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On a family of weighted convolution algebras. (English) Zbl 0752.43001
Let \(G\) be a locally compact abelian group and \(\widehat G\) its dual. Let \(w\) be a Beurling weight on \(G\) and \(\omega\) a Beurling weight on \(\widehat G\). Let \(L^ p_ w(G)\) be the weighted Banach space under the natural norm: \(\| f\|_{p,w}=\left(\int_ G| f(x)|^ pw^ p(x)dx\right)^{1/p}\), \(1\leq p<\infty\). Let \(A^ p_{w,\omega}(G)=\{f:f\in L^ 1_ w(G),\;f\in L^ p_ \omega(\widehat G)\}\) and \(\| f\|^ p_{w,\omega}=\| f\|_{1,w}+\|\widehat f\|_{p,\omega}\) where \(\widehat f\) is the Fourier transform of \(f\).
The authors begin by showing (among other things) that \(A^ p_{w,\omega}(G)\), with the above norm, is a Banach ideal in \(L^ 1_ w(G)\), hence a Banach algebra with respect to convolution. Also, \(A^ p_{w,\omega}(G)\) is translation and character invariant. A weight \(w\) is said to satisfy the (BD) condition, i.e. the Beurling-Domar condition, if one has \(\sum_{n\geq 1}n^{-2}\log(w(x^ n))<\infty\) for all \(x\) in \(G\). The authors prove the following Theorem. Assume \(w\) satisfies the (BD) condition and that \(\omega(t)\) goes to infinity as \(t\) goes to infinity in \(\widehat G\). Then \(A^ p_{w,\omega}(G)\subseteq A^ q_{w,\omega}(G)\) if and only if \(p\leq q\). The authors show that the algebras in the above theorem are equal if and only if “all parameters are equal”. This paper also discusses approximate identities in \(A^ p_{w,\omega}(G)\) and the nonfactorization in \(A^ p_{w,\omega}(G)\).

43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A70 Analysis on specific locally compact and other abelian groups
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