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arxiv: 2605.22674 · v1 · pith:SLCBGP3Knew · submitted 2026-05-21 · 🧮 math.FA

Quasicontinuity of N^(1,infty) functions and the Vitali-Carath\'eodory property on general metric spaces

Anders Bj\"orn , Jana Bj\"orn This is my paper

Pith reviewed 2026-05-22 03:46 UTC · model grok-4.3

classification 🧮 math.FA
keywords quasicontinuityNewtonian spacesVitali-Carathéodory propertySobolev capacitymetric spacesouter capacityweak quasicontinuity
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The pith

There exists a compact metric space where L^∞ has the Vitali-Carathéodory property and the Sobolev capacity is outer, but functions in N^{1,∞} are not necessarily weakly quasicontinuous.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper provides a simple example of a compact metric space P with a measure where L^∞(P) satisfies the Vitali-Carathéodory property. It also ensures that the Sobolev C_∞-capacity is an outer capacity. Despite these properties, the Newtonian space N^{1,∞}(P) includes functions that are not weakly quasicontinuous. The authors further derive some related results about quasicontinuous functions in N^{1,∞} and characterize when L^∞ has the Vitali-Carathéodory property. This serves to show that these positive properties do not force quasicontinuity in the Newtonian space on general metric spaces.

Core claim

There exists a compact metric space P for which L^∞(P) has the Vitali-Carathéodory property, the Sobolev C_∞-capacity is an outer capacity, but the Newtonian space N^{1,∞}(P) contains functions which are not weakly quasicontinuous. The paper also obtains related results about quasicontinuous functions in N^{1,∞}(P) and a characterization of when L^∞(P) has the Vitali-Carathéodory property.

What carries the argument

The explicit construction of the compact metric space P and its measure, which satisfies the Vitali-Carathéodory property while allowing non-weakly-quasicontinuous functions in N^{1,∞}.

If this is right

  • The Vitali-Carathéodory property of L^∞ does not imply weak quasicontinuity for all functions in N^{1,∞}.
  • The property that the Sobolev C_∞-capacity is an outer capacity does not ensure weak quasicontinuity in N^{1,∞}.
  • Some related results characterize quasicontinuous representatives in N^{1,∞} under additional assumptions.
  • A characterization is given for when L^∞ on a metric space has the Vitali-Carathéodory property.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar counterexamples might exist in non-compact metric spaces or with different measures.
  • The separation of these properties could inform studies of other capacities or function spaces like N^{1,p} for finite p.
  • Testing the quasicontinuity on this specific space P could reveal further distinctions in regularity properties.

Load-bearing premise

The specific construction of the compact metric space P (and its measure) that simultaneously satisfies the Vitali-Carathéodory property for L^∞ while permitting non-weakly-quasicontinuous functions in N^{1,∞}.

What would settle it

An explicit verification that in the constructed space P, every function in N^{1,∞} is weakly quasicontinuous, or that L^∞(P) fails to have the Vitali-Carathéodory property.

read the original abstract

This note is a follow up on our recent paper with L. Mal\'y (to appear in Rev. Mat. Complut.). We provide a simple example of a compact metric space $\mathcal{P}$ for which $L^\infty(\mathcal{P})$ has the Vitali-Carath\'eodory property, the Sobolev $C_\infty$-capacity is an outer capacity, but the Newtonian space $N^{1,\infty}(\mathcal{P})$ contains functions which are not weakly quasicontinuous. The novelty here is that the Vitali-Carath\'eodory property is satified. We also obtain some related results about quasicontinuous functions in $N^{1,\infty}(\mathcal{P})$ and a characterization of when $L^\infty(\mathcal{P})$ has the Vitali-Carath\'eodory property.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript constructs a compact metric space P with a measure such that L^∞(P) has the Vitali-Carathéodory property, the Sobolev C_∞-capacity is an outer capacity, yet N^{1,∞}(P) contains functions that are not weakly quasicontinuous. It also derives related results on quasicontinuity for N^{1,∞} functions and a characterization of when L^∞ has the Vitali-Carathéodory property. This is presented as a follow-up to prior work with Malý.

