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arxiv: 2502.16209 · v2 · submitted 2025-02-22 · 🧮 math.CV

On Caratheodory theorem for open discrete unclosed mappings

Zarina Kovba , Evgeny Sevost'yanov This is my paper

Pith reviewed 2026-05-23 02:39 UTC · model grok-4.3

classification 🧮 math.CV
keywords inverse Poletsky inequalityopen discrete mappingsprime endsequicontinuityOrlicz-Sobolev classesCarathéodory theoremquasiregular mappings
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The pith

Open discrete mappings obeying the inverse Poletsky inequality are equicontinuous at prime ends even when the image domain lacks local connectedness at the boundary and the mappings do not preserve the boundary.

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

The paper establishes equicontinuity for families of open and discrete mappings that satisfy the inverse Poletsky-type inequality, measured with respect to prime ends in the closure of the domain. This holds although the image domain is not assumed to be locally connected on its boundary and the mappings need not send the boundary to the boundary. The argument relies on the modulus control supplied by the inequality to obtain uniform continuity statements at prime ends. The same conclusion is then transferred to the corresponding Orlicz-Sobolev classes of mappings.

Core claim

Families of open and discrete mappings satisfying the inverse Poletsky inequality are equicontinuous in the closure of the definition domain when continuity is understood in the topology of prime ends. The result applies even though the image domain may fail to be locally connected on its boundary and the mappings may fail to preserve the boundary.

What carries the argument

The inverse Poletsky-type inequality, which supplies a lower bound on the modulus of the image of a path family, used together with the theory of prime ends to control boundary behavior.

If this is right

  • The families remain equicontinuous at every prime end of the image domain.
  • The equicontinuity statement holds for the corresponding Orlicz-Sobolev classes.
  • Boundary continuity can be obtained without assuming local connectedness of the image domain.
  • The result applies to mappings that do not necessarily map the boundary onto the boundary.

Where Pith is reading between the lines

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

  • Prime ends may serve as a substitute for local connectedness when studying modulus inequalities for non-closed mappings.
  • The same technique could be tested on other classes of mappings controlled by modulus inequalities in higher dimensions.
  • The approach separates the modulus condition from any requirement that the mapping be closed or homeomorphic onto its image.

Load-bearing premise

The mappings satisfy the inverse Poletsky-type inequality while remaining open and discrete.

What would settle it

A sequence of open discrete mappings obeying the inverse Poletsky inequality that fails to be equicontinuous at some prime end would disprove the claim.

Figures

Figures reproduced from arXiv: 2502.16209 by Evgeny Sevost'yanov, Zarina Kovba.

Figure 1
Figure 1. Figure 1: To the statement of Lemma 2.2 The version of the following Lemma was proved in [ISS] for homeomorphisms and in [Sev1] for boundary preserving mappings. Now we need it in the situation, when mappings may not preserve the boundary of a domain. Lemma 2.3. Let D and D ′ be domains in R n , n > 2, let E be a set in D ′ , which is closed in Rn and such that ∂D ′ ⊂ E and that D ′ \ E consists of finite number of … view at source ↗
Figure 2
Figure 2. Figure 2: The formulation of Lemma 2.4 3 Proof of Theorem 1.1 The possibility of a boundary extension which is continuous at ∂D with respect to D\f −1 (E∩ D ′ ) follows by [SDK, Theorem 1.1]. We need to prove the equicontinuity of SP E,δ,Q(D, D ′ ) at ∂D with respect to D \ f −1 (E ∩ D ′ ) by the metric ρ in (1.10). Let us prove this statement by the contradiction, namely, assume that there is x0 ∈ ∂D, a number ε0 >… view at source ↗
Figure 3
Figure 3. Figure 3: To the proof of theorem 1.1. f m(xm) = fm(xm). In addition, there is a sequence ym ∈ D \f −1 (E ∩D ′ ) such that ym → x0 as m → ∞ and ρ(fm(ym), f m(x0)) → 0 as m → ∞. Since the space ( S N i=1 DiP , ρ) is compact, we may assume that fm(xm) and fm(x0) converge as m → ∞ to some points P1 and P2. By the continuity of the metric ρ, (3.1) implies that P1 6= P2. By the assumption 2), D ′ \ C(f, ∂D)) consists of … view at source ↗
Figure 4
Figure 4. Figure 4: To the proof of Theorem 1.1, case III. a) point from the condition of the theorem. Without loss of generalization, we may assume that ai1 = a1. Consequently joining the points gM1 (xM1 ), gM1+1(xM1+1), . . . , gm(xm), . . . , going to the subsequence if it needs and reasoning similarly to the proof of Lemma 5.2 in [DS], we obtain a path αe : [1/2, 1] → K1 starting at the point gM1 (xM1 ) and ending at the … view at source ↗
Figure 5
Figure 5. Figure 5: To the proof of Theorem 1.1, case III. b) |β| ∩ B(z1, ε1) = ∅, see [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
read the original abstract

