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On mappings generating embedding operators in Sobolev classes on metric measure spaces

Alexander Menovschikov, Alexander Ukhlov

TL;DR

The article characterizes when bi-measurable homeomorphisms $\varphi$ between domains in a doubling metric measure space $X$ generate bounded composition operators $\varphi^{\ast}$ on Newtonian–Sobolev spaces via $\varphi^{\ast}(f)=f\circ\varphi$, providing necessary and sufficient conditions in terms of the Reshetnyak–Sobolev class and dilatation metrics. It develops capacitary Luzin $N^{-1}$ properties and capacity-transform estimates to connect preimage and image sets, then proves Sobolev-type embedding theorems for weak $(p,q)$-quasiconformal $\alpha$-regular domains, including explicit bounds and compactness criteria. The results extend composition-operator theory and Sobolev embeddings to general metric measure spaces, enabling potential-theoretic and elliptic-problem analyses beyond Euclidean settings. Overall, the work unifies geometric, analytic, and capacity-based tools to derive embedding results governed by $p$-capacity, metric Jacobians, and $\alpha$-regularity in non-Riemannian spaces.

Abstract

In this article, we study homeomorphisms $\varphi: Ω\to \widetildeΩ$ that generate embedding operators in Sobolev classes on metric measure spaces $X$ by the composition rule $\varphi^{\ast}(f)=f\circ\varphi$. In turn, this leads to Sobolev type embedding theorems for a wide class of bounded domains $\widetildeΩ\subset X$.

On mappings generating embedding operators in Sobolev classes on metric measure spaces

TL;DR

The article characterizes when bi-measurable homeomorphisms between domains in a doubling metric measure space generate bounded composition operators on Newtonian–Sobolev spaces via , providing necessary and sufficient conditions in terms of the Reshetnyak–Sobolev class and dilatation metrics. It develops capacitary Luzin properties and capacity-transform estimates to connect preimage and image sets, then proves Sobolev-type embedding theorems for weak -quasiconformal -regular domains, including explicit bounds and compactness criteria. The results extend composition-operator theory and Sobolev embeddings to general metric measure spaces, enabling potential-theoretic and elliptic-problem analyses beyond Euclidean settings. Overall, the work unifies geometric, analytic, and capacity-based tools to derive embedding results governed by -capacity, metric Jacobians, and -regularity in non-Riemannian spaces.

Abstract

In this article, we study homeomorphisms that generate embedding operators in Sobolev classes on metric measure spaces by the composition rule . In turn, this leads to Sobolev type embedding theorems for a wide class of bounded domains .

Paper Structure

This paper contains 12 sections, 15 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Sobolev classes on metric measure spaces
  3. Sobolev functions on metric measure spaces
  4. Sobolev mappings on metric measure spaces
  5. Homogeneous metric measure space
  6. Composition operators on Sobolev spaces
  7. The capacitary characteristics of mappings generating composition operators
  8. The analytical characteristics of mappings generating composition operators
  9. Sobolev type embedding theorems
  10. Composition operators on normed Sobolev spaces
  11. Relative embedding theorem
  12. Embedding theorem in weak $(p,q)$-quasiconformal $\alpha$-regular domains

Key Result

Theorem 2.1

Let $\varphi: \Omega \to \widetilde{\Omega}$ be a bi-measurable homeomorphism of domains $\Omega,\widetilde{\Omega}\subset X$. Then a function $f: \widetilde{\Omega} \to [0, \infty]$ is measurable if and only if a function $(f\circ \varphi)\cdot J(\cdot,\varphi): \Omega \to [0, \infty]$ is measurab holds.

Theorems & Definitions (22)

  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 12 more