On some properties of generalized Marcinkiewicz spaces
We give a full solution of the following problems concerning the spaces Mφ(~ X): (i) to what extent two functions φ and ψ should be different in order to ensure that Mφ(~ X) 6= Mψ(~ X) for any nontrivial Banach couple ~ X; (ii) when an embedding Mφ(~ X) Mψ(~ X) can (or cannot) be dense; (iii) which Banach space can be regarded as an Mφ(~ X)-space for some (unknown beforehand) Banach couple ~ X. Introduction. The generalized Marcinkiewicz spaces Mφ( ~ X) (sometimes denoted also by ~ Xφ,∞) apparently are the simplest intermediate (and interpolation) spaces for every Banach couple ~ X = (X0,X1). The only condition for x ∈ Σ( ~ X) = X0 +X1 to belong to the unit ball of Mφ is the inequality K(t, x, ~ X) = ‖x‖X0+tX1 ≤ φ(t) for all t > 0. Such simplicity makes these spaces a good “touchstone” for various hypotheses on global properties of intermediate (or interpolation) spaces for abstract Banach couples. Very often they turn out to be extreme spaces having a required property and can be used for description of other spaces. The spaces Mφ( ~ X) appeared for the first time in [D] just in such a role. Together with their dual spaces Λφ( ~ X) they were used for description of spaces which are interpolation with respect to one-dimensional operators. Further they were systematically studied in [P1], where their extreme position was stated for many other problems. It was also proved there that any real interpolation space can be represented as the sum of some collection of Mφ( ~ X)-spaces. At the same time the paper [DKO] associated these spaces with certain extreme interpolation functors. Notice that, for some special couples, these spaces were known and intensively studied previously, 2000 Mathematics Subject Classification: Primary 46B20.