Galerkin Approximation of the Fractional Hardy Constant

Andreea Dima; Liviu I. Ignat

Galerkin Approximation of the Fractional Hardy Constant

T0 review · 0 major / 5 minor · reviewed 2026-07-30 · grok-4.5

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Pith's one-line read Piecewise-linear Galerkin schemes recover the fractional Hardy constant at the same slow logarithmic rate as the classical case.

desk verdict Solid, expected fractional extension of the classical Hardy FEM-rate program; the ~1/|log h|^2 asymptotic is real and carefully proved. read the letter →

arxiv 2607.26830 v1 pith:7C2Q2G3K submitted 2026-07-29 math.NA cs.NAmath.AP

Andreea Dima , Liviu I. Ignat This is my paper · ORCID

classification math.NAcs.NAmath.AP MSC 65N3046E35

keywords FractionalHardyInequalityGalerkinapproximationFiniteElementMethodLogarithmicimprovementDiscreteoptimalconstantGround-staterepresentation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks how well a standard finite-element method can capture the sharp constant in the fractional Hardy inequality. On any bounded convex smooth domain that contains the origin, and for piecewise-linear elements on a quasi-uniform mesh of size h, the discrete constant lies above the true constant by a quantity that is asymptotically comparable to 1 over the square of the logarithm of h. The same rate was already known for the ordinary (local) Hardy inequality; the authors show that the non-local fractional version behaves identically. A sympathetic reader cares because the continuous constant is never attained, so the rate at which a numerical scheme approaches it is a concrete probe of how the deficit concentrates near the origin. The result also supplies a benchmark against which more exotic approximation spaces can later be tested.

What carries the argument

A logarithmic improvement of the fractional Hardy inequality on bounded domains (used for the matching lower bound) together with an explicit cut-off competitor built from the pseudo-minimizer |x|^{-(N-2s)/2} and a ground-state representation of the deficit (used for the upper bound).

What would settle it

Compute the discrete fractional Hardy constant on the unit ball for a sequence of successively refined quasi-uniform meshes and check whether the difference from the known continuous constant decays like 1/|log h|²; any other asymptotic would refute the theorem.

Watch

Extended reading notes

Core claim

For every dimension N ≥ 1 and every fractional order s in (0, min{1, N/2}), the Galerkin constant λ_{N,s,h}(Ω) computed with continuous piecewise-linear elements on a regular quasi-uniform triangulation of a bounded convex smooth domain Ω containing the origin satisfies λ_{N,s,h}(Ω) − λ_{N,s} ∼ 1/|log h|² as h → 0.

Load-bearing premise

The lower bound stands or falls with a continuous logarithmic-improvement inequality that the paper imports as a black box; if that remainder is weaker than claimed, the discrete lower bound of order 1/|log h|² disappears.

Editorial extensions

If this is right

Standard linear finite-element codes cannot beat a logarithmic rate when approximating the fractional Hardy constant.
The same rate already known for the local Hardy inequality persists under non-local fractional diffusion.
Any future scheme that claims a faster rate must exploit structure beyond piecewise-linear approximation on quasi-uniform meshes.
The explicit near-minimizing sequence constructed in the paper can be reused to test other discrete spaces.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The logarithmic rate is likely intrinsic to the fractional Hardy deficit itself rather than an artifact of piecewise-linear elements, so nonlinear approximation classes may still be limited by the same barrier.
Matching upper and lower bounds for the L^p fractional Hardy deficit would immediately yield analogous discrete rates once a logarithmic improvement on domains is available.
The same cut-off competitor technique should transfer, with only technical changes, to other non-local inequalities whose extremals concentrate at a point.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The one thing to know: they get the sharp discrete rate λ_{N,s,h}−λ_{N,s}∼1/|log h|² for the fractional Hardy constant (p=2, s small) under piecewise-linear Galerkin on quasi-uniform meshes in a smooth convex domain containing the origin. Same rate as the classical s=1 papers they extend.

What is actually new is the fractional statement itself. The strategy is not new—it is the Ignat–Zuazua / Della Pietra playbook (log improvement + ground-state identity + cut-off competitor + standard interpolation)—but carrying it through for the nonlocal seminorm is real work. The lower bound is short and clean once you grant Tzirakis’s continuous log improvement; the upper bound is long (lots of case splits on the double integral for the competitor) but structured, and the appendix lemmas on interpolation error and elementary Hardy cut-offs look correct. They do not hide the free parameters (α, γ, μ); they just choose them in ranges that close the estimates.

