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On the generalized weighted Lebesgue spaces of locally compact groups. (English) Zbl 1227.43002

The article contains a definition of a topological weighted convolution algebra \(L^p(G,\Omega)\) on a locally compact group \(G\), where \(p>1\) and \(\Omega\) is a family of weights. Such algebras were studied before for \(G=\mathbb R^n\) in a paper of El Kinani and Benassouz which the authors cite, and in another one [A. El Kinani, Rend. Circ. Mat. Palermo (2) 56, No. 3, 369–380 (2007; Zbl 1145.46031)] which they seem to be unaware of. Basic properties of such algebras are proved, such as description of their dual space, maximal ideals, translation invariant subspaces.
Theorem 4.9 claims that \(L^p(G,\Omega)\) is an algebra if and only if the well-known convolution inequality holds for the weights; namely, let \(1/p+1/q=1\), then for every \(\omega\in \Omega\) there is \(\nu\in\Omega\) such that \(\nu^{-q}*\nu^{-q}\leq \omega^{-q}\) locally almost everywhere. For a unique weight, this would just imply that there exists a constant \(C\) such that \(\omega^{-q}*\omega^{-q}\leq C\omega^{-q}\).
But this statement is known to be false [Yu. N. Kuznetsova, Funct. Anal. Appl. 40, No. 3, 234–236 (2006; Zbl 1117.43002), example following Theorem 1]. The error in the present article should lie in the authors’ belief that the projective tensor product of \(L^p(G,\omega)\) with itself, for one fixed weight, is again an \(L^p\)-space (this is claimed in the proof of Lemma 4.8). But this fact is not true [see e.g. A. Ya. Helemskii, Lectures and exercises on functional analysis. Translations of Mathematical Monographs 233. Providence, RI: American Mathematical Society (AMS) (2006; Zbl 1123.46001), p. 183].

MSC:

43A10 Measure algebras on groups, semigroups, etc.
46H99 Topological algebras, normed rings and algebras, Banach algebras
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
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