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arxiv: 2507.00222 · v2 · pith:5OMGDX4Qnew · submitted 2025-06-30 · 🧮 math.CV

On equicontinuity of mappings with inverse moduli inequalities by prime ends of variable domains

Zarina Kovba , Evgeny Sevost'yanov This is my paper

Pith reviewed 2026-05-22 00:25 UTC · model grok-4.3

classification 🧮 math.CV
keywords equicontinuityprime endsmoduli inequalitiesPoletskii inequalityboundary behaviorvariable domainsmappings in the plane
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The pith

Mappings obeying integrable inverse moduli inequalities are equicontinuous with respect to prime ends of variable domains.

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

The paper examines mappings that satisfy inverse moduli inequalities of Poletskii type, under which the images of the original domain are allowed to change. It proves that if the majorant function in the inequality is integrable, then these mappings are equicontinuous when boundary behavior is measured using prime ends of a reference domain. A sympathetic reader would care because this equicontinuity supplies uniform control on how points near the boundary are mapped, even when the target domains vary. The result therefore supplies a tool for studying limits and continuity properties without fixing the image domain in advance.

Core claim

Mappings satisfying an inverse modulus inequality of Poletskii type with integrable majorant are equicontinuous relative to the prime ends of some domain. The proof uses the integrability of the majorant to produce uniform estimates on the modulus of families of curves that connect interior points to the boundary, ensuring that prime ends lying close together are sent to sets that remain close in the image.

What carries the argument

The inverse moduli inequality of Poletskii type with integrable majorant, which supplies a lower bound for the modulus of the image curve family in terms of an integral involving the majorant.

If this is right

  • Equicontinuous families permit extraction of convergent subsequences when boundary values are considered via prime ends.
  • The conclusion applies directly to classes of mappings whose images are permitted to vary.
  • Boundary continuity statements can be obtained without assuming a fixed target domain.

Where Pith is reading between the lines

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

  • Analogous integrability hypotheses might produce equicontinuity statements for mappings in higher-dimensional spaces.
  • The result could support theorems on boundary correspondence for sequences of mappings under variable domains.
  • Explicit majorants such as logarithmic or power functions could be checked to obtain concrete examples.

Load-bearing premise

The majorant function in the inverse moduli inequality is integrable over the domain.

What would settle it

A mapping that satisfies the inverse Poletskii inequality with an integrable majorant yet fails to be equicontinuous at some prime end would disprove the claim.

Figures

Figures reproduced from arXiv: 2507.00222 by Evgeny Sevost'yanov, Zarina Kovba.

