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Traces of vanishing Hölder spaces

Kaushik Mohanta, Carlos Mudarra, Tuomas Oikari

TL;DR

The paper develops a unified theory for extending vanishing Hölder spaces defined on arbitrary subsets $E\subset\mathbb{R}^n$ to the whole space. It introduces three vanishing scales (small, large, far) for $\dot C^{0,\omega}$ and corresponding vanishing jet spaces, and proves that a single linear Whitney extension operator preserves these scales simultaneously. The authors give necessary and sufficient conditions for extendability in each scale via vanishing jet conditions and obtain complete characterizations of approximability of Hölder functions by Lipschitz and boundedly supported functions. The results remove geometric restrictions on $E$ and provide a robust framework for extending Hölder-regular objects alongside their jets.

Abstract

For an arbitrary subset $E\subset\mathbb{R}^n,$ we introduce and study the three vanishing subspaces of the Hölder space $\dot{C}^{0,ω}(E)$ consisting of those functions for which the ratio $|f(x)-f(y)|/ω(|x-y|)$ vanishes, when $(1)$ $|x-y|\to 0$ , $(2)$ $|x-y|\to\infty$ or $(3)$ $\min(|x|,|y|)\to\infty.$ We prove that the Whitney extension operator maps each of these vanishing subspaces from $E$ to the corresponding vanishing spaces defined on the whole ambient space $\mathbb{R}^n.$ In fact, this follows as the zeroth order special case of a more general problem involving higher order derivatives. As a consequence, we obtain complete characterizations of approximability of Hölder functions $\dot{C}^{0,ω}(E)$ by Lipschitz and boundedly supported functions.

Traces of vanishing Hölder spaces

TL;DR

The paper develops a unified theory for extending vanishing Hölder spaces defined on arbitrary subsets to the whole space. It introduces three vanishing scales (small, large, far) for and corresponding vanishing jet spaces, and proves that a single linear Whitney extension operator preserves these scales simultaneously. The authors give necessary and sufficient conditions for extendability in each scale via vanishing jet conditions and obtain complete characterizations of approximability of Hölder functions by Lipschitz and boundedly supported functions. The results remove geometric restrictions on and provide a robust framework for extending Hölder-regular objects alongside their jets.

Abstract

For an arbitrary subset we introduce and study the three vanishing subspaces of the Hölder space consisting of those functions for which the ratio vanishes, when , or We prove that the Whitney extension operator maps each of these vanishing subspaces from to the corresponding vanishing spaces defined on the whole ambient space In fact, this follows as the zeroth order special case of a more general problem involving higher order derivatives. As a consequence, we obtain complete characterizations of approximability of Hölder functions by Lipschitz and boundedly supported functions.
Paper Structure (5 sections, 6 theorems, 103 equations)

This paper contains 5 sections, 6 theorems, 103 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Basic definitions and main results
  4. Necessity of Theorem \ref{['thm:main:ext']}
  5. Sufficiency of Theorem \ref{['thm:main:ext']}

Key Result

Theorem 1.7

Let $\omega$ be a modulus of continuity satisfying eq:mod:intro, $E\subset \mathbb{R}^n$ an arbitrary set, $m \in \mathbb{N} \cup \lbrace 0 \rbrace$, and $V$ a Banach space. For an $m$-jet $\lbrace A_k \rbrace_{k=0}^m \in \dot{\operatorname{J}}^{m,\omega}(E,V)$, the following hold. Moreover, when $E$ is closed, the result holds when $V$ is merely a normed space, and these extensions can be defin

Theorems & Definitions (18)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Remark 2.1
  • proof
  • ...and 8 more