Explicit constants in $L^p$-Hardy inequalities for Aharonov-Bohm potentials

Durvudkhan Suragan

An explicit L^p-Hardy constant for the Aharonov-Bohm potential strictly exceeds the free constant and is controlled by the distance of the flux to the integers.

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2026-07-15 04:16 UTC pith:ARV3QWET

load-bearing objection Explicit constructive AB Hardy constant answering two open questions; the load-bearing angular bound is only asserted, not checkable from the abstract. the 2 major comments →

arxiv 2607.12671 v1 pith:ARV3QWET submitted 2026-07-14 math.AP math.FA

Explicit constants in L^p-Hardy inequalities for Aharonov-Bohm potentials

Durvudkhan Suragan This is my paper

classification math.AP math.FA MSC 35A2326D1035J1081Q10

keywords Aharonov-Bohm potentialL^p-Hardy inequalityexplicit constantstwisted angular constantmagnetic Hardy inequalityflux quantizationcompactness-free proof

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes a constructive, compactness-free proof of the L^p-Hardy inequality for the two-dimensional Aharonov-Bohm magnetic potential when the flux is non-integer and 1

Core claim

For the Aharonov-Bohm potential A_β with β not an integer and 1<p<2 the L^p-Hardy inequality holds with the explicit constant [((2-p)/p)^2+(sin(π dist(β,ℤ))/π)^2]^{p/2}, which is strictly larger than the free constant ((2-p)/p)^p and is comparable to a quantity that depends only on dist(β,ℤ).

What carries the argument

A compactness-free two-sided bound on the twisted angular constant associated with the Aharonov-Bohm flux; the bound is controlled by sin(π dist(β,ℤ))/π and replaces the earlier non-constructive compactness argument.

Load-bearing premise

The twisted angular constant admits a genuine two-sided bound controlled by sin(π times the distance of the flux to the integers); if that angular estimate fails or is only one-sided, the explicit constant and the comparability claim both collapse.

What would settle it

Compute or rigorously re-derive the two-sided bound on the twisted angular constant for a concrete non-integer flux (e.g., β=1/2) and check whether its lower and upper estimates match sin(π dist(β,ℤ))/π up to absolute constants independent of β; any mismatch falsifies the claimed explicit Hardy constant.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

The best Hardy constant λ_β(p) is now known to be at least the displayed explicit expression and is comparable to a function of dist(β,ℤ) alone.
When p≥2 the same magnetic field yields an L^p-Hardy inequality with the ordinary homogeneous weight |x|^{-p}.
Complex Aharonov-Bohm potentials inherit new L^p-Hardy inequalities with fully explicit constants.
Any estimate that previously used only the free Hardy constant can be improved by the magnetic increment whenever the flux is non-integer.

Where Pith is reading between the lines

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

The same angular bound should produce explicit remainder terms or improved constants for related magnetic inequalities (e.g., Rellich-type or weighted Sobolev inequalities) once the corresponding free constants are known.
Because the constant depends continuously on dist(β,ℤ) and vanishes exactly at integers, the magnetic improvement can be used as a quantitative probe of flux quantization in numerical or spectral experiments.
An analogous two-sided angular estimate for higher-dimensional or multi-flux Aharonov-Bohm systems would immediately yield explicit multi-particle Hardy constants.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Explicit constructive AB Hardy constant answering two open questions; the load-bearing angular bound is only asserted, not checkable from the abstract.

read the letter

Punchline: this answers the two concrete open questions left by Cazacu–Krejčiřík–Lam–Laptev—constructive proof and explicit dist(β,ℤ) dependence—for the L^p-Hardy constant of the 2D Aharonov–Bohm potential when 1

What is actually new is the compactness-free two-sided bound on the twisted angular constant that produces the closed form, plus the byproduct that for p≥2 the AB field still yields an L^p-Hardy inequality with the usual homogeneous weight |x|^{-p}, and the extension to complex AB potentials with explicit constants. Prior work only got the strict improvement via compactness; the formula and the constructive route are the real increments. The abstract is clean, the free constant is the classical external benchmark, and there is no data-fitting or free parameters, so circularity risk is low.

The soft spot is exactly the one the stress-test flags: we have only the abstract, so the load-bearing two-sided angular estimate controlled by sin(π dist(β,ℤ))/π is asserted rather than exhibited. If that estimate holds (even one-sided of the same order), the rest of the claims follow; if it fails, the explicit constant and the comparability collapse. Everything else—mode reduction, p≥2 observation, complex case—is secondary. That is a verification issue, not a conceptual red flag, and it is proportionate to an abstract-only read.

This is for people who already work on magnetic Hardy inequalities or spectral theory of AB operators. A serious referee in that subfield will get value from checking the angular bound and the constants. It does not reorganize a major branch of analysis, but it is a solid, honest constructive extension of known theory. I would send it to peer review rather than desk-reject; the questions it answers are real and the claimed evidence is of the right type.

