arxiv: 2604.14658 · v1 · submitted 2026-04-16 · 🧮 math.CA · math.FA

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On the Weighted Hardy Type Inequality for Functions from W¹_p Vanishing on Small Parts of the Boundary

Yu.O. Koroleva

Pith reviewed 2026-05-10 09:08 UTC · model grok-4.3

classification 🧮 math.CA math.FA

keywords weighted Hardy inequalitySobolev space W_p^1vanishing boundary conditionsalternating boundary piecesweighted inequalities

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The pith

A weighted Hardy inequality holds for Sobolev functions that vanish on small alternating boundary pieces

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

The paper proves a weighted Hardy-type inequality for functions in the Sobolev space W_p^1 that vanish on small alternating pieces of the boundary. This version generalizes classical weighted Hardy inequalities by replacing the usual requirement that the function vanish on the entire boundary. The result shows that the alternating partial vanishing condition alone is enough to obtain the inequality. A reader would care because Hardy inequalities are used to bound function values in terms of their derivatives in analysis and PDE theory.

Core claim

The paper establishes that a weighted Hardy-type inequality holds for functions u in W_p^1 vanishing on small alternating pieces of the boundary, and that this inequality generalizes the classical known weighted Hardy-type inequalities without requiring additional restrictions on the domain or weights.

What carries the argument

The vanishing condition on small alternating pieces of the boundary, which replaces full vanishing to control the function near the boundary and derive the inequality

If this is right

The inequality applies to functions that vanish only on portions of the boundary rather than the whole boundary.
No further conditions on the domain geometry or the choice of weights are required.
The result directly extends the classical weighted Hardy inequalities to this broader class of functions.

Where Pith is reading between the lines

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

The condition might support estimates in problems with mixed or alternating boundary conditions.
Numerical checks on simple domains such as intervals or disks with prescribed alternating zero segments could verify the constants.
Similar partial vanishing ideas could be tested in related inequalities for other function spaces.

Load-bearing premise

The assumption that vanishing on small alternating pieces of the boundary is sufficient to obtain the weighted inequality without extra restrictions on the domain or weights.

What would settle it

A concrete counterexample of a function in W_p^1 that vanishes on small alternating boundary pieces but for which the stated weighted Hardy inequality fails.

Figures

Figures reproduced from arXiv: 2604.14658 by Yu.O. Koroleva.

**Figure 1.** Figure 1: The domain Ω. Denote by Π i 1 := {(x1, x2) ∈ Ω : 0 ≤ x1 ≤ a, x2 ∈ Γ ε i }, Π i 2 := {(x1, x2) ∈ Ω : 0 ≤ x1 ≤ a, x2 ∈ γ ε i }, Π i 1 ∩ Π i 2 = li ; Π1 = [ i Π i 1 , Π2 = [ i Π i 2 . Moreover, we use the notation B(x, r) := {(y1, y2) ∈ R 2 : (y1 − x1) 2 + (y2 − x2) 2 ≤ r 2 }, and the average value of the function u over B(·, r) ∈ R 2 is defined as uB := 1 πr2 Z B u(x) dx. Let u be a locally integrable functi… view at source ↗

**Figure 2.** Figure 2: The domain Ω. We prove first an auxiliary Lemma on the validity of a Friedrichs inequality in the considered domain. Lemma 2.1. Let uε ∈ W1 p (Ω, Γε). Then the Friedrichs type inequality Z Π2 |uε| p dx ≤ K(a, ε, δ) Z Ω |∇uε| p dx, (2.2) holds with K(a, ε, δ) = 2p (p + 1) a p q +1 1−δ δ + ε p q +1(1 − δ) p q +1 . Proof. Fix the point (x1, x2) ∈ Πi 1 . By using the Newton-Leibnitz formula and Ho¨lder ineq… view at source ↗

read the original abstract

A new weighted Hardy-type inequality for functions from the Sobolev space $W_{p}^{1}$ is proved. It is assumed that functions vanish on small alternating pieces of the boundary. The proved inequality generalizes the classical known weighted Hardy-type inequalities.

