}_\{L^p\}$ and that strong operator sequential convergence and norm convergence in $\{\cal T\}_\{\{\cal M\}^p_r\}$ coincide. We also obtain some results concerning convolution operators under the Wiener transformation. These are to improve a Tauberian theorem of Wiener on $\{\cal M\}^2$.},
author = {Lau, Ka-Sing},
journal = {Annales de l'institut Fourier},
keywords = {class of convolution operators; Marcinkiewicz space; Wiener transformation; Tauberian theorem},
language = {eng},
number = {3},
pages = {225-243},
publisher = {Association des Annales de l'Institut Fourier},
title = {The class of convolution operators on the Marcinkiewicz spaces},
url = {http://eudml.org/doc/74505},
volume = {31},
year = {1981},
}

TY - JOUR
AU - Lau, Ka-Sing
TI - The class of convolution operators on the Marcinkiewicz spaces
JO - Annales de l'institut Fourier
PY - 1981
PB - Association des Annales de l'Institut Fourier
VL - 31
IS - 3
SP - 225
EP - 243
AB - Let ${\cal T}_X$ denote the operator-norm closure of the class of convolution operators $\Phi _\mu : X\rightarrow X$ where $X$ is a suitable function space on $R$. Let ${\cal M}^p_r$ be the closed subspace of regular functions in the Marinkiewicz space ${\cal M}^p$, $1\le p&lt; \infty $. We show that the space ${\cal T}_{{\cal M}^p_r}$ is isometrically isomorphic to ${\cal T}_{L^p}$ and that strong operator sequential convergence and norm convergence in ${\cal T}_{{\cal M}^p_r}$ coincide. We also obtain some results concerning convolution operators under the Wiener transformation. These are to improve a Tauberian theorem of Wiener on ${\cal M}^2$.
LA - eng
KW - class of convolution operators; Marcinkiewicz space; Wiener transformation; Tauberian theorem
UR - http://eudml.org/doc/74505
ER -