Embedding Functions and Their Role in Interpolation Theory

The embedding functions of an intermediate space A into a Banach couple (A0, A1) are defined as its embedding constants into the couples ( 1 α A0, 1 β A1), ∀α, β > 0. Using these functions, we study properties and interrelations of different intermediate spaces, give a new description of all real interpolation spaces, and generalize the concept of weak-type interpolation to any Banach couple to obtain new interpolation theorems. 0. Introduction The interpolation theory arose from a general problem of studying linear operators on large collections of Banach spaces. First those are spaces with a good analytical description (expression of norm) depending on a numerical parameter, such as Lp, Lipα, W k p etc. [11]. As a natural generalization, one then took families of spaces with a function parameter [17] or some other common characteristics — for example, symmetric (rearrangement invariant) spaces [21]. Impetuous development of the interpolation theory generated soon a problem of this theory fundamentals and basic notions (see e.g. a classical work [1]). So one get the totality of all intermediate spaces for a Banach couple as a basic and largest object, from which one may extract spaces with different interpolation properties. A Banach spaceA is called intermediate for a Banach couple A = (A0, A1) if ∆( A) ⊂ A ⊂ Σ( A), where ∆( A) = A0 ∩ A1, Σ( A) = A0 + A1 with the standard definition of norms. The totality of all such spaces will be denoted by π( A). The indicated embeddings are always accompanied by the norm inequalities (1) ‖x‖A ≥ D ‖x‖Σ( A) (∀x ∈ A), ‖x‖A ≤ C ‖x‖∆( A) (∀x ∈ ∆( A)). 1991 Mathematics Subject Classification. Primary: 46B70.