Soft spots, in proportion: the lower bound is a black-box citation of Tzirakis [16, Thm 5]. That is standard and not a hidden flaw—the matching upper bound is self-contained via Frank–Lieb–Seiringer plus an explicit u_ε—so the ∼ relation stands on the same footing as the classical results. Novelty and significance are moderate: expected next case in a narrow program, not a reorganization of the field. Section 5 on L^p and Bregman variants is honest about what is missing. Citation pattern is appropriate; self-cites are to the direct predecessors.

This is for people already working on FEM approximation of sharp constants or fractional Hardy. A serious referee in math.NA or nonlocal analysis should see it. I would send it to peer review without hesitation; it is careful, reproducible on the page, and free of load-bearing gaps. Engage if that line is on your desk; otherwise file as the fractional counterpart you expected.

Referee Report

0 major / 5 minor

Summary. The paper proves that the discrete fractional Hardy constant obtained by piecewise-linear Galerkin approximation on a quasi-uniform regular mesh of a bounded convex smooth domain Ω containing the origin satisfies λ_{N,s,h}(Ω)−λ_{N,s}∼1/|log h|² for s∈(0,min{1,N/2}) and all sufficiently small h. The lower bound is deduced from Tzirakis’s logarithmic improvement of the fractional Hardy inequality on domains, combined with elementary cut-offs near the origin, inverse estimates, and two Hardy-type lemmas in the appendix. The matching upper bound is obtained by constructing an explicit near-minimizer (a cut-off logarithmic perturbation of the pseudo-minimizer |x|^{-(N−2s)/2}), applying the Frank–Lieb–Seiringer ground-state representation, and controlling the piecewise-linear interpolant error via Gagliardo–Nirenberg and standard FEM estimates.

Significance. The result cleanly extends the classical (s=1) Galerkin approximation theory for the Hardy constant developed by Ignat–Zuazua and Della Pietra et al. to the fractional setting, and shows that the same logarithmic rate persists. The upper-bound construction is self-contained and quantitative; the lower bound rests on a standard continuous deficit estimate used only qualitatively. The work fits a coherent program (Sobolev, fractional Sobolev, Hardy, fractional Hardy) and supplies a complete asymptotic rather than a one-sided bound. The detailed competitor analysis and the appendix lemmas are reusable technical contributions.

minor comments (5)

[Section 4, opening paragraph] The reduction of the upper bound from a general convex smooth Ω to the unit ball is stated in one sentence (“by comparison”). A short remark clarifying that λ_{N,s} is domain-independent and that the competitor can be transplanted (or that the discrete constant on a subdomain is at least as large) would make the argument self-contained for readers who have not seen the classical papers.
[Section 4] The free parameters α≥1, γ∈(0,1) and μ∈(0,1/2) are introduced without a brief summary of the admissible range that makes all estimates close simultaneously. A single sentence collecting the constraints (e.g. after Lemma 4.1 or before the choice ε=h^{γ/2}) would help.
[Section 3] In (3.3)–(3.5) the passage from the weighted L² mass on B_h to the unweighted mass on B_{2h} via Lemmas A.1–A.2 and the inverse estimate is correct but dense. Signposting that the only role of the inverse estimate is to absorb the gradient term would improve readability.
[References] Several references appear as arXiv preprints with 2025–2026 identifiers (including the authors’ own related works). If any have since been accepted or published, the bibliographic data should be updated at proof stage.
[Sections 3–4] Typographical consistency: the manuscript mixes “log” and “|log|” and occasionally writes log²(e/|x|) without parentheses; a uniform convention would be preferable. Also, the constant D=sup_{x∈Ω}|x| is introduced in Prop. 3.1 and immediately specialized to D≤1; stating the normalization 0∈Ω⊂B once at the beginning of Section 3 would avoid repetition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the |log h|^{-2} rate is derived from external continuous inequalities plus standard FEM estimates, not assumed or fitted.

