Extremes of interpolation scales of Banach spaces

TL;DR

This paper investigates conditions under which the spheres of Banach spaces within complex interpolation scales are uniformly homeomorphic, focusing on the extremes of these scales and their applications to classical spaces.

Contribution

It provides new sufficient conditions for uniform homeomorphism of spheres at the extremes of complex interpolation scales of Banach spaces.

Findings

Established conditions for uniform homeomorphism at the scale extremes

Applied results to spheres of ll_2 and other Banach spaces

Extended previous work on interpolation space geometry

Abstract

M. Daher gave conditions so that the spheres of the spaces in the interior of a complex interpolation scale are uniformly homeomorphic. We look for sufficient conditions for the validity of this result and related ones on the extremes of the scale, with applications to uniform homeomorphism between spheres of Banach spaces and the sphere of $\ell_2$.