Nonlinear Lebesgue spaces: Curves and geometry

TL;DR

This paper explores the geometric and analytic properties of nonlinear Lebesgue spaces, focusing on their length, curvature bounds, and curve speeds, through a nonlinear Fubini–Lebesgue theorem and pointwise analysis.

Contribution

It formalizes the pointwise geometric structure of nonlinear Lebesgue spaces and introduces a nonlinear Fubini–Lebesgue theorem for L^p curves.

Findings

Identification of L^p curves with mappings into L^p curve spaces

Pointwise description of length and curvature bounds

Definition of curve speed despite lack of differential structure

Abstract

This paper is the second in a series by the author and collaborators devoted to the study of geometric and analytic properties of nonlinear Lebesgue spaces, that is, L^p spaces of mappings taking values in arbitrary metric spaces. The present article formalizes the pointwise description of their geometric properties -- their length structure, bounds on their Alexandrov curvature as well as the definition of a speed for absolutely continuous curves despite the lack of differential structure. To obtain this pointwise description, we first prove a nonlinear analogue of the Fubini--Lebesgue theorem, which yields an identification of L^p curves in nonlinear Lebesgue spaces to mappings taking values in the space of L^p curves. This identification of L^p curves then enables a similar identification for absolutely continuous curves, from which the pointwise description of the geometric properties of nonlinear Lebesgue spaces follows.