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Nonlinear Lebesgue spaces: Curves and geometry

Guillaume Sérieys

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.

Nonlinear Lebesgue spaces: Curves and geometry

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.
Paper Structure (29 sections, 45 theorems, 152 equations)

This paper contains 29 sections, 45 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonlinear Lebesgue spaces
  4. Basic assumptions
  5. Measurable mappings
  6. Lebesgue mappings
  7. Curves in metric spaces
  8. Basic assumptions
  9. Sobolev curves and continuous representations
  10. Curves of bounded variation and càdlàg representations
  11. Main results
  12. Characterization of Lp curves in nonlinear Lebesgue spaces
  13. Characterization of absolutely continuous curves for p> 1
  14. Characterization of càdlàg curves of bounded variation for p=1
  15. Applications
  16. ...and 14 more sections

Key Result

Proposition 2.4

Let $(f_n)_{n\in\mathbb{N}}$ be a sequence in $\mathcal{L}^0_{\mathrm{s}}(M,N)$ and assume that $f(x)\coloneqq \lim_{n\to\infty} f_n(x)$ exists in $N$ for all $x\in M$. Then, $f\in \mathcal{L}^0_{\mathrm{s}}(M,N)$.

Theorems & Definitions (147)

  • Definition 2.2: Measurable mappings
  • Remark 2.3: "Borel measurable" mappings
  • Proposition 2.4: $\mathcal{L}^0_{\mathrm{s}}(M,N)$ is closed under pointwise limit
  • Definition 2.5: Simple mappings
  • Proposition 2.6: Equivalence relation on $\mathcal{L}^0(M,N)$
  • Definition 2.7: Equivalence classes of measurable mappings
  • Definition 2.8: $\mathcal{L}^p$ semi-metrics
  • Proposition 2.9: $\mathcal{L}^p$ semi-metrics separate equivalence classes
  • Remark 2.10: $\mathcal{L}^p$ semi-metrics metrize $L^0(M,N)$
  • Definition 2.11: $\mathcal{L}^p$ spaces
  • ...and 137 more