Uniform homeomorphisms between spheres induced by interpolation methods

TL;DR

This paper demonstrates that uniform homeomorphisms between spheres of interpolation spaces are a general phenomenon across various interpolation methods, extending previous results beyond specific cases.

Contribution

It generalizes the uniform homeomorphism result for spheres in interpolation spaces to a broad class of interpolation methods within a discrete framework.

Findings

Uniform homeomorphisms exist between spheres of interpolation spaces across various methods.

The result extends beyond regular couples of uniformly convex spaces.

The phenomenon is shown to be quite general within the interpolation framework.

Abstract

M. Daher [9] showed that if $(X_0, X_1)$ is a regular couple of uniformly convex spaces then the unit spheres of the complex interpolation spaces $X_{\theta}$ and $X_{\eta}$ are uniformly homeomorphic for every $0 < \theta, \eta < 1$. We show that this is a rather general phenomenon of interpolation methods described by the discrete framework of interpolation of [15].