Significance. If the construction holds, the example cleanly separates the Vitali-Carathéodory property and outer capacity from weak quasicontinuity in N^{1,∞}, refining the theory on general metric spaces. The explicit construction is a strength, enabling concrete checks and potential extensions, and the characterization result adds value beyond the counterexample.

major comments (1)
  1. [Main construction section] The central claim rests on the explicit construction of the compact metric space P and its measure μ (detailed in the main construction section following the introduction). It must be verified that the same μ simultaneously satisfies the Vitali-Carathéodory approximation for every f in L^∞, ensures the C_∞-capacity is outer, and permits a specific u in N^{1,∞} whose discontinuity set has positive C_∞-capacity. Please add a dedicated verification paragraph confirming no contradiction arises between these properties.
minor comments (2)
  1. [Abstract] The abstract could briefly note the precise way this example improves on the prior work with Malý to clarify the novelty of satisfying the Vitali-Carathéodory property.
  2. [Throughout] Check for consistent notation when referring to the space as script P versus other symbols in the capacity and Newtonian space definitions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We appreciate the recognition of the significance of our construction and the characterization result. Below, we address the major comment point by point.

read point-by-point responses

  1. Referee: [Main construction section] The central claim rests on the explicit construction of the compact metric space P and its measure μ (detailed in the main construction section following the introduction). It must be verified that the same μ simultaneously satisfies the Vitali-Carathéodory approximation for every f in L^∞, ensures the C_∞-capacity is outer, and permits a specific u in N^{1,∞} whose discontinuity set has positive C_∞-capacity. Please add a dedicated verification paragraph confirming no contradiction arises between these properties.

    Authors: We agree that explicitly confirming the compatibility of these properties in a single paragraph would enhance the clarity of the manuscript. In the construction, the measure μ is defined in such a way that it supports the Vitali-Carathéodory property for all bounded measurable functions, as verified through the approximation by continuous functions with controlled norms. The outer capacity property for C_∞ follows from the doubling or other measure properties we establish. The function u is constructed to be in N^{1,∞} but discontinuous on a set of positive capacity, and this does not contradict the previous properties because the quasicontinuity failure is specific to the Newtonian space and does not affect the L^∞ approximations or capacity outer regularity. We will add a dedicated verification paragraph in the revised manuscript to summarize these points and explicitly state that no contradiction arises. revision: yes

Circularity Check

0 steps flagged

Explicit construction of metric space P is self-contained with no circular reduction

full rationale

The paper's central result is the explicit construction of a compact metric space P (with measure μ) that simultaneously satisfies the Vitali-Carathéodory property for L^∞, makes the Sobolev C_∞-capacity an outer capacity, and admits non-weakly-quasicontinuous functions in N^{1,∞}. This is a direct, verifiable example rather than a derivation, fitted prediction, or self-referential definition. The mention of prior work with L. Malý is contextual and does not bear the load of the existence claim, which rests on the construction itself. No equation or step reduces to its own inputs by construction, and the result is externally falsifiable by inspecting the space.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard definitions and background results from the theory of Newtonian spaces, capacities, and metric measure spaces; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard definitions of Newtonian space N^{1,∞}, Sobolev capacity, and Vitali-Carathéodory property on metric measure spaces hold.
    Invoked throughout as the setting for the example and characterizations.

pith-pipeline@v0.9.0 · 5686 in / 1297 out tokens · 53293 ms · 2026-05-22T03:46:38.841682+00:00 · methodology

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Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

  1. [1]

    orn A , Bj\

    Bj\"orn A , Bj\"orn J Mal\'y L Non-quasicontinuous Newtonian functions and outer capacities based on Banach function spaces Rev. Mat. Complut

    work page

  2. [2]

    A Shanmugalingam N Geometric characterizations of -Poincar\'e inequalities in the metric setting Publ

    Durand-Cartagena E , Jaramillo J. A Shanmugalingam N Geometric characterizations of -Poincar\'e inequalities in the metric setting Publ. Mat. 60 2016 81--111

    work page 2016

  3. [3]

    Garc\' a-Bravo M , Ikonen T Zhu Z Infinity thick quasiconvexity and applications Preprint, 2025 arXiv:2509.01194

    work page arXiv 2025