Figure 1
Figure 1. Figure 1: To proof of Lemma 3.1 We now prove that the relation (3.5) contradicts the definition of fm in (1.3)–(1.4). Let ∆m := fm(Γm). Since fm is a closed mapping, it preserves the boundary (see [Vu, The￾orem 3.3]), so that C(fm, ∂D) ⊂ ∂fm(D). Thus, C(βm(t), tm) ⊂ ∂fm(D), where βm(t) = fm(αm(t)) = fm(γ|[0,tm)), γ ∈ Γ(Am, E0, Rn) and tm = sup γ(t)∈D t. Observe that, dist (K0, ∂fm(D)) > ε > 0 for some ε > 0 and all … view at source ↗
Figure 2
Figure 2. Figure 2: To the proof of Theorem 1.1 fm(γ(t1)) ∈ S(z0, r1) and fm(γ)|1 := fm(γ)|[t1,1]. Without loss of generality, we may consider fm(γ)|1 ⊂ R n \ d1. Arguing similarly for fm(γ)|1, we may consider a point t2 ∈ (t1, 1) such that fm(γ(t2)) ∈ S(z0, r0). We set f(γ)|2 := f(γ)|[t1,t2] . Then fm(γ)|2 is a subpath of fm(γ) and, in addition, fm(γ)|2 ∈ Γ(S(z0, r1), S(z0, r0), D0). Without loss of generality we may assume … view at source ↗
Figure 3
Figure 3. Figure 3: Illustration for Example 2. Then under notions of the statement of Theorem 1.2, the following condition is true: for any x0 ∈ D there is y0 := f(x0) ∈ D0 such that, for any ε > 0 there is δ = δ(ε) > 0 and M = M(ε) ∈ N such that ρ(fmk (x), y0) < ε for all x ∈ B(x0, δ) ∩ D and k > M0. Moreover, f(D) = D0 and f(D) = D0P . Proof of Theorem 6.1 directly follows by Theorem 4.1 and Lemma 5.1. ✷ Example 2. Let D0 … view at source ↗
Figure 4
Figure 4. Figure 4: Illustration for Example 3. Now, δτ = δr < 1 for x 2 n−1 < 1 − h 2 , so that kg ′ (x)k = 1, |J(x, g)| =  √ h 1−x2 n n−1 for x 2 n−1 < 1 − h 2 . If x 2 n−1 > 1 − h 2 , then kg ′ (x)k = J(x, g) = 1. Thus, KO(x, g) = kg ′ (x)k n |J(x, g)| = p 1 − x 2 n h !n−1 6 1 hn−1 . Finally, the mapping ϕ(x) = g −1◦f ◦g transforms B n onto B n + = {x = (x1, . . . xn) ∈ B n : xn > 0} and is a quasiconformal because all t… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration for Example 6. where fm and f are from Example 2, and for n > 3 Gm(x) =    (h1 ◦ Fm)(x), x ∈ F −1 m (B(x0, r0)), fm(x), x 6∈ f −1 m (B(x0, r0)) , G(x) =    (h1 ◦ F)(x), x ∈ F −1 (B(x0, r0)), F(x), x 6∈ F −1 (B(x0, r0)) , where Fm and F are from Example 3. By the construction, gm and Gm satisfy the re￾lations (1.3)–(1.4) at any point y0 ∈ D0 for Q = C · Q(y) = logn−1  r0e |y|  , where C… view at source ↗
read the original abstract

The paper is devoted to the study of the boundary behavior of mappings. We consider mappings that satisfy inverse moduli inequalities of Poletskii type, under which the images of the domain under the mappings may change. It is proved that a classes of such mappings are equicontinuous with respect to prime ends of some domain if the majorant in the indicated modulus inequality is integrable.

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 / 3 minor

Summary. The manuscript studies boundary behavior of mappings satisfying inverse moduli inequalities of Poletskii type, allowing the image domains to vary with the mapping. It proves that a class of such mappings is equicontinuous with respect to prime ends of a target domain whenever the majorant function appearing in the modulus inequality is integrable.

Significance. If the central claim holds, the result supplies a natural extension of equicontinuity criteria from fixed-domain settings to families of mappings on variable domains, using the prime-end topology. This is relevant to the study of boundary distortion and cluster sets in geometric function theory.

minor comments (3)
  1. The abstract and introduction should state the precise integrability condition (e.g., membership in L^p or Orlicz class) and the exact form of the inverse Poletskii inequality that is assumed.
  2. Notation for prime ends, the variable domains, and the modulus of curve families should be introduced with a short self-contained paragraph before the main theorem.
  3. Any auxiliary lemmas used to control the modulus of families approaching a prime end should be clearly separated from the main argument and their hypotheses listed explicitly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and for recommending minor revision. The referee's summary accurately captures the main contribution concerning equicontinuity of mappings satisfying inverse Poletskii-type modulus inequalities with respect to prime ends when the majorant is integrable.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external integrability hypothesis

full rationale

The paper establishes equicontinuity of mappings satisfying an inverse Poletskii-type modulus inequality with integrable majorant, with respect to prime ends on variable domains. This conclusion follows from standard modulus distortion estimates once the integrability condition makes the modulus of curve families approaching a prime end arbitrarily small. No step reduces by the paper's own equations to a fitted parameter, self-definition, or self-citation chain; the integrability assumption is an independent external input, and the argument structure invokes no uniqueness theorem or ansatz from the authors' prior work as load-bearing. The result is therefore not equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the mappings obey an inverse Poletskii-type modulus inequality whose majorant is integrable; no free parameters, new entities, or additional ad-hoc axioms are visible from the abstract.

axioms (1)
  • domain assumption Mappings satisfy inverse moduli inequalities of Poletskii type with integrable majorant.
    This is the defining property of the class for which equicontinuity is claimed.

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