Quantitative stability for fractional Hardy inequalities: Rearrangement-free techniques and Emden-Fowler analysis

Avas Banerjee; Debdip Ganguly; Vivek Sahu

arxiv: 2605.15748 · v1 · pith:4W4AZ3DDnew · submitted 2026-05-15 · 🧮 math.AP

Quantitative stability for fractional Hardy inequalities: Rearrangement-free techniques and Emden-Fowler analysis

Avas Banerjee , Debdip Ganguly , Vivek Sahu This is my paper

Pith reviewed 2026-05-20 17:07 UTC · model grok-4.3

classification 🧮 math.AP

keywords fractional Hardy inequalityquantitative stabilityvirtual extremalsMarcinkiewicz spaceEmden-Fowler analysisnonlocal inequalitiesPoincaré-Sobolev inequality

0 comments

The pith

The fractional Hardy deficit is at least the distance to virtual extremals raised to the power max{4, 2p} under weighted normalization.

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

The paper proves a quantitative stability estimate for the sharp fractional Hardy inequality. With the integral of |u(x)|^p over |x|^{sp} normalized to one, the positive deficit is bounded from below by the alpha power of the distance from u to the family of virtual extremals, where alpha equals the maximum of four and twice p and distance is measured in Marcinkiewicz space. The argument proceeds by decomposing the nonlocal energy and controlling it via a localized Poincaré-Sobolev inequality together with rescaling and Lorentz embeddings. This yields an improvement on existing stability exponents for the local case s equals one and produces a new nonlocal Hardy-Heisenberg uncertainty principle.

Core claim

Under the normalization integral of |u(x)|^p / |x|^{sp} dx equals one, the deficit delta_{s,p}(u) is comparable from below to (dist_{s,p}(u, Z))^alpha with alpha equal to max{4, 2p}, where Z denotes the family of virtual extremals and the distance is taken in the Marcinkiewicz weak-L^{p_s^*} space. For p equals two an Emden-Fowler change of variables together with pseudo-differential analysis shows the nonlocal deficit coincides with its local counterpart, allowing stability to be read off from the diagonalization of the quadratic form on the cylinder R times S^{N-1}.

What carries the argument

Localized Poincaré-Sobolev inequality combined with rescaling and Lorentz embeddings that decompose the nonlocal energy and bound it by the Marcinkiewicz distance to the virtual extremals.

If this is right

The same stability estimate holds for the local Hardy inequality when s equals one, with an improved exponent over previous results.
When p equals two the nonlocal deficit is shown to equal the corresponding local deficit via the Emden-Fowler correspondence.
A Hardy-Heisenberg-type uncertainty principle holds in the nonlocal fractional setting.

Where Pith is reading between the lines

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

The rearrangement-free method may adapt to quantitative stability questions for other nonlocal inequalities such as the fractional Sobolev embedding.
Numerical tests in low dimensions could check whether the exponent alpha is optimal by constructing near-extremal sequences.
The reduction for p equals two suggests possible links to spectral theory on cylinders for other quadratic forms with radial weights.

Load-bearing premise

A localized Poincaré-Sobolev inequality together with suitable rescaling and Lorentz embeddings must hold so that the nonlocal energy difference can be controlled by the distance to the virtual extremals.

What would settle it

A sequence of functions normalized so that the weighted integral equals one, with Marcinkiewicz distance to Z tending to zero, yet whose deficit decays faster than any positive power with exponent max{4, 2p}.

