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On a family of Wiener type spaces. (English) Zbl 0843.43002

The authors study Wiener type spaces \(W(A^{p, q}_{w, \omega}, L^r_\nu (G))\) where \(G\) is a locally compact abelian group (non-compact, non-discrete). For example, they show the following theorem: Let \(U_1\) and \(U_2\) be the weight functions in the construction of the Wiener type spaces \(W(A^{p, q}_{w_1 \omega_1} (G)\), \(L^{r_1}_{\nu_1} (G))\) and \(W(A^{p, q}_{w_2, \omega_2} (G)\), \(L^{r_2}_{\nu_2} (G))\), respectively. Also assume that \(w_1, w_2, \nu_1, \nu_2\) are weights on \(G\); \(\omega_1, \omega_2\) are weights on \(\widehat G\) and \(1 \leq p\), \(q, r_1, r_2 < \infty\). Let \(U_1 \sim U_2\), \(w_1 < w_2\) and \(\omega_1 < \omega_2\). Then \(W (A^{p, q}_{w_2, \omega_2} (G)\), \(L^r_{\nu_2} (G)) \hookrightarrow W (A^{p, q}_{w_1, \omega_1} (G)\), \(L^r_{\nu_1} (G))\) if and only if \(\nu_1 < \nu_2\).
Reviewer: T.Nakazi (Sapporo)

MSC:

43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
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