Isometries and geometric liftings for Alexiewicz-normed $L^\infty$ spaces

TL;DR

This paper characterizes the structure of isometries in Alexiewicz-normed spaces of bounded functions, linking them to measure-preserving transformations and geometric properties of underlying sets.

Contribution

It provides a classification of surjective linear isometries and geometric criteria for lifting interval maps to homeomorphisms in these function spaces.

Findings

Surjective linear isometries are weighted composition operators with specific properties.

Interval maps can be characterized as homeomorphisms or bi-Lipschitz maps based on geometric criteria.

The study connects measure-theoretic and geometric aspects of function spaces.

Abstract

We study spaces of essentially bounded functions on compact subsets of the real line, equipped with the Alexiewicz norm given by the supremum norm of the primitive. Using the associated measure projection, we classify their surjective linear isometries as weighted composition operators determined by a sign and an increasing bi-Lipschitz map between the corresponding measure intervals. We also give geometric criteria characterizing when this interval-level map lifts to a homeomorphism or to a bi-Lipschitz homeomorphism between the underlying compact sets.