We study mappings satisfying the inverse Poletsky-type inequality in a domain of the Euclidean space. Such inequalities are well known and play an important role in the study of quasiconformal and quasiregular mappings. We consider the case when the mapped domain, generally speaking, is not locally connected on its boundary. At the same time, we consider the situation when the mapping is open and discrete, but may not preserve the boundary of the domain. In terms of prime ends, we obtain results on the equicontinuity of families of such mappings in the closure of the definition domain. As a consequence, we also obtain the corresponding statement for Orlicz-Sobolev classes.

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 extends Carathéodory-type equicontinuity results to open discrete mappings in Euclidean domains that satisfy an inverse Poletsky-type inequality. The setting allows domains that are not locally connected at the boundary and mappings that need not preserve the boundary. The central results establish equicontinuity of families of such mappings (in the closure of the domain) via prime ends; a consequence is stated for mappings belonging to Orlicz-Sobolev classes.

Significance. If the derivations hold, the work broadens the scope of equicontinuity theorems in geometric function theory by relaxing boundary-connectivity and boundary-preservation assumptions while retaining control via the inverse Poletsky inequality. The prime-end formulation and the Orlicz-Sobolev corollary are standard directions in the field and would be of interest to researchers working on distortion-controlled mappings.

minor comments (3)
  1. The abstract states the main claims but supplies no derivation outline, error estimates, or verification steps; the full manuscript should include at least a sketch of the key estimates that produce equicontinuity from the inverse Poletsky inequality.
  2. Notation for the inverse Poletsky inequality and for the prime-end topology should be introduced with explicit references to prior literature (e.g., the precise form of the inequality and the definition of prime ends used).
  3. The transition from the general open-discrete case to the Orlicz-Sobolev corollary needs a clear statement of the embedding or integrability condition that places the mappings inside the Orlicz-Sobolev class.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the summary of our manuscript and for recognizing its potential to broaden equicontinuity results by relaxing local connectivity and boundary-preservation assumptions while retaining control via the inverse Poletsky inequality. The report contains no enumerated major comments, so we have no specific points to address.

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard modulus estimates from given inequality

full rationale

The paper applies the inverse Poletsky-type inequality (a known condition in the literature) to obtain modulus estimates for open discrete mappings, then derives equicontinuity in the closure via prime ends and a consequence for Orlicz-Sobolev classes. No equations or steps reduce by construction to the input inequality itself, no fitted parameters are relabeled as predictions, and no load-bearing self-citation chain is indicated in the provided text. The central claim remains an extension with independent content from the inequality and topological assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no access to the body of the paper, so free parameters, axioms, and invented entities cannot be extracted or verified.

pith-pipeline@v0.9.0 · 5640 in / 1091 out tokens · 27571 ms · 2026-05-23T02:39:20.918669+00:00 · methodology

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Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · 1 internal anchor

  1. [1]

    Carath\'eodory boundary extensions for generalized quasiregular mappings

    Desyatka, V., E. Sevost'yanov: On the boundary behavior of unclosed mappings with the inverse Poletsky inequality. https://arxiv.org/abs/2403.11023

    work page internal anchor Pith review Pith/arXiv arXiv

  2. [2]

    - Springer, Berlin etc., 1969

    Federer, H.: Geometric Measure Theory. - Springer, Berlin etc., 1969

    work page 1969

  3. [3]

    Sevost'yanov, S

    Ilkevych, N., E. Sevost'yanov, S. Skvortsov: On the global behavior of inverse mappings in terms of prime ends. - Annales Fennici Mathematici 46, 2021, 371--388

    work page 2021

  4. [4]

    Kovtonyuk, D.A. and V.I. Ryazanov: On the theory of prime ends for space mappings. - Ukr. Math. J. 67:4, 2015, 528--541

    work page 2015

  5. [5]

    Sevost'yanov: On prime ends on Riemannian manifolds

    Ilyutko, D.P., E.A. Sevost'yanov: On prime ends on Riemannian manifolds. - J. Math. Sci. 241:1, 2019. 47--63

    work page 2019

  6. [6]

