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arxiv: 1907.00186 · v1 · pith:TCC6F4LYnew · submitted 2019-06-29 · 🧮 math.AP

Integral representation using Green function for fractional Hardy equation

Mousomi Bhakta , Anup Biswas , Debdip Ganguly , Luigi Montoro This is my paper

Pith reviewed 2026-05-25 13:05 UTC · model grok-4.3

classification 🧮 math.AP
keywords Green functionfractional Hardy operatorintegral representationweak solutionsfractional LaplacianHardy inequality
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The pith

The Green function for the fractional Hardy operator represents its weak solutions as integrals.

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

The authors study the Green function of the operator P equals (-Delta)^s minus theta over |x| to the 2s in R^N for theta below the best constant in the fractional Hardy inequality. They establish that this Green function exists with sufficient regularity. They then use the Green function to show that weak solutions admit an integral representation. A sympathetic reader would care because the representation converts the nonlocal equation into an explicit integral formula involving only the right-hand side data.

Core claim

The Green function for the fractional Hardy operator P exists and is sufficiently regular when 0 less than theta less than Lambda sub N,s, and every weak solution u of P u equals f satisfies u(x) equals the integral of the Green function G(x,y) times f(y) dy.

What carries the argument

The Green function for the operator P equals (-Delta)^s minus theta over |x| to the 2s, which converts weak solutions into explicit integrals.

Load-bearing premise

The Green function for the operator P exists and is sufficiently regular in the stated range of theta to permit the integral representation.

What would settle it

Finding a weak solution u of P u equals f that cannot be recovered as the integral of any candidate Green function against f would disprove the representation claim.

read the original abstract

Our main aim is to study Green function for the fractional Hardy operator $P:=(-\Delta)^s -\frac{\theta}{|x|^{2s}}$ in $\mathbb{R}^N$, where $0<\theta<\Lambda_{N,s}$ and $\Lambda_{N,s}$ is the best constant in the fractional Hardy inequality. Using Green function, we also show that the integral representation of the weak solution holds.

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 / 0 minor

Summary. The paper studies the Green function for the fractional Hardy operator P = (-Δ)^s - θ/|x|^{2s} in R^N for 0 < θ < Λ_{N,s}, where Λ_{N,s} is the best constant in the fractional Hardy inequality. It claims that this Green function yields an integral representation for weak solutions of the associated equation.

Significance. If the result holds, the construction of a sufficiently regular Green function for this perturbed fractional Laplacian and the resulting integral representation formula would supply a useful analytic tool for studying weak solutions to fractional Hardy equations, extending classical representation techniques to this setting.

major comments (1)
  1. The central claim that weak solutions admit the integral representation u(x) = ∫ G(x,y) f(y) dy rests entirely on the existence and sufficient regularity of the Green function G for P in the range 0 < θ < Λ_{N,s}; the abstract supplies no proof sketch, technical conditions, or verification steps for this step.

Simulated Author's Rebuttal

1 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's report. We address the major comment below.

read point-by-point responses

  1. Referee: The central claim that weak solutions admit the integral representation u(x) = ∫ G(x,y) f(y) dy rests entirely on the existence and sufficient regularity of the Green function G for P in the range 0 < θ < Λ_{N,s}; the abstract supplies no proof sketch, technical conditions, or verification steps for this step.

    Authors: The manuscript constructs the Green function for the fractional Hardy operator P in the range 0 < θ < Λ_{N,s} and establishes its key properties, including sufficient regularity. Using this, we prove the integral representation formula for weak solutions. While the abstract does not include a proof sketch (consistent with its purpose as a summary), the full paper details the technical conditions, construction, and verification steps in the subsequent sections. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract states the goal of studying the Green function for P = (-Δ)^s - θ/|x|^{2s} (0 < θ < Λ_{N,s}) and showing that weak solutions admit an integral representation via that Green function. No equations, fitted parameters, self-citations, or derivations are supplied that reduce the claimed representation to a definition or input by construction. The result is presented as depending on an independent existence and regularity proof for the Green function, which is not shown to be self-referential. This is the most common honest finding for an abstract-only description of a technical existence result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, no invented entities, and only the standard domain assumption that the best Hardy constant Λ_{N,s} is known from prior work.

axioms (1)
  • domain assumption Λ_{N,s} is the best constant in the fractional Hardy inequality
    Used to delimit the admissible range 0 < θ < Λ_{N,s} for the potential coefficient.

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Reference graph

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