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 $0<q\le p\le 1$, 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 $0<p<\infty$.

References in corpus (4)

• Cwikel estimates revisited
• Calderón-Lozanovskii interpolation on quasi-Banach lattices
• Arazy-Cwikel property for quasi-Banach couples
• Arazy-Cwikel and Calderón-Mityagin type properties of the couples $({\ell }^p,{\ell }^q)$, $0\le p<q\le \infty$

Cited by in corpus (1)

• Arazy-Cwikel and Calderón-Mityagin type properties of the couples $({\ell }^p,{\ell }^q)$, $0\le p<q\le \infty$