full rationale

Theorem 1.1 is obtained by two independent arguments. The lower bound invokes Tzirakis’s external logarithmic improvement (Prop. 3.1 / [16, Thm 5]) and converts the continuous deficit into a discrete lower bound via elementary cut-offs near the origin and inverse estimates (Lemmas A.1–A.2); nothing in that chain is defined in terms of the discrete constant. The upper bound constructs an explicit competitor u_ε from the known pseudo-minimizer, applies the external Frank–Lieb–Seiringer ground-state identity ([9, Prop. 4.1]), and controls the FEM interpolation error by standard Gagliardo–Nirenberg and interpolation lemmas (A.3–A.5). Self-citations to the authors’ classical-Hardy and fractional-Sobolev papers supply only motivation and comparison of rates; they are not used as uniqueness theorems or as load-bearing premises. No parameter is fitted to data and then re-predicted, and the target asymptotic is not smuggled in by definition. The derivation is therefore self-contained against its stated external inputs.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The paper is a pure-math rate proof. It imports two substantial external analytic facts (the continuous logarithmic improvement and the ground-state representation) and standard FEM approximation theory; it introduces no fitted physical parameters and no new ontological entities. Technical cut-off parameters (α,γ,μ,δ) are free choices inside open intervals that are later optimized, not data-fitted constants.

free parameters (3)

α≥1 (log-power in the competitor) = any real ≥1
Chosen by hand in the definition of ũ_1; any α≥1 works for the upper bound. Not fitted to data.
γ∈(0,1) (relation ε=h^{γ/2}) = any real in (0,1)
Free mesh-to-cut-off scaling chosen so that the interpolation error is o(1/|log h|²). Not data-fitted.
μ∈(0,1/2) (cut-off transition width) = any real in (0,1/2)
Technical width of the smooth cut-off ξ; any value in the open interval works.

assumptions (6)

domain assumption Logarithmic improvement of fractional Hardy on bounded domains (Prop. 3.1 / Tzirakis [16, Thm 5])
Black-box input to the entire lower bound in Section 3; the discrete lower rate is inherited from this continuous deficit.
domain assumption Ground-state representation of the fractional Hardy deficit (Frank–Lieb–Seiringer [9, Prop. 4.1])
Identity (4.4) that converts the energy deficit of u_ε into a weighted Gagliardo seminorm of θ_ε; used throughout Section 4.
standard math Standard inverse estimates and interpolation error bounds for piecewise-linear elements on quasi-uniform regular meshes (Brenner–Scott; [14, Lemma 3.1])
Used for (3.4) and Lemmas A.4–A.5; classical FEM theory.
standard math Homogeneous Gagliardo–Nirenberg interpolation inequality (Leoni [15, Thm 7.45])
Converts L^q and W^{1,p} interpolation errors into the Ḣ^s error of I_h u_ε−u_ε.
domain assumption Domain Ω bounded, convex, smooth, containing the origin; mesh regular and quasi-uniform
Standing geometric hypotheses of Theorem 1.1 and of the discrete space V_h.
standard math Exact value of the continuous fractional Hardy constant λ_{N,s} (Herbst / Frank–Seiringer)
Used as the continuum benchmark; classical.

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Cite this review

Pith. "Pith review of Galerkin Approximation of the Fractional Hardy Constant." pith.science (2026). https://pith.science/paper/7C2Q2G3K

@misc{pith2026260726830,
  author       = {Pith},
  title        = {Pith review of: Galerkin Approximation of the Fractional Hardy Constant},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7C2Q2G3K}},
  note         = {Machine review of arXiv:2607.26830}
}

abstract

We establish sharp estimates for the discrete optimal constant of the fractional Hardy Inequality in dimension $N\geq 1$, with fractional exponent $s\in \left(0,\min\left\{1,\frac{N}{2}\right\}\right)$. The convergence rates that we establish take place for the Galerkin approximation with piecewise linear elements, when the computations are carried out in a bounded, convex and smooth domain containing the origin, for which we employ a quasi-uniform and regular mesh.

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Pith tools

Reviewed July 30, 2026 · model on record in the stance chip above.

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Paola Antonietti and Aldo Pratelli. “Finite element approximation of the Sobolev con- stant”. In:Numerische Mathematik117 (Dec. 2011), pp. 37–64

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Avas Banerjee, Debdip Ganguly, and Vivek Sahu. “Quantitative stability for fractional Hardy inequalities: Rearrangement-free techniques and Emden-Fowler analysis”. In:arXiv preprint arXiv:2605.15748(2026)

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Finite Element Approxi- mation for the Fractional Eigenvalue Problem

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Finite Element Approximation of the Hardy Constant

Francesco Della Pietra, Giovanni Fantuzzi, Liviu I. Ignat, Alba Lia Masiello, Gloria Paoli, and Enrique Zuazua Iriondo. “Finite Element Approximation of the Hardy Constant”. In:Journal of Convex Analysis31 (2024), pp. 497–523

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