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Lipschitz spaces over non-porous sets

Ramón J. Aliaga

Abstract

Let $M$ be a subset of $\mathbb{R}^n$. If $M$ is not porous, in particular if it has positive $n$-dimensional Lebesgue measure, we prove that the Lipschitz spaces $\mathrm{Lip}_0(M)$ and $\mathrm{Lip}_0(\mathbb{R}^n)$ are linearly isomorphic. The result also holds more generally if $\mathbb{R}^n$ is replaced with a Carnot group equipped with its Carnot-Carathéodory metric.

Lipschitz spaces over non-porous sets

Abstract

Let be a subset of . If is not porous, in particular if it has positive -dimensional Lebesgue measure, we prove that the Lipschitz spaces and are linearly isomorphic. The result also holds more generally if is replaced with a Carnot group equipped with its Carnot-Carathéodory metric.

Paper Structure

This paper contains 6 sections, 18 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Porous sets and density in balls
  3. Proof of the main result
  4. The Banach space case
  5. The Carnot group case
  6. Questions and discussion

Key Result

Theorem \ref{th:porous}

Suppose that $M\subset\mathbb{R}^n$ is not porous in $\mathbb{R}^n$. Then $\mathop{\mathrm{Lip}}\nolimits_0(M)$ is isomorphic to $\mathop{\mathrm{Lip}}\nolimits_0(\mathbb{R}^n)$.

Theorems & Definitions (33)

  • Theorem \ref{th:porous}
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • proof : Proof of Lemma \ref{['lm:porous balls']}
  • Definition 2.5
  • Proposition 2.6
  • Lemma 2.7
  • ...and 23 more