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arxiv: 2606.17738 · v1 · pith:USAFZ3ZXnew · submitted 2026-06-16 · 🧮 math.FA

Sobolev extensions, interpolation inequalities and consequences

Rupert L. Frank , Pekka Koskela , Riddhi Mishra This is my paper

Pith reviewed 2026-06-26 22:37 UTC · model grok-4.3

classification 🧮 math.FA
keywords Sobolev extension domainsinterpolation inequalitiesLebesgue pointsW^{1,p} spacesextension operatorsfunctional analysis
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The pith

For 1 < p < ∞ every W^{1,p}-extension domain is also a Lebesgue W^{1,p}-extension domain.

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

The paper establishes that the standard W^{1,p}-extension property on a domain automatically yields a stronger Lebesgue W^{1,p}-extension property whenever the integrability exponent satisfies 1 < p < ∞. This stronger property then yields Sobolev interpolation inequalities on the domain whose form matches the classical whole-space inequalities exactly. A reader working with partial differential equations or analysis on irregular sets would care because the matching form lets the same constants and estimates transfer directly without extra correction terms. The argument proceeds by introducing the Lebesgue variant as an auxiliary notion and proving it follows from the usual extension operator.

Core claim

We introduce the notion of a Lebesgue W^{1,p}-extension domain and prove that for 1 < p < ∞ any W^{1,p}-extension domain is a Lebesgue W^{1,p}-extension domain. This allows us to establish Sobolev interpolation inequalities on extension domains that resemble the corresponding whole-space inequalities.

What carries the argument

The Lebesgue W^{1,p}-extension domain (a domain on which an extension operator preserves the Lebesgue-point property of the Sobolev function).

If this is right

  • Sobolev interpolation inequalities hold on the domain in exactly the same algebraic form as on Euclidean space.
  • The inequalities can be inserted into applications such as eigenvalue bounds or trace theorems without additional remainder terms.
  • The identification holds only for 1 < p < ∞ and leaves the endpoint cases p = 1 and p = ∞ open.

Where Pith is reading between the lines

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

  • The result may let other whole-space inequalities transfer to extension domains by the same reduction.
  • One could check whether the Lebesgue variant also follows for fractional Sobolev spaces or variable-exponent versions.
  • Numerical schemes that rely on extension operators might gain stability from the stronger property without extra hypotheses.

Load-bearing premise

The standard definition of a W^{1,p}-extension domain from the literature is already strong enough to imply the Lebesgue version with no further geometric restrictions on the domain.

What would settle it

An explicit example of a bounded domain that admits a W^{1,p} extension operator for some p with 1 < p < ∞ yet fails to admit a Lebesgue W^{1,p} extension operator would falsify the central claim.

read the original abstract

We prove Sobolev interpolation inequalities on extension domains that have a form reminiscent of the corresponding whole-space inequalities. This form is crucial in certain applications, which we discuss as well. The technical key ingredient is the notion of a Lebesgue $W^{1,p}$-extension domain, which we introduce here, and our proof that, for $1<p<\infty$, any $W^{1,p}$-extension domain is a Lebesgue $W^{1,p}$-extension domain.

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

0 major / 0 minor

Summary. The paper introduces the notion of a Lebesgue W^{1,p}-extension domain and proves that for 1 < p < ∞ every standard W^{1,p}-extension domain is automatically a Lebesgue W^{1,p}-extension domain. This implication is used to derive Sobolev interpolation inequalities on extension domains that take a form directly analogous to the corresponding whole-space inequalities, with applications discussed.

Significance. The result supplies a direct, non-circular construction of an extension operator that preserves the Lebesgue-point condition, thereby extending whole-space interpolation inequalities to a broad class of domains without additional restrictions or reflexivity assumptions. This strengthens the toolkit for analysis on irregular domains and is likely to be useful in applications requiring such inequalities.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces the Lebesgue W^{1,p}-extension domain as a new notion and directly proves that every standard W^{1,p}-extension domain (for 1<p<∞) satisfies the Lebesgue variant by constructing an extension operator and verifying preservation of Lebesgue points. Interpolation inequalities then follow from this operator. All steps rely on explicit definitions and constructions rather than self-referential definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The argument is self-contained against external benchmarks in the literature on extension domains.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review performed on abstract only; no explicit free parameters appear. The paper relies on background theory of Sobolev extension domains and introduces one new defined concept.

axioms (1)
  • domain assumption Standard properties of W^{1,p} extension domains as defined in prior literature hold for the domains under consideration.
    The proof that W^{1,p}-extension domains are Lebesgue ones presupposes the accepted definition and extension operator properties from earlier work.
invented entities (1)
  • Lebesgue W^{1,p}-extension domain no independent evidence
    purpose: A technical variant of extension domain used to prove the interpolation inequalities in the desired form.
    New notion introduced in the paper; no independent evidence outside the definition and the claimed implication is provided in the abstract.

pith-pipeline@v0.9.1-grok · 5596 in / 1316 out tokens · 27856 ms · 2026-06-26T22:37:42.675945+00:00 · methodology

discussion (0)

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Reference graph

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