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Metric Sobolev spaces II: dual energies and divergence measures

Luigi Ambrosio, Toni Ikonen, Danka Lučić, Enrico Pasqualetto

TL;DR

The paper advances a comprehensive vector-calculus framework on general metric measure spaces for all $p\in[1,\infty)$, introducing cotangent/tangent modules, two notions of derivations, and a divergence theory captured by dual energies ${\sf F}_p$ and ${\sf D}_q$. It forges deep connections between Lipschitz and Sobolev derivations, dynamic plans, and metric currents, and defines a $p$-Laplacian as the divergence of a gradient, with reflexivity and duality results for Sobolev spaces. By establishing the equality ${\sf F}_p={\sf D}_q$ and relating divergence measures to Cap$_p$, the work extends duality in Sobolev spaces and provides a robust potential-theoretic toolkit, including condenser capacities and capacitary potentials. The results yield a non-smooth analytic foundation with implications for geometric analysis and potential theory on metric measure spaces, independent of doubling or Poincaré assumptions, and illuminate the interactions between variational energies, currents, and derivations in this setting.

Abstract

This is the second of two works concerning the Sobolev calculus on metric measure spaces and its applications. In this work, we focus on several approaches to vector calculus in the non-smooth setting of complete and separable metric spaces equipped with a boundedly-finite Borel measure. More precisely, we study different notions of (co)vector fields and derivations appearing in the literature, as well as their mutual relation. We also carry forward a thorough investigation of gradients, divergence measures, and Laplacian measures, together with their applications in potential analysis (for example, regarding the condenser capacity) and in the study of duality properties of Sobolev spaces. Most of the results are obtained for the full range of exponents $p\in[1,\infty)$ and without finiteness assumption on the measure.

Metric Sobolev spaces II: dual energies and divergence measures

TL;DR

The paper advances a comprehensive vector-calculus framework on general metric measure spaces for all , introducing cotangent/tangent modules, two notions of derivations, and a divergence theory captured by dual energies and . It forges deep connections between Lipschitz and Sobolev derivations, dynamic plans, and metric currents, and defines a -Laplacian as the divergence of a gradient, with reflexivity and duality results for Sobolev spaces. By establishing the equality and relating divergence measures to Cap, the work extends duality in Sobolev spaces and provides a robust potential-theoretic toolkit, including condenser capacities and capacitary potentials. The results yield a non-smooth analytic foundation with implications for geometric analysis and potential theory on metric measure spaces, independent of doubling or Poincaré assumptions, and illuminate the interactions between variational energies, currents, and derivations in this setting.

Abstract

This is the second of two works concerning the Sobolev calculus on metric measure spaces and its applications. In this work, we focus on several approaches to vector calculus in the non-smooth setting of complete and separable metric spaces equipped with a boundedly-finite Borel measure. More precisely, we study different notions of (co)vector fields and derivations appearing in the literature, as well as their mutual relation. We also carry forward a thorough investigation of gradients, divergence measures, and Laplacian measures, together with their applications in potential analysis (for example, regarding the condenser capacity) and in the study of duality properties of Sobolev spaces. Most of the results are obtained for the full range of exponents and without finiteness assumption on the measure.
Paper Structure (25 sections, 70 theorems, 303 equations)

This paper contains 25 sections, 70 theorems, 303 equations.

Table of Contents

  1. Introduction
  2. General overview
  3. Contents of this work
  4. Acknowledgements.
  5. Preliminaries
  6. Metric measure spaces
  7. Lp-Banach Linfty-modules
  8. Choquet integral
  9. Metric 1-currents
  10. Metric Sobolev spaces
  11. Quasicontinuity of Sobolev functions
  12. Sobolev capacities
  13. Quasicontinuity and quasiuniform convergence
  14. Derivations and tangent modules
  15. Tangent and cotangent modules
  16. ...and 10 more sections

Key Result

Proposition 2.11

Let $({\rm X},\Sigma,\mathfrak{m})$ be a $\sigma$-finite measure space and $p\in[1,\infty)$. Let $\mathscr M$ be an $L^p(\mathfrak{m})$-Banach $L^\infty(\mathfrak{m})$-module. Let us define the operator $\text{\sc Int}_{\mathscr M}\colon\mathscr M^*\to\mathscr M'$ as Then $\text{\sc Int}_\mathscr M$ is an isomorphism of Banach spaces.

Theorems & Definitions (188)

  • Definition 2.1: Metric measure space
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4: Boundedly-finite signed Borel measure
  • Definition 2.5: Modulus
  • Definition 2.6: Plan
  • Definition 2.7: Barycenter of a plan
  • Definition 2.8: $L^p(\mathfrak{m})$-Banach $L^\infty(\mathfrak{m})$-module
  • Remark 2.9
  • Definition 2.10: Dual of an $L^p(\mathfrak{m})$-Banach $L^\infty(\mathfrak{m})$-module
  • ...and 178 more