## 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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