[Paper Review] Sobolev and BV spaces on metric measure spaces via derivations and integration by parts

Simone Di Marino|arXiv (Cornell University)|Sep 19, 2014

Geometric Analysis and Curvature Flows12 references12 citations

TL;DR

This paper establishes a new characterization of Sobolev and BV spaces on metric measure spaces using integration by parts with Weaver’s metric derivations, showing equivalence to all major existing definitions. It introduces a derivation-based weak gradient and proves that the total variation of the distributional derivative coincides across different formulations, unifying the theory without requiring doubling or Poincaré conditions.

ABSTRACT

We develop a theory of BV and Sobolev Spaces via integration by parts formula in abstract metric spaces; the role of vector fields is played by Weaver's metric derivations. The definition hereby given is shown to be equivalent to many others present in literature.

Motivation & Objective

• To develop a theory of Sobolev and BV spaces on general metric measure spaces using integration by parts with derivations, avoiding reliance on upper gradients.
• To bridge the gap between classical Sobolev theory and modern metric space analysis by recovering the weak gradient via duality and integration by parts.
• To show that the proposed derivation-based definition of Sobolev and BV spaces is equivalent to all major existing formulations, including those based on upper gradients, relaxed slopes, and plan-based definitions.
• To establish the equivalence of the total variation measure |Df| across different definitions, including BV*, w-BV, and the derivation-based BV space.
• To demonstrate that the theory holds in complete, separable metric measure spaces without assuming doubling or Poincaré conditions.

Proposed method

• The paper defines a function f ∈ L^p to be in W^{1,p} if there exists a linear functional L_f on derivations such that ∫ f · div b d𝔪 = -∫ L_f(b) d𝔪 for all derivations b with |b|, div b ∈ L^q.
• It introduces a differential df: Der^{q,q} → L^1, which allows defining a pointwise gradient |∇f| satisfying |df(b)| ≤ |∇f| · |b| almost everywhere.
• The theory uses Weaver’s derivations as a generalization of vector fields, leveraging their Liebniz rule and weak locality to define weak derivatives.
• It establishes equivalence between the derivation-based Sobolev space and the standard definition via q-relaxed slope and minimal q-upper gradient, using duality and approximation.
• For BV spaces, the paper defines |Df| via the duality ∫ f · div b d𝔪 ≤ ∫ |b|^* d|Df| for all derivations b ∈ Der_L, and proves that this measure coincides with the total variation in BV* and w-BV spaces.
• The proof of equivalence relies on the correspondence between ∞-plans (measures on curves) and derivations, using the fact that Lipschitz functions have well-defined derivatives along curves.

Experimental results

Research questions

• RQ1 Can Sobolev spaces on metric measure spaces be defined via an integration by parts formula using derivations, rather than via upper gradients or relaxed slopes?
• RQ2 Is the derivation-based definition of the weak gradient equivalent to the classical notion of the gradient in R^n and to other modern definitions in metric measure spaces?
• RQ3 Does the total variation measure |Df| defined via integration by parts with derivations coincide with the total variation in the BV* and w-BV spaces?
• RQ4 Can the equivalence between BV* and w-BV spaces be recovered and extended using the derivation framework?
• RQ5 What is the role of ∞-plans and curve-based derivations in connecting the derivation-based BV space to the plan-based BV definitions?

Key findings

• The paper proves that the derivation-based definition of W^{1,p} is equivalent to the standard definition via minimal q-upper gradients and relaxed slopes, even without doubling or Poincaré conditions.
• The differential df defined on derivations gives a pointwise gradient |∇f| that satisfies |df(b)| ≤ |∇f| · |b| almost everywhere, and this |∇f| coincides with the minimal weak upper gradient from previous literature.
• The total variation measure |Df| defined via integration by parts with derivations satisfies |Df| ≤ |Df|_* and |Df|(X) ≥ |Df|_w(X), and equality holds across all three definitions.
• The equivalence |Df| = |Df|_* = |Df|_w is established for all f ∈ BV, showing that the derivation-based BV space coincides with BV* and w-BV spaces.
• The theory unifies multiple approaches to Sobolev and BV spaces in metric measure spaces, providing a common foundation via derivations and integration by parts.