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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 $$A^ p_{w,\omega}(G)$$.

##### MSC:
 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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