Interpolation of nonlinear positive or order preserving operators on Banach lattices

TL;DR

This paper explores the characterization of exact interpolation spaces for various classes of positive and order-preserving operators within Banach lattices, extending classical results to broader operator classes.

Contribution

It generalizes the theory of exact interpolation spaces to include order-preserving Lipschitz operators and positive Gagliardo-Peetre operators in Banach lattices.

Findings

Characterization of exact interpolation spaces for order-preserving Lipschitz operators.

Extension of Bénilan and Crandall's results to broader operator classes.

Connection between interpolation spaces and partially K-monotone spaces.

Abstract

We study the relationship between exact interpolation spaces for positive, linear operators, for order preserving, Lipschitz continuous operators, and for positive Gagliardo-Peetre operators, and exact partially $K$-monotone spaces in interpolation couples of compatible Banach lattices. By general Banach lattice theory we recover a characterisation of exact interpolation spaces for order preserving, Lipschitz continuous operators in the couple $(L^1,L^\infty )$ due to B\'enilan and Crandall.