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arxiv: 2309.02928 · v2 · submitted 2023-09-06 · 🧮 math.AP · math.CA· math.FA

Equivalence of Sobolev norms in Lebesgue spaces for Hardy operators in a half-space

The Anh Bui , Konstantin Merz This is my paper

Pith reviewed 2026-05-24 06:59 UTC · model grok-4.3

classification 🧮 math.AP math.CAmath.FA
keywords Hardy operatorsSobolev normshalf-spacesquare function estimatesheat kernelsfractional LaplacianSchrödinger operatorspotential
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The pith

Homogeneous L^p-Sobolev norms for Hardy operators in a half-space match those of the pure Laplacian when the potential is present.

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 homogeneous Sobolev spaces generated by Hardy operators, formed by the Laplacian or fractional Laplacian plus a potential depending only on the distance to the boundary, coincide with the spaces generated by the operators without that potential. The comparison is carried out in L^p for p in the admissible range by means of new square function estimates that apply when heat kernels decay slowly. A reader would care because the equivalence lets results on regularity, boundedness, or solvability transfer directly between the potential and potential-free cases. The statements cover all admissible coupling constants in the local setting and repulsive potentials in the fractional setting, and they extend earlier L^2 comparisons.

Core claim

We compare the scales of homogeneous L^p-Sobolev spaces generated by Hardy operators consisting of the ordinary or fractional Laplacian in a half-space plus a distance-dependent potential with the scales generated by the same operators without the potential. The comparison is proved by establishing new square function estimates for operators whose heat kernels decay slowly. The results hold for all admissible coupling constants in the local case and for repulsive potentials in the fractional case; they also cover attractive potentials in the fractional case once the expected heat kernel estimates are available.

What carries the argument

Square function estimates for operators with slowly decaying heat kernels, which are used to prove equivalence of the Sobolev norms with and without the potential.

If this is right

  • The equivalence of the Sobolev scales transfers boundedness and regularity properties between the potential and potential-free settings.
  • The L^2 equivalences obtained recently extend to the full range of L^p spaces.
  • In the fractional case the equivalence holds for all repulsive potentials and, conditionally, for attractive potentials once the corresponding heat kernel bounds are verified.

Where Pith is reading between the lines

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

  • The square function method may apply to Hardy-type operators on domains other than the half-space whenever comparable heat kernel decay is available.
  • Once the attractive fractional case is settled, the equivalence would immediately give L^p well-posedness results for fractional Schrödinger equations with inverse-power boundary potentials.
  • The same estimates could be used to compare homogeneous and inhomogeneous Sobolev spaces generated by these operators.

Load-bearing premise

The new square function estimates for operators with slowly decaying heat kernels hold and suffice to prove the norm equivalences.

What would settle it

An explicit function in L^p for which the homogeneous Sobolev norm computed with the Hardy operator differs from the norm computed without the potential, for some admissible coupling constant and p, would disprove the claimed equivalence.

read the original abstract

We consider Hardy operators, i.e., homogeneous Schr\"odinger operators consisting of the ordinary or fractional Laplacian in a half-space plus a potential, which only depends on the appropriate power of the distance to the boundary of the half-space. We compare the scales of homogeneous $L^p$-Sobolev spaces generated by these Hardy operators with and without potential with each other. To that end, we prove and use new square function estimates for operators with slowly decaying heat kernels. Our results hold for all admissible coupling constants in the local case and for repulsive potentials in the fractional case, and extend those obtained recently in $L^2$. They also cover attractive potentials in the fractional case, once expected heat kernel estimates are available.

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 / 2 minor

Summary. The paper establishes the equivalence of homogeneous L^p-Sobolev norms generated by Hardy operators (the Laplacian or fractional Laplacian plus a potential depending only on the distance to the boundary) with and without the potential, in the half-space setting. The equivalence is obtained via new square-function estimates for operators whose heat kernels decay slowly; the results hold unconditionally for all admissible couplings in the local case and for repulsive potentials in the fractional case, and conditionally on heat-kernel bounds for attractive fractional potentials. The work extends recent L^2 results to the full range of Lebesgue exponents.

Significance. If the square-function estimates are valid, the paper supplies a useful comparison between the scales of homogeneous Sobolev spaces associated with these Hardy operators, thereby extending the L^2 theory to L^p and furnishing a technical tool (square functions for slowly decaying kernels) that may apply more broadly. The explicit separation of unconditional and conditional cases is a strength.

minor comments (2)
  1. [Abstract] Abstract, p. 1: the phrase 'once expected heat kernel estimates are available' is left without a precise reference or statement of the expected bound; adding a short parenthetical description of the anticipated decay would clarify the conditional clause.
  2. [Introduction] The manuscript cites prior L^2 work but does not indicate whether any of the new L^p estimates reduce to the L^2 case by specialization of p; a brief remark in the introduction would make the extension explicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation of minor revision. No major comments appear in the report, so we have no specific points to address point by point at this stage. We remain available to incorporate any minor changes requested by the editor.

Circularity Check

0 steps flagged

Minor self-citation of L^2 results; new estimates introduced for L^p

full rationale

The paper extends prior L^2 results on Hardy operators to L^p spaces by proving new square-function estimates for operators with slowly decaying heat kernels. The abstract states these estimates are proved for the local case and repulsive fractional case (with attractive case conditional on heat-kernel bounds), and the norm equivalences follow from them. The reference to 'those obtained recently in L^2' is a standard extension step rather than a load-bearing self-citation that reduces the central claim to a prior result by construction. No fitted inputs, self-definitional steps, or ansatz smuggling are present; the derivation chain contains independent analytic content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; ledger populated from stated assumptions in the abstract. Relies on standard properties of the Laplacian and heat kernels plus the existence of the new square-function estimates.

axioms (2)
  • standard math Standard properties of the ordinary and fractional Laplacian on half-spaces
    Invoked implicitly as background for defining the Hardy operators.
  • domain assumption Existence of heat kernel estimates (slow decay) for the operators considered
    Required to prove the square-function estimates used for the equivalence.

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discussion (0)

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Reference graph

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