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Abstract

We give a full solution of the following problems concerning the spaces $M_\varphi ( \vec X )$: (i) to what extent two functions $\varphi $ and $\psi $ should be different in order to ensure that $M_\varphi ( \vec X )\not =M_\psi ( \vec X )$ for any nontrivial Banach couple $ \vec X $; (ii) when an embedding $M_\varphi ( \vec X )\varsubsetneq M_\psi ( \vec X )$ can (or cannot) be dense; (iii) which Banach space can be regarded as an $M_\varphi ( \vec X )$-space for some (unknown beforehand) Banach couple $ \vec X $.