A description of interpolation spaces for quasi-Banach couples by real $K$-method

Abstract

The main aim of this paper is to develop a general approach, which allows to extend the basics of Brudnyi-Kruglyak interpolation theory to the realm of quasi-Banach lattices. We prove that all K-monotone quasi-Banach lattices with respect to a L-convex quasi-Banach lattice couple have in fact a stronger property of the so-called K(p, q)-monotonicity for some 0 < q ≤ p ≤ 1, which allows us to get their description by the real K-method. Moreover, we obtain a refined version of the K-divisibility property for Banach lattice couples and then prove an appropriate version of this property for L-convex quasi-Banach lattice couples. The results obtained are applied to refine interpolation properties of couples of sequence l p -and function L p -spaces, considered for the full range 0 < p < ∞.