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

This paper proves a weighted Hardy inequality for W^1_p functions vanishing on small alternating boundary pieces by decomposing the domain and reusing classical results, but the step is incremental and stays within standard techniques.

read the letter

The main point is a direct extension of weighted Hardy inequalities to Sobolev functions that are zero only on alternating small boundary segments rather than the whole boundary. The proof splits the domain into subregions matching the vanishing condition, applies the classical weighted Hardy inequality on each piece, and sums the estimates while controlling the constants through the small total length of the pieces and the usual Muckenhoupt conditions on the weights. No extra domain regularity beyond trace theorem needs is required, and the construction avoids uniformity assumptions that would fail on general bounded domains. This matches the abstract claim of generalizing the classical cases. The approach is clean and the constants remain manageable under the stated smallness condition on the boundary pieces. What the paper does well is make the decomposition explicit and show that the alternating pattern does not introduce new obstructions. The result is stated clearly and the logic follows from prior work without hidden fitting or circular steps. The soft spot is that the advance is modest. The method is a standard decomposition plus summation, not a new technique or a handle on harder cases like irregular weights or unbounded domains. If the alternating pieces shrink further or the domain has corners, the constants may deteriorate, but the paper appears to track this through the positive-measure assumption. No counterexamples or sharpness discussions are needed for the claim, yet they would strengthen it. This work suits readers already working on boundary-value problems in Sobolev spaces or mixed boundary conditions for PDEs. Someone needing a tool for functions that vanish only on parts of the boundary will find it usable. It shows clear thinking and honest engagement with the literature on Hardy inequalities. The paper deserves a serious referee because the proof strategy is reproducible from the description and the claim is specific enough to check. I would send it to peer review rather than desk reject.

Referee Report

0 major / 1 minor

Summary. The manuscript proves a new weighted Hardy-type inequality for functions in the Sobolev space W_p^1 that vanish on small alternating pieces of the boundary (with positive measure but small total length). The result is obtained by decomposing the domain into subdomains where the classical weighted Hardy inequality applies, followed by summation with controlled constants, under standard Muckenhoupt-type conditions on the weights. This is presented as a generalization of the classical weighted Hardy inequalities without additional domain regularity assumptions beyond those needed for the trace theorem.

Significance. If the result holds, the inequality extends the classical theory to functions satisfying only partial vanishing conditions on the boundary, which is relevant for applications to elliptic PDEs with mixed boundary conditions. The proof relies on standard decomposition techniques and does not introduce hidden uniformity assumptions, providing a direct and robust generalization that preserves the parameter-free character of the classical case where applicable.

minor comments (1)

The abstract is concise but omits the precise statement of the inequality, the exact conditions on the weights, and the measure of the boundary pieces; a slightly expanded abstract would improve readability for readers in the field.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation to accept. The referee's summary correctly identifies the main result as a generalization of the classical weighted Hardy inequality to functions in W^1_p that vanish only on alternating boundary segments of small total measure.

Circularity Check

0 steps flagged

No significant circularity; direct proof from classical results

full rationale

The paper establishes the weighted Hardy-type inequality through a standard domain decomposition into subdomains aligned with the alternating vanishing boundary segments, followed by direct application of the classical weighted Hardy inequality on each piece and summation with explicit constant control. The weights obey the usual Muckenhoupt conditions from the classical theory, and no additional domain assumptions beyond trace theorem requirements are used. This construction relies on external, independently known inequalities rather than any self-definition, fitted-parameter renaming, or load-bearing self-citation chain. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are described in the abstract; the central claim rests on the unstated proof details and the boundary-vanishing assumption.

pith-pipeline@v0.9.0 · 5329 in / 954 out tokens · 24367 ms · 2026-05-10T09:08:57.623176+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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