Interpolation of compact Lipschitz operators

TL;DR

This paper proves that certain nonlinear Lipschitz operators, under specific compactness and approximation conditions, map real interpolation spaces compactly between Banach couples.

Contribution

It establishes the compactness of nonlinear Lipschitz operators on real interpolation spaces under new approximation and compactness conditions.

Findings

T maps (A_0,A_1)_{ heta,p} compactly into (B_0,B_1)_{ heta,p}

Results apply to nonlinear Lipschitz operators with specific compactness conditions

Provides a framework for interpolation of compact Lipschitz operators

Abstract

Let (A_0,A_1) and (B_0,B_1) be Banach couples such that A_0 is contained in A_1 and (B_0,B_1) satisfies Arne Persson's approximation condition (H). Let T:A_1 --> B_1 be a possibly nonlinear Lipschitz mapping which also maps A_0 into B_0 and satisfies the following quantitative compactnesss condition: Ta \in ||a||_{A_0} K for each a \in A_0, where K is a fixed compact subset of B_0. We show that T maps the real interpolation space (A_0,A_1)_{\theta,p} compactly into its counterpart (B_0,B_1)_{\theta,p} for each \theta \in (0,1) and p \in [1,\infty].