Referee Report

2 major / 0 minor

Summary. The manuscript addresses L^p-Hardy inequalities for the two-dimensional Aharonov-Bohm potential A_β with flux β∉ℤ. For 1<p<2 it claims that a compactness-free two-sided bound on the twisted angular constant yields the explicit Hardy constant [((2-p)/p)^2+(sin(π dist(β,ℤ))/π)^2]^{p/2}, which strictly exceeds the free constant ((2-p)/p)^p and is comparable to a quantity depending only on dist(β,ℤ), thereby answering questions of Cazacu–Krejčiřík–Lam–Laptev. Byproducts asserted include an L^p-Hardy inequality with the homogeneous weight |x|^{-p} for p≥2 and new explicit inequalities for complex AB potentials.

Significance. If the two-sided angular bound is correctly established and produces the stated constant, the work supplies a constructive, explicit improvement of the free Hardy constant in the magnetic setting and removes reliance on compactness arguments. Explicit constants and compactness-free proofs are of genuine interest for Hardy inequalities and magnetic Schrödinger operators; the byproducts for p≥2 and complex potentials would further increase the paper’s utility. The contribution is therefore potentially substantial, but its value is entirely contingent on the correctness of the angular estimate.

major comments (2)

The load-bearing analytic input is a compactness-free two-sided bound on the twisted angular constant controlled by sin(π dist(β,ℤ))/π. The abstract asserts this bound and inserts it into the displayed constant, yet supplies neither the precise definition of the twisted angular constant nor any intermediate estimate. Without that derivation (or at least a one-sided lower bound of the same order), neither the strict improvement over ((2-p)/p)^p nor the claimed comparability with a quantity depending only on dist(β,ℤ) can be verified. This is the single point on which the central claim rests.
Only the abstract is available for review. Consequently the reduction to angular modes, the passage from the angular bound to the full Hardy inequality, the byproduct for p≥2, and the treatment of complex AB potentials remain unchecked. A full manuscript is required before any definitive assessment of correctness can be made.

Circularity Check

0 steps flagged

No significant circularity; abstract-only analytic claim with external classical benchmark and no fitted parameters or load-bearing self-definition.

full rationale

Only the abstract is available. It frames an explicit L^p-Hardy constant for the Aharonov-Bohm potential as obtained from a compactness-free two-sided bound on the twisted angular constant, answering open questions posed by Cazacu-Krejčiřík-Lam-Laptev. The free constant ((2-p)/p)^p is the classical external benchmark; the new constant is written in closed form involving that quantity plus (sin(π dist(β,ℤ))/π)^2. No parameters are fitted to data, no quantity is defined in terms of the claimed output, and no uniqueness theorem or ansatz is imported from the authors' own prior work as a load-bearing step. The citation to prior work merely states the problem that used compactness; the present claim is that a different, constructive estimate replaces it. Without the full text one cannot inspect intermediate equations, but nothing in the supplied abstract exhibits a self-definitional reduction, a fitted input renamed as prediction, or a self-citation chain that forces the result by construction. Per the analyzer rules this is the ordinary non-finding: score 0, empty steps.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Abstract-only review: free parameters are absent (the constant is closed-form in p and β). Axioms are the standard analytic background for magnetic Hardy inequalities plus the unproved (here) two-sided angular bound. No new physical entities are introduced; the Aharonov-Bohm potential is classical.

axioms (3)

domain assumption Classical free L^p-Hardy inequality with constant ((2-p)/p)^p on R^2 for 1<p<2.
Used as the external benchmark that the magnetic constant is claimed to exceed; standard in the literature the abstract cites.
ad hoc to paper The two-dimensional Aharonov-Bohm vector potential A_β with flux β∉ℤ is well-defined and induces a twisted angular operator whose ground-state constant admits a two-sided bound controlled by sin(π dist(β,ℤ))/π.
This is the paper's key analytic input replacing the prior compactness argument; asserted in the abstract without displayed proof.
standard math Standard functional-analytic setting for magnetic L^p-Hardy inequalities (Sobolev-type spaces, distributional gradients with magnetic potential).
Background framework assumed throughout the abstract and the cited prior work.

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0 comments

read the original abstract

For the two-dimensional Aharonov-Bohm potential $A_\beta$ with flux $\beta\notin\mathbb{Z}$ and $1<p<2$, Cazacu, Krej\v{c}i\v{r}\'{\i}k, Lam and Laptev proved by a compactness argument that their constant $\lambda_\beta(p)$ in the $L^p$-Hardy inequality strictly exceeds the free constant $\big(\tfrac{2-p}{p}\big)^p$, and asked for a constructive proof with explicit estimates and for comparability of $\lambda_\beta(p)$ with a quantity depending on $\text{dist}(\beta,\mathbb{Z})$. We answer both questions by using a compactness-free two-sided bound for the twisted angular constant. Our explicit Hardy constant is $$\big[\big(\tfrac{2-p}{p}\big)^{2}+\big(\tfrac{\sin(\pi\text{dist}(\beta,\mathbb{Z}))}{\pi}\big)^{2}\big]^{p/2},\quad 1<p<2.$$ As a byproduct we observe that when $p\ge 2$ the Aharonov--Bohm field produces an $L^p$-Hardy inequality with the usual homogeneous weight $|x|^{-p}$. Our approach also provides new $L^p$-Hardy inequalities with explicit constants for the complex AB potentials.