read the original abstract

A classical result due to Frank and Seiringer asserts that for $1\leq p<\frac Ns$, there exists a sharp constant $\mathcal{C}_{N,s,p}>0$ such that $$ \delta_{s,p}(u):=\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}\,dx\,dy-\mathcal{C}_{N,s,p}\int_{\mathbb{R}^N}\frac{|u(x)|^p}{|x|^{sp}}\,dx\ge0, $$ for all $u\in W^{s,p}(\mathbb{R}^N)$. The optimal constant is explicitly known. We investigate quantitative refinements of this inequality. Our first result shows that, under the normalization $ \int_{\mathbb{R}^N}\frac{|u(x)|^p}{|x|^{sp}}\,dx=1,$ the inequality \[ \delta_{s,p}(u)\gtrsim\bigl(\mathrm{dist}_{s,p}(u,\mathcal{Z})\bigr)^\alpha, \] holds, where $\alpha=\max\{4,2p\}$, $\mathcal{Z}$ denotes the family of ``virtual'' extremals, and the distance is measured in Marcinkiewicz (weak-$L^{p_s^*}$) space. The stability exponent remains constant for $p\le2$, while it depends on $p$ for $p>2$. Our approach is based on a localized Poincar\'e-Sobolev inequality combined with suitable rescaling and Lorentz embeddings. We exploit a decomposition of the nonlocal energy together with Lorentz estimates, which enables us to control the deficit $\delta_{s,p}(u)$ in terms of the distance to $\mathcal{Z}$. The method also applies to the local case $s=1$, the argument is rearrangement-free and the exponent in the stability estimate improves the existing literature. For $p=2$, via an Emden-Fowler correspondence and pseudo-differential operators, we show that the nonlocal Hardy deficit coincides with the local one and obtain quantitative stability on $\mathbb{R}\times\mathbb{S}^{N-1}$ using the diagonalization of the fractional Hardy quadratic form due to Frank, Lieb, and Seiringer. As an application, we establish a Hardy-Heisenberg-type uncertainty principle in the nonlocal setting, which appears to be new in the literature.

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.

Desk Editor's Note private letter to a colleague

The paper gives explicit quantitative stability for the fractional Hardy inequality with exponent max{4,2p} that stays fixed for p≤2, using a rearrangement-free method based on localized inequalities and Lorentz embeddings, plus a new nonlocal uncertainty principle via Emden-Fowler for p=2.

read the letter

Hi, The main thing to know about this paper is that it establishes a quantitative stability inequality for the fractional Hardy deficit with an explicit exponent α = max{4, 2p} that remains constant when p is at most 2, and they achieve this without using rearrangements. For the case p=2 they use an Emden-Fowler reduction to show the nonlocal deficit is the same as the local one and derive a new nonlocal version of the Hardy-Heisenberg uncertainty principle. What stands out as new is the stability exponent itself and the fact that it doesn't worsen for smaller p, along with the rearrangement-free technique based on localized Poincaré-Sobolev inequalities, rescaling, and Lorentz embeddings. This lets them control the deficit directly from the distance to the family of virtual extremals in the Marcinkiewicz space. They also handle the local case s=1 with the same method. The Emden-Fowler analysis for p=2 connects nicely to existing diagonalization results from Frank, Lieb, and Seiringer, which keeps the argument from being too self-contained in a bad way. The application to the uncertainty principle is a direct consequence and seems fresh for the nonlocal setting. On the potential weaknesses, the approach depends on the constants in the localized inequalities and the Lorentz estimates remaining uniform under rescaling, especially near the origin where the singular weight |x|^{-sp} can cause problems. The stress-test concern about scale-dependent deterioration in the weak-L tails or localization radius is worth checking carefully in the proofs. If those constants blow up or depend on the scale parameter, the claimed lower bound with that specific α might not hold for all functions satisfying the normalization. The abstract outlines the steps plausibly, but the full error estimates would need to confirm no hidden dependencies. Measuring distance in Marcinkiewicz space rather than a stronger norm is a choice that makes the result weaker in some sense, though probably necessary. This paper is for specialists in fractional Sobolev spaces, Hardy inequalities, and quantitative functional inequalities in PDE theory. Someone working on stability versions of classical inequalities or on uncertainty principles in nonlocal settings would get value from the explicit exponent and the new principle. It improves on the Frank-Seiringer result by adding the quantitative layer with a different proof strategy. I would recommend sending it to peer review. The claims are specific, the method differs from prior work, and even if revisions are needed on the uniformity, the core ideas are worth a careful look by experts.

