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Zbl 0576.43003
Bourgain, Jean
Translation invariant forms on $L\sp p(G)$ $(1<p<\infty)$. (English)
[J] Ann. Inst. Fourier 36, No.1, 97-104 (1986). ISSN 0373-0956; ISSN 1777-5310/e

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 $L\sp p(G)$ is a scalar multiple of the Haar measure. This result extends the theorem of {\it G. H. Meisters} and {\it W. M. Schmidt} [J. Funct. Anal. 11, 407-424 (1972; Zbl 0247.43004)] on $L\sp 2(G)$. Our method permits in fact to consider any superreflexive translation invariant Banach lattice on G, which is the adopted point of view. \par 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.

MSC 2000:: *43A15 Lp-spaces and other function spaces on groups, etc.

Keywords: connected metrizable compact abelian group; translation invariant linear form; scalar multiple; Haar measure; boundedness; random convolution operators

Citations: Zbl 0247.43004

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