<p>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 <i>X</i> as the function spaces defining the smoothness of the Sobolev-type functions. These results are in contrast to the case <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$X=L^p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>=</mo> <msup> <mi>L</mi> <mi>p</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$1\le p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, where all Sobolev-type functions in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$N^{1,p}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>N</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> are known to be quasicontinuous, provided that the underlying metric space <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">P</mi> </math></EquationSource> </InlineEquation> is locally complete. In most of our examples, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">P</mi> </math></EquationSource> </InlineEquation> is a compact subset of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textbf{R}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="bold">R</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$X=L^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>=</mo> <msup> <mi>L</mi> <mi>∞</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. 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.</p>

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

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2026年4月24日

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Anders Björn,
Jana Björn,
Lukáš Malý

摘要

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 $X=L^p$ $X = L^{p}$ with $1\le p<\infty $ $1 \leq p < \infty$ , where all Sobolev-type functions in $N^{1,p}$ $N^{1, p}$ are known to be quasicontinuous, provided that the underlying metric space $\mathcal {P}$ $P$ is locally complete. In most of our examples, $\mathcal {P}$ $P$ is a compact subset of $\textbf{R}^2$ $R^{2}$ and $X=L^\infty $ $X = L^{\infty}$ . 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.