Referee Report

1 major / 2 minor

Summary. The paper establishes quantitative stability for the fractional Hardy inequality of Frank-Seiringer. Under the normalization ∫ |u(x)|^p / |x|^{sp} dx = 1, the deficit δ_{s,p}(u) satisfies δ_{s,p}(u) ≳ (dist_{s,p}(u, Z))^α with α = max{4, 2p}, where Z denotes the family of virtual extremals and distance is taken in the Marcinkiewicz (weak-L^{p_s^*}) space. The proof proceeds via a localized Poincaré-Sobolev inequality, rescaling, and Lorentz embeddings that decompose the nonlocal energy; for p = 2 an Emden-Fowler change of variables reduces the problem to a local quadratic form on ℝ × S^{N-1} whose diagonalization is known. The method is rearrangement-free, recovers the local case s = 1, and yields a new nonlocal Hardy-Heisenberg uncertainty principle.

Significance. If the claimed lower bound holds with the stated exponent, the result would strengthen the quantitative theory of Hardy inequalities by supplying an explicit stability exponent without rearrangements and by improving the exponent range relative to prior work. The explicit reduction for p = 2 via Emden-Fowler and the resulting uncertainty principle constitute concrete technical contributions.

major comments (1)

[§4] §4 (rescaling and localized Poincaré-Sobolev step): the argument invokes a localized Poincaré-Sobolev inequality whose constant is asserted to be uniform after rescaling. However, when the test function is concentrated at radii r ≪ 1, the localization radius must shrink proportionally to r; the subsequent Lorentz-space tail estimates then produce constants that may deteriorate with r. No explicit bound independent of the scale parameter is supplied, which is load-bearing for the claimed exponent α = max{4, 2p} holding uniformly for all normalized u.

minor comments (2)

[Introduction] The precise definition of the Marcinkiewicz distance dist_{s,p}(·, Z) and the family Z of virtual extremals should be stated explicitly (rather than only referenced) in the introduction and in the statement of the main theorem.
[§5] In the Emden-Fowler section, add a short paragraph clarifying how the pseudo-differential operator arising from the change of variables exactly reproduces the nonlocal deficit, so that the diagonalization result of Frank-Lieb-Seiringer applies directly.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to confirm scale-independence of constants in the localized estimates of §4. We address the concern directly below.

read point-by-point responses

Referee: [§4] §4 (rescaling and localized Poincaré-Sobolev step): the argument invokes a localized Poincaré-Sobolev inequality whose constant is asserted to be uniform after rescaling. However, when the test function is concentrated at radii r ≪ 1, the localization radius must shrink proportionally to r; the subsequent Lorentz-space tail estimates then produce constants that may deteriorate with r. No explicit bound independent of the scale parameter is supplied, which is load-bearing for the claimed exponent α = max{4, 2p} holding uniformly for all normalized u.

Authors: We agree that an explicit verification of uniformity is required. In the rescaling, for a concentration point x0 with |x0| = r ≪ 1 we set v(y) := r^γ u(r y + x0) with γ chosen so that the normalization ∫ |v(y)|^p / |y|^{sp} dy = 1 is preserved. The localized Poincaré-Sobolev inequality is then applied on a ball of unit radius in the y-variable. Because both the fractional Gagliardo seminorm and the weighted L^p term are homogeneous of the same degree under this scaling, the constant in the localized inequality is identical to the one obtained on the unit scale and therefore independent of r. The subsequent Lorentz-space tail estimates rely on the scale-invariance of the weak-L^{p_s^*} quasi-norm together with the specific choice of exponents in the decomposition; these estimates likewise carry constants independent of r. We will insert a short auxiliary lemma (or expanded remark) in §4 that records the explicit scaling factors and confirms the r-independence. This addition strengthens the presentation without changing the overall argument or the claimed exponent α. revision: yes

Circularity Check

0 steps flagged

No significant circularity; quantitative bound derived from standard embeddings and external diagonalization

full rationale

The paper derives the stability estimate δ_{s,p}(u) ≳ (dist_{s,p}(u,Z))^α from a localized Poincaré-Sobolev inequality, rescaling arguments, and Lorentz embeddings that decompose the nonlocal energy. These are classical tools independent of the target result. For the p=2 case, the Emden-Fowler reduction invokes the diagonalization of the fractional Hardy quadratic form due to Frank, Lieb, and Seiringer (external prior work, not self-citation). No equation or claim reduces by construction to a fitted input, self-defined quantity, or load-bearing self-citation chain. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard background results in fractional Sobolev spaces and embeddings; no free parameters are introduced and the only new entity is the set of virtual extremals whose independent evidence is not supplied beyond the stability statement itself.

