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arxiv: 2606.05477 · v1 · pith:7LNCOHTPnew · submitted 2026-06-03 · 🧮 math.AP

Robust interpolation inequalities via Chebyshev-type integral inequalities

Guy Foghem This is my paper

Pith reviewed 2026-06-28 04:48 UTC · model grok-4.3

classification 🧮 math.AP
keywords interpolation inequalitiesGagliardo seminormsChebyshev integral inequalitiesregional fractional p-Laplaciannonlocal-to-local limitstability of solutionsSobolev spaces
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The pith

Robust log-convex interpolation inequalities hold for Gagliardo seminorms via Chebyshev-type integral inequalities.

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

The paper establishes robust log-convex interpolation inequalities within the scale of Gagliardo seminorms by deriving Chebyshev-type integral inequalities that hold for general non-synchronous functions. This work is motivated by the need to analyze the stability of weak solutions to the regional fractional p-Laplacian Dirichlet problem as the order s approaches the local case from below. Under the hypothesis that the forcing term f_s and boundary data g_s converge appropriately, the solutions u_s converge to the local solution u_1 in the space W^{η,p}(Ω) for every η strictly less than 1. A sympathetic reader cares because the inequalities supply a quantitative bridge between nonlocal and local regimes without requiring synchronous behavior in the integrands.

Core claim

By deriving Chebyshev-type integral inequalities for general non-synchronous functions, robust log-convex interpolation inequalities are obtained within the scale of Gagliardo seminorms. These inequalities are then applied to the boundary-value problem for the regional fractional p-Laplacian: if u_s in W^{s,p}(Ω) satisfies (−Δ)_{p,Ω}^s u_s = f_s in Ω and γ_0^s(u_s) = g_s on ∂Ω for 1/p < s ≤ 1, and if the data converge suitably as s → 1^−, then ||u_s − u_1||_{W^{η,p}(Ω)} → 0 for all 0 ≤ η < 1.

What carries the argument

Chebyshev-type integral inequalities for general non-synchronous functions, which are used to prove the log-convex interpolation inequalities in Gagliardo seminorms.

Load-bearing premise

The data f_s and g_s converge in an appropriate sense as s approaches 1 from below.

What would settle it

An explicit pair of sequences f_s and g_s that converge as s → 1^− together with a corresponding family of solutions u_s that fail to converge in some W^{η,p}(Ω) for η < 1.

Figures

Figures reproduced from arXiv: 2606.05477 by Guy Foghem.

Figure 1
Figure 1. Figure 1: Two-dimensional illustration with θ = 0.6. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
read the original abstract

We establish robust log-convex interpolation inequalities within the scale of Gagliardo seminorms. We achieve this by deriving some Chebyshev-type integral inequalities for general non-synchronous functions. Our primary motivation for establishing these robust interpolation inequalities stems from the study of the asymptotic nonlocal-to-local stability of weak solutions to the boundary Dirichlet problem associated with the regional fractional $p$-Laplacian. More precisely, if $u_s \in W^{s,p}(\Omega)$ weakly satisfies $(-\Delta)_{p, \Omega}^s u_s = f_s $ in $\Omega$ and $ \gamma^s_0(u_s) = g_s$ on $\partial\Omega,$ with $\frac{1}{p} < s \leq 1$ and $\Omega \subset \mathbb{R}^d$ is bounded Lipschitz, then, under appropriate convergence of the data $f_s$ and $g_s$ as $s \to 1^-$, we establish that $\| u_s - u_1 \|_{W^{\eta,p}(\Omega)} \xrightarrow{s \to 1^-} 0 $ for all $0 \leq \eta < 1$.

