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Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential

Sylvester Eriksson-Bique, Elefterios Soultanis

Abstract

We represent minimal upper gradients of Newtonian functions, in the range $1\le p<\infty$, by maximal directional derivatives along "generic" curves passing through a given point, using plan-modulus duality and disintegration techniques. As an application we introduce the notion of $p$-weak charts and prove that every Newtonian function admits a differential with respect to such charts, yielding a linear approximation along $p$-almost every curve. The differential can be computed curvewise, is linear, and satisfies the usual Leibniz and chain rules. The arising $p$-weak differentiable structure exists for spaces with finite Hausdorff dimension and agrees with Cheeger's structure in the presence of a Poincaré inequality. It is moreover compatible with, and gives a geometric interpretation of, Gigli's abstract differentiable structure, whenever it exists. The $p$-weak charts give rise to a finite dimensional $p$-weak cotangent bundle and pointwise norm, which recovers the minimal upper gradient of Newtonian functions and can be computed by a maximization process over generic curves. As a result we obtain new proofs of reflexivity and density of Lipschitz functions in Newtonian spaces, as well as a characterization of infinitesimal Hilbertianity in terms of the pointwise norm.

Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential

Abstract

We represent minimal upper gradients of Newtonian functions, in the range , by maximal directional derivatives along "generic" curves passing through a given point, using plan-modulus duality and disintegration techniques. As an application we introduce the notion of -weak charts and prove that every Newtonian function admits a differential with respect to such charts, yielding a linear approximation along -almost every curve. The differential can be computed curvewise, is linear, and satisfies the usual Leibniz and chain rules. The arising -weak differentiable structure exists for spaces with finite Hausdorff dimension and agrees with Cheeger's structure in the presence of a Poincaré inequality. It is moreover compatible with, and gives a geometric interpretation of, Gigli's abstract differentiable structure, whenever it exists. The -weak charts give rise to a finite dimensional -weak cotangent bundle and pointwise norm, which recovers the minimal upper gradient of Newtonian functions and can be computed by a maximization process over generic curves. As a result we obtain new proofs of reflexivity and density of Lipschitz functions in Newtonian spaces, as well as a characterization of infinitesimal Hilbertianity in terms of the pointwise norm.

Paper Structure

This paper contains 31 sections, 46 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Curvewise characterization of minimal upper gradients
  4. Application: $p$-weak differentiable structure
  5. Connections to Cheeger's and Gigli's differentiable structures
  6. Acknowledgements
  7. Preliminaries
  8. Plans and modulus
  9. Sobolev spaces and functions
  10. Curvewise (almost) optimality of minimal upper gradients
  11. Upper gradients with respect to plans
  12. Proof of Theorem \ref{['thm:curvewise-p=1']}
  13. Alternative curvewise characterizations of upper gradients when $p>1$
  14. Charts and differentials
  15. Notational remarks
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $1\le p<\infty$, and let $1<q\le \infty$ satisfy $1/p+1/q=1$. Suppose $f\in N^{1,p}(X)$, $g_f$ is a Borel representative of the minimal $p$-weak upper gradient of $f$, and $D:=\{ g_f>0 \}$. There exists a $q$-plan $\bm\eta$ with $\mu|_D \ll \bm\eta^\#$ so that the disintegration $\{\bm\pi_x\}$ o for $\mu$-almost every $x\in D$.

Theorems & Definitions (101)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Definition 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 91 more