axioms (2)

standard math Existence and sharpness of the constant C_{N,s,p} in the Frank-Seiringer inequality
Invoked in the definition of the deficit δ_{s,p}(u) and the family Z.
standard math Validity of Lorentz embeddings and Marcinkiewicz space properties for controlling the deficit
Used to relate the distance to Z with the nonlocal energy.

invented entities (1)

virtual extremals Z no independent evidence
purpose: Set of functions achieving equality in the Hardy inequality for the purpose of measuring stability distance
Introduced to formulate the quantitative lower bound; no independent falsifiable prediction is given outside the stability statement.

pith-pipeline@v0.9.0 · 5983 in / 1695 out tokens · 102019 ms · 2026-05-20T17:07:17.078918+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Our approach is based on a localized Poincaré-Sobolev inequality combined with suitable rescaling and Lorentz embeddings... decomposition of the nonlocal energy together with Lorentz estimates
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

via an Emden–Fowler correspondence and pseudo-differential operators, we show that the nonlocal Hardy deficit coincides with the local one

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Reference graph

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[1] [1]

Chaudhuri, and M

Adimurthi, N. Chaudhuri, and M. Ramaswamy,An improved Hardy-Sobolev inequality and its application, Proc. Amer. Math. Soc.130(2002), no. 2, 489–505

work page 2002

[2] [2]

Jana, and P

Adimurthi, P. Jana, and P. Roy,Boundary fractional Hardy’s inequality in dimension one: the critical case, Commun. Contemp. Math.28(2026), no. 4, Paper No. 2550051, 14

work page 2026

[3] [3]

Roy, and V

Adimurthi, P. Roy, and V. Sahu,Fractional boundary Hardy inequality for the critical cases, J. Funct. Anal. 290(2026), no. 8, Paper No. 111351

work page 2026

[4] [4]

Roy, and V

Adimurthi, P. Roy, and V. Sahu,Fractional Hardy inequality with singularity on submanifold, Calc. Var. Partial Differential Equations65(2026), no. 2, Paper No. 62, 40

work page 2026

[5] [5]

Roy, and V

Adimurthi, P. Roy, and V. Sahu,Boundary Hardy inequality on functions of bounded variation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) (Mar. 2025), p. 26

work page 2025

[6] [6]

Akutagawa and H

K. Akutagawa and H. Kumura,Geometric relative Hardy inequalities and the discrete spectrum of Schr¨ odinger operators on manifolds, Calc. Var. Partial Differential Equations48(2013), no. 1-2, 67–88

work page 2013

[7] [7]

W. Ao, A. DelaTorre, and M. del M. Gonz´ alez,Symmetry and symmetry breaking for the fractional Caffarelli- Kohn-Nirenberg inequality, J. Funct. Anal.282(2022), no. 11, Paper No. 109438, 58

work page 2022

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K. Bal, K. Mohanta, P. Roy, and F. Sk,Hardy and Poincar´ e inequalities in fractional Orlicz-Sobolev spaces, Nonlinear Anal.216(2022), Paper No. 112697, 22

work page 2022

[9] [9]

Berchio, D

E. Berchio, D. Ganguly, and G. Grillo,Sharp Poincar´ e-Hardy and Poincar´ e-Rellich inequalities on the hy- perbolic space, J. Funct. Anal.272(2017), no. 4, 1661–1703

work page 2017

[10] [10]

Berchio, D

E. Berchio, D. Ganguly, G. Grillo, and Y. Pinchover,An optimal improvement for the Hardy inequality on the hyperbolic space and related manifolds, Proc. Roy. Soc. Edinburgh Sect. A150(2020), no. 4, 1699–1736

work page 2020

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