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

Annales de l'institut Fourier (1986)

- Volume: 36, Issue: 1, page 97-104
- ISSN: 0373-0956

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topBourgain, Jean. "Translation invariant forms on $L^p(G)(1<p<\infty )$." 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< p< \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 $L^p(G)(1<p<\infty )$},

url = {http://eudml.org/doc/74707},

volume = {36},

year = {1986},

}

TY - JOUR

AU - Bourgain, Jean

TI - Translation invariant forms on $L^p(G)(1<p<\infty )$

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< p< \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 -

## References

top- [1] A. CONNES, Private communication.
- [2] B. JOHNSON, A proof of the translation invariant form conjecture for L²(G), Bull. de Sciences Math., 107, n° 3, (1983), 301-310. Zbl0529.43003MR85k:43003
- [3] J. LINDENSTRAUSS, L. TZAFRIRI, Classical Banach spaces II, Springer, 97 (1979). Zbl0403.46022MR81c:46001
- [4] G.H. MEISTERS, W.M. SCHMIDT, Translation invariant linear forms on L²(G) for compact Abelian group G, J. Funct. Anal., n° 13, (1972), 407-424. Zbl0247.43004
- [5] G. PISIER, Some applications of the complex interpolation method to Banach lattices, J. d'Analyse Math. de Jérusalem, Vol 35 (1979), 264-281. Zbl0427.46048MR80m:46020
- [6] J. BERGH, J. LOFSTRÖM, Interpolation Spaces.

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