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On a family of weighted spaces. (English) Zbl 0862.43002
Let \(G\) be a locally compact abelian group with dual group \(\widehat G\) and \(w\) a real-valued, measurable, locally bounded function on \(G\) such that \(1\leq w(x)\), \(w(x +y) \leq w(x) w(y)\) for \(x,y\in G\). Let \(L^p_w(G)\) \((1\leq p< \infty)\) be the set of functions \(f\) such that \(wf \in L^p(G)\) with norm \(|f|_{p,w} = |wf |_p\). The paper is devoted to study the weighted spaces \(A^{p,q}_{w,\omega} (G)\) of functions \(f\in L^p_\omega (G)\) with generalized Fourier transform \(\widehat f\in L^q_\omega (\widehat G)\) \((1\leq q< \infty)\) equipped with the sum norm \(|f|^{p,q}_{w, \omega} (G)= |f|^p_w + |\widehat f|^q_\omega\). Classical and generalized Fourier transforms are used to prove the properties of spaces \(A^{p,q}_{w,\omega} (G)\). Some of the results obtained generalize well-known assertions for the case of non-weighted space \(A_p(G) = A_{1,1}^{1,p'} (G)\).
Reviewer: A.A.Kilbas (Minsk)

43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
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