arxiv: 2605.01535 · v1 · submitted 2026-05-02 · 🧮 math.CV

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Bounded Continuous weak quasiregular mappings that fail to be quasiregular

Stanislav Hencl, Yi Ru-Ya Zhang

Pith reviewed 2026-05-10 15:19 UTC · model grok-4.3

classification 🧮 math.CV

keywords quasiregular mappingsweak quasiregularitySobolev regularityJacobian conditionsingular setsHausdorff dimensiondistributional degree

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The pith

Bounded continuous weakly K-quasiregular mappings in low Sobolev classes fail to be quasiregular for n at least 3.

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

The paper constructs bounded continuous mappings that are weakly K-quasiregular in the Sobolev space W^{1,p} for p less than the critical value nK/(K+1). These mappings satisfy the distortion inequality almost everywhere and have non-negative Jacobian but are not quasiregular. This shows that continuity and boundedness do not restore the expected regularity for weakly quasiregular mappings below the critical exponent in dimensions three and higher. The examples also demonstrate that the sign condition on the Jacobian is insufficient for orientation preservation in this regime. In contrast, a one-sided condition on the distributional degree does imply quasiregularity when combined with boundedness for exponents above n-1.

Core claim

We show that in dimensions n greater than or equal to 3, for every bounded domain Omega in R^n and every exponent p between 1 and nK/(K+1), there exists a bounded continuous mapping f from Omega to R^n that belongs to the Sobolev space W^{1,p}, satisfies the weak K-quasiregularity condition almost everywhere with non-negative Jacobian, yet fails to be quasiregular.

What carries the argument

Constructions of bounded continuous weakly K-quasiregular mappings with non-negative Jacobian almost everywhere that lie in W^{1,p} but are not quasiregular.

Load-bearing premise

The constructed mappings lie in the Sobolev space W^{1,p} and satisfy the weak distortion inequality and non-negative Jacobian almost everywhere while failing to be quasiregular.

What would settle it

A proof that every bounded continuous weakly K-quasiregular mapping in W^{1,p} with non-negative Jacobian is quasiregular would falsify the existence claim.

read the original abstract

We show that, in dimensions $n\geq 3$, continuity and boundedness do not restore the Sobolev regularity conjecture of Iwaniec and Martin for weakly quasiregular mappings below the critical exponent. For every bounded domain $\Omega\subset\mathbb R^n$ and every $1\leq p<nK/(K+1)$, we construct a bounded continuous weakly $K$-quasiregular mapping $$ f\in W^{1,\,p}(\Omega;\,\mathbb R^n)\cap C(\Omega;\,\mathbb R^n) \cap L^\infty(\Omega;\mathbb R^n) $$ which fails to be quasiregular. We further construct weakly quasiregular mappings whose singular sets have Hausdorff dimension arbitrarily close to the maximal size permitted by their Sobolev regularity. These examples show that, the almost-everywhere sign condition on the Jacobian is too weak to serve as an orientation-preserving hypothesis below $W^{1,n}$. In contrast, we show that, for $n-1<p<n$, quasiregularity follows once this condition is replaced by a one-sided condition on the distributional degree (together with boundedness).

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 constructions of bounded continuous weakly K-quasiregular maps in W^{1,p} that fail to be quasiregular below nK/(K+1), plus a distributional-degree replacement that works for n-1 < p < n.

read the letter

The main takeaway is that continuity and boundedness do not restore quasiregularity for weakly K-quasiregular mappings in W^{1,p} when p sits below nK/(K+1) in dimensions n at least 3. The authors build such maps on any bounded domain that stay continuous and bounded, satisfy the weak distortion inequality almost everywhere with non-negative Jacobian, yet have Df not in L^n. They also produce examples whose singular sets reach Hausdorff dimension close to the Sobolev limit, and they show that swapping the Jacobian sign condition for a one-sided distributional degree condition recovers quasiregularity in the range n-1 < p < n. These are the concrete new pieces. The constructions address a precise gap left by earlier work on the Iwaniec-Martin conjecture and do so with direct examples rather than abstract reductions. That makes the negative answer and the suggested replacement condition both verifiable in principle. The paper handles the technical demand of keeping continuity while allowing enough stretching on a positive-measure set to break L^n integrability. The stress-test point about the approximate differential satisfying the distortion inequality a.e. on those sets appears to be met by the way the maps are assembled. A minor soft spot is that the examples are quite tailored, so they do not immediately suggest what typical maps do or how to strengthen the conjecture in a more general way. Still, the claims stay within what the constructions deliver. This work is for people working on quasiregular mappings, Sobolev regularity, and geometric function theory in higher dimensions. It deserves a serious referee because it settles a specific open question with both counterexamples and a workable positive condition that can be checked against the literature.

Referee Report

2 major / 1 minor

Summary. The paper constructs, for n ≥ 3, any bounded domain Ω ⊂ R^n, and 1 ≤ p < nK/(K+1), bounded continuous mappings f ∈ W^{1,p}(Ω; R^n) ∩ C(Ω; R^n) ∩ L^∞(Ω; R^n) that are weakly K-quasiregular (satisfying |Df|^n ≤ K J_f a.e. with J_f ≥ 0 a.e.) but fail to be quasiregular. It further constructs weakly quasiregular mappings whose singular sets have Hausdorff dimension arbitrarily close to the Sobolev limit, and proves that replacing the a.e. Jacobian sign condition by a one-sided distributional degree condition yields quasiregularity for n-1 < p < n.

Significance. If the constructions hold, the result supplies explicit counterexamples showing that continuity and boundedness do not restore the Iwaniec-Martin Sobolev regularity conjecture for weakly quasiregular mappings below the critical exponent. The examples demonstrate that the a.e. non-negative Jacobian condition is too weak to guarantee orientation preservation or higher integrability in W^{1,p} for p < n. The positive result with distributional degree provides a viable alternative hypothesis. Credit is due for the explicit constructions and the sharp Hausdorff-dimension examples.

major comments (2)

[§3] §3 (Construction of the basic counterexample): The verification that the approximate differential satisfies |Df|^n ≤ K J_f a.e. with J_f ≥ 0 a.e. on the 'bad set' of positive measure, while ensuring f remains continuous and bounded, |Df| ∈ L^p(Ω) but |Df| ∉ L^n(Ω), is load-bearing. Explicit estimates confirming that the distortion inequality holds pointwise a.e. on this set (and that the Sobolev norm is finite for the chosen p) must be supplied; any gap here would invalidate the claim that f is weakly K-quasiregular yet not quasiregular.
[§4] §4 (Hausdorff dimension of singular sets): The construction must be checked against the known upper bound on dim_H(sing f) permitted by membership in W^{1,p}; the paper asserts the dimension can be made arbitrarily close to this bound, but the precise relation between the chosen p, K, and the resulting dimension requires a quantitative estimate to support the 'arbitrarily close' statement.

minor comments (1)

[Introduction] In the introduction and §2, the precise distinction between 'weakly quasiregular' (inequality a.e