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

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