S-Divisibility Property and a Holmstedt Type Formula

Abstract

Given a cone P of positive functions and an operator S: U Ä P with U an additive group, we extend the concept of K-divisibility to get some new formulas for the K-functional of finite families of lattices. Applications are given in the setting of rearrangement invariant spaces and weighted Lorentz spaces. As a consequence of our results, we also obtain a generalized Holmstedt formula due to Asekritova (1980, Yaroslav. Gos. Univ. 165, 15 18). Academic Press

Real interpolation for divisible cones

“…Let (g 0 , g { ) be a minimal element in . 4 which is a contradiction. Therefore we conclude that £" = 0 for every 0 < a < 1 and hence f -g o + 9i which finishes the proof.…”

“…The purpose of this section is to compute some X-functionals for several divisible cones and for the couples (L"°(w 0 ), L"(w,)) and (X, L 00 ) where X is a lattice. We shall use formula (4) from Section 3 and the fact that the ^-functional for the couple E , can sometimes be explicitly computed as follows (see [4]):…”

“…We shall use formula (4) from Section 3 and the fact that the ^-functional for the couple E , can sometimes be explicitly computed as follows (see [4]): …”

“…Sequences (a n ) n so that a n /b n is decreasing for some fixed sequence (&"). 4. The cone of all functions/ so that//i is decreasing for a fixed function h.…”

Function cones and interpolation

“…These facts were at the origin of the paper [9], and also of [6] and [7], where it was seen that a certain decomposition property for functions belonging to the cone is needed to obtain an acceptable interpolation theory, and further conditions are needed to get the desirable identity (Q ∩ E 0 , Q ∩ E 1 ) θ,q = Q ∩ (E 0 , E 1 ) θ,q for a cone Q and a couple (E 0 , E 1 ) of function spaces.…”