    Kuratowski, K.: Topology, v. 1. -- Academic Press, New York--London, 1968

    work page 1968

  7. [7]

    Kuratowski, K.: Topology, v. 2. -- Academic Press, New York--London, 1968

    work page 1968

  8. [8]

    a is\" a l\

    Martio, O., S. Rickman, and J. V\" a is\" a l\" a : Topological and metric properties of quasiregular mappings. - Ann. Acad. Sci. Fenn. Ser. A1. 488, 1971, 1--31

    work page 1971

  9. [9]

    Ryazanov, U

    Martio, O., V. Ryazanov, U. Srebro, and E. Yakubov: Moduli in modern mapping theory. - Springer Science + Business Media, LLC, New York, 2009

    work page 2009

  10. [10]

    N\" a kki, R.: Prime ends and quasiconformal mappings. - J. Anal. Math. 35, 1979, 13--40

    work page 1979

  11. [11]

    N\" a kki, R. and B. Palka: Uniform equicontinuity of quasiconformal mappings. - Proc. Amer. Math. Soc. 37:2, 1973, 427--433

    work page 1973

  12. [12]

    Springer-Verlag, Berlin, 1993

    Rickman, S.: Quasiregular mappings. Springer-Verlag, Berlin, 1993

    work page 1993

  13. [13]

    - Dover Publ

    Saks, S.: Theory of the Integral. - Dover Publ. Inc., New York, 1964

    work page 1964

  14. [14]

    - Ukrainian Mathematical Journal 73:7, 2021,1107-1121

    Sevost'yanov, E.A: Boundary extensions of mappings satisfying the Poletsky inverse in terms of prime ends. - Ukrainian Mathematical Journal 73:7, 2021,1107-1121

    work page 2021

  15. [15]

    Developments in Mathematics (DEVM, volume 78), Springer Nature Switzerland AG, Cham, 2023

    Sevost'yanov E.: Mappings with Direct and Inverse Poletsky Inequalities. Developments in Mathematics (DEVM, volume 78), Springer Nature Switzerland AG, Cham, 2023

    work page 2023

  16. [16]

    Desyatka, Z

    Sevost'yanov, E., V. Desyatka, Z. Kovba: On the prime ends extension of unclosed inverse mappings. https://arxiv.org/abs/2409.02956

    work page arXiv

  17. [17]

    Skvortsov: On the Convergence of Mappings in Metric Spaces with Direct and Inverse Modulus Conditions

    Sevost'yanov, E.A., S.A. Skvortsov: On the Convergence of Mappings in Metric Spaces with Direct and Inverse Modulus Conditions. - Ukr. Math. J. 70:7, 2018, 1097--1114

    work page 2018

  18. [18]

    Skvortsov: On the local behavior of a class of inverse mappings

    Sevost'yanov, E.A., S.A. Skvortsov: On the local behavior of a class of inverse mappings. - J. Math. Sci. 241:1, 2019, 77--89

    work page 2019

  19. [19]

    Skvortsov: On mappings whose inverse satisfy the Poletsky inequality

    Sevost'yanov, E.A., S.A. Skvortsov: On mappings whose inverse satisfy the Poletsky inequality. - Ann. Acad. Scie. Fenn. Math. 45, 2020, 259--277

    work page 2020

  20. [20]

    Skvortsov: Logarithmic H\" o lder continuous mappings and Beltrami equation

    Sevost'yanov, E.A., S.A. Skvortsov: Logarithmic H\" o lder continuous mappings and Beltrami equation. - Analysis and Mathematical Physics 11:3, 2021, 138

    work page 2021

  21. [21]

    Skvortsov, O.P

    Sevost'yanov, E.A., S.O. Skvortsov, O.P. Dovhopiatyi: On nonhomeomorphic mappings with the inverse Poletsky inequality. - Journal of Mathematical Sciences 252:4, 2021, 541--557

    work page 2021

  22. [22]

    a is\" a l\

    V\" a is\" a l\" a J.: Lectures on n -dimensional quasiconformal mappings. - Lecture Notes in Math. 229, Springer-Verlag, Berlin etc., 1971

    work page 1971

  23. [23]

    Vuorinen, M.: Exceptional sets and boundary behavior of quasiregular mappings in n -space. - Ann. Acad. Sci. Fenn. Ser. A 1. Math. Dissertationes 11, 1976, 1--44

    work page 1976