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Non-quasicontinuous Newtonian functions and outer capacities based on Banach function spaces

Anders Björn, Jana Björn, Lukáš Malý

Abstract

We construct various examples of Sobolev-type functions, defined via upper gradients in metric spaces, that fail to be quasicontinuous or weakly quasicontinuous. This is done with quasi-Banach function lattices $X$ as the function spaces defining the smoothness of the Sobolev-type functions. These results are in contrast to the case $X=L^p$ with $1\le p<\infty$, where all Sobolev-type functions in $N^p$ are known to be quasicontinuous, provided that the underlying metric space $\mathcal{P}$ is locally complete. In most of our examples, $\mathcal{P}$ is a compact subset of $\mathbf{R}^2$ and $X=L^\infty$. Four particular examples are the damped topologist's sine curve, the von Koch snowflake curve, the Cantor ternary set and the Sierpiński carpet. We also discuss several related properties, such as whether the Sobolev capacity is an outer capacity, and how these properties are related. A fundamental role in these considerations is played by the lack of the Vitali--Carathéodory property.

Non-quasicontinuous Newtonian functions and outer capacities based on Banach function spaces

Abstract

We construct various examples of Sobolev-type functions, defined via upper gradients in metric spaces, that fail to be quasicontinuous or weakly quasicontinuous. This is done with quasi-Banach function lattices as the function spaces defining the smoothness of the Sobolev-type functions. These results are in contrast to the case with , where all Sobolev-type functions in are known to be quasicontinuous, provided that the underlying metric space is locally complete. In most of our examples, is a compact subset of and . Four particular examples are the damped topologist's sine curve, the von Koch snowflake curve, the Cantor ternary set and the Sierpiński carpet. We also discuss several related properties, such as whether the Sobolev capacity is an outer capacity, and how these properties are related. A fundamental role in these considerations is played by the lack of the Vitali--Carathéodory property.

Paper Structure

This paper contains 4 sections, 7 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Banach function lattices and Newtonian spaces
  3. The proofs of Theorems \ref{['thm-qcont-char']} and \ref{['thm-qcont-char-cont-dense']}
  4. The proof of Theorem \ref{['thm-VC']}

Key Result

Theorem 1.1

Assume that there are $L^\infty$-almost no nonconstant rectifiable curves in the metric space $\mathcal{P}$ and that there is a point $x_0 \in \mathcal{P}$ with $\mu(\{x_0\})=0$. Then $N^{1,\infty}(\mathcal{P})= L^\infty(\mathcal{P})$ and the following are true: In particular, all of these facts hold if $\mathcal{P}$ is the von Koch snowflake curve equipped with the $\frac{\log4}{\log3}$-dimension

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 5 more