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

2 major / 2 minor

Summary. The manuscript claims to derive Chebyshev-type integral inequalities for non-synchronous functions that yield robust log-convex interpolation inequalities in the scale of Gagliardo seminorms. These are then applied to establish that weak solutions u_s to the regional fractional p-Laplacian Dirichlet problem (-Δ)_{p,Ω}^s u_s = f_s in Ω with γ_0^s(u_s)=g_s on δΩ converge in W^{η,p}(Ω) to the local solution u_1 as s o1^-, provided the data f_s and g_s converge appropriately.

Significance. If the Chebyshev-type inequalities are new and hold with the stated robustness (parameter-free or with explicit constants independent of certain parameters), they could serve as a technical tool for nonlocal-to-local passage in fractional Sobolev spaces. The stability statement, however, remains conditional on external data convergence and therefore does not by itself resolve the question of whether the weak formulation automatically produces the required convergence of f_s and g_s.

major comments (2)
  1. [Abstract] Abstract (paragraph beginning 'More precisely, if u_s ∈ W^{s,p}(Ω)…'): the stability conclusion ||u_s - u_1||_{W^{η,p}} o 0 is asserted only under the external hypothesis that f_s and g_s converge appropriately as s o1^-. No derivation is supplied showing that the weak form of the equation forces this convergence of the data; the interpolation inequalities therefore support only a conditional stability result rather than an unconditional one.
  2. [Abstract] The manuscript does not appear to contain an a priori estimate or compactness argument that would upgrade the assumed data convergence into the required convergence of the solutions without the external hypothesis; this leaves the central application claim dependent on an assumption that is not derived from the new inequalities.
minor comments (2)
  1. Notation for the regional fractional p-Laplacian (-Δ)_{p,Ω}^s and the trace operator γ_0^s should be defined explicitly in the introduction or preliminaries section rather than assumed known.
  2. [Abstract] The precise sense in which f_s and g_s are required to converge (e.g., in L^{p'}(Ω) or in the dual of the trace space) should be stated explicitly to make the stability statement verifiable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the comments on the abstract. We address the points below, noting that the result is intentionally presented as conditional on data convergence, which is explicitly stated in the abstract and is the intended scope of the application of the new inequalities.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph beginning 'More precisely, if u_s ∈ W^{s,p}(Ω)…'): the stability conclusion ||u_s - u_1||_{W^{η,p}} o 0 is asserted only under the external hypothesis that f_s and g_s converge appropriately as s o1^-. No derivation is supplied showing that the weak form of the equation forces this convergence of the data; the interpolation inequalities therefore support only a conditional stability result rather than an unconditional one.

    Authors: The manuscript explicitly states the result under the assumption of appropriate convergence of f_s and g_s as s → 1^-. It does not claim or attempt to derive that the weak formulation automatically forces convergence of the data; the Chebyshev-type inequalities are used to establish solution convergence assuming the data converge. This conditional stability is the precise claim supported by the new inequalities. revision: no

  2. Referee: [Abstract] The manuscript does not appear to contain an a priori estimate or compactness argument that would upgrade the assumed data convergence into the required convergence of the solutions without the external hypothesis; this leaves the central application claim dependent on an assumption that is not derived from the new inequalities.

    Authors: The central application is the conditional stability result, for which the new inequalities supply the key tool to pass to the limit once data convergence is assumed. An a priori estimate removing the external hypothesis on the data would constitute a separate result outside the scope of the present work, whose focus is the derivation of the robust interpolation inequalities and their application in the conditional setting. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivations are independent

full rationale

The paper derives Chebyshev-type integral inequalities for general non-synchronous functions from first principles and applies them to obtain robust log-convex interpolation inequalities in Gagliardo seminorms. These are then used for a conditional stability result on weak solutions to the regional fractional p-Laplacian, which explicitly requires an external hypothesis of data convergence f_s, g_s as s→1^− rather than deriving that convergence. No step reduces by construction to a fitted input, self-citation chain, or definitional equivalence; the central claims rest on independently derived inequalities.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the statements rely on standard properties of Gagliardo seminorms and weak solutions but these are not itemized.

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