Translation invariant forms on $L^p\left(G\right)(1\<p\<\infty )$

Bourgain, Jean. "Translation invariant forms on $L^p(G)(1&lt;p&lt;\infty )$." Annales de l'institut Fourier 36.1 (1986): 97-104. <http://eudml.org/doc/74707>.

@article{Bourgain1986,
abstract = {It is shown that if $G$ is a connected metrizable compact Abelian group and $1&lt; p&lt; \infty $, any (possibly discontinuous) translation invariant linear form on $L^ p(G)$ is a scalar multiple of the Haar measure. This result extends the theorem of G.H. Meisters and W.M. Schmidt (J. Funct. Anal. 13 (1972), 407-424) on $L^ 2(G)$. Our method permits in fact to consider any superreflexive translation invariant Banach lattice on $G$, which is the adopted point of view. We study the representation of an element $f$ of this invariant lattice $X$ as a sum of a bounded number of elements of the form $g-\tau (a)g$, where $g$ in $X$, $a$ in $G$ and $\tau (a)$ the corresponding translation operator. Our approach consists in proving the boundedness of certain random convolution operators using interpolation techniques.},
author = {Bourgain, Jean},
journal = {Annales de l'institut Fourier},
keywords = {connected metrizable compact abelian group; translation invariant linear form; scalar multiple; Haar measure; boundedness; random convolution operators},
language = {eng},
number = {1},
pages = {97-104},
publisher = {Association des Annales de l'Institut Fourier},
title = {Translation invariant forms on $L^p(G)(1&lt;p&lt;\infty )$},
url = {http://eudml.org/doc/74707},
volume = {36},
year = {1986},
}

TY - JOUR
AU - Bourgain, Jean
TI - Translation invariant forms on $L^p(G)(1&lt;p&lt;\infty )$
JO - Annales de l'institut Fourier
PY - 1986
PB - Association des Annales de l'Institut Fourier
VL - 36
IS - 1
SP - 97
EP - 104
AB - It is shown that if $G$ is a connected metrizable compact Abelian group and $1&lt; p&lt; \infty $, any (possibly discontinuous) translation invariant linear form on $L^ p(G)$ is a scalar multiple of the Haar measure. This result extends the theorem of G.H. Meisters and W.M. Schmidt (J. Funct. Anal. 13 (1972), 407-424) on $L^ 2(G)$. Our method permits in fact to consider any superreflexive translation invariant Banach lattice on $G$, which is the adopted point of view. We study the representation of an element $f$ of this invariant lattice $X$ as a sum of a bounded number of elements of the form $g-\tau (a)g$, where $g$ in $X$, $a$ in $G$ and $\tau (a)$ the corresponding translation operator. Our approach consists in proving the boundedness of certain random convolution operators using interpolation techniques.
LA - eng
KW - connected metrizable compact abelian group; translation invariant linear form; scalar multiple; Haar measure; boundedness; random convolution operators
UR - http://eudml.org/doc/74707
ER -