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Density problem for Sobolev spaces on Gehring Hayman domains with the ball separation condition in metric measure spaces

Jesse Koivu

Abstract

We prove that for a domain $Ω$ in a PI space $X$ such that $Ω$ satisfies the Gehring Hayman condition and the ball separation condition, the Newtonian Sobolev space $N^{1,\infty}(Ω)$ is dense in the space $N^{1,p}(Ω)$ for $1 < p < \infty$.

Density problem for Sobolev spaces on Gehring Hayman domains with the ball separation condition in metric measure spaces

Abstract

We prove that for a domain in a PI space such that satisfies the Gehring Hayman condition and the ball separation condition, the Newtonian Sobolev space is dense in the space for .

Paper Structure

This paper contains 7 sections, 11 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ball separation and the Gehring-Hayman condition
  4. Gromov hyperbolic domains in PI spaces
  5. Whitney-type covering
  6. Decomposing the C-GHS domain
  7. The proof of density

Key Result

Theorem 1.1

Let $(X,{\sf d},\mu)$ be a doubling metric measure space satisfying a local Poincaré inequality. Let $\Omega \subset X$ be a C-GHS space and $1 < p < \infty$. Then $N^{1,\infty}(\Omega)$ is dense in $N^{1,p}(\Omega)$.

Theorems & Definitions (30)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Definition 2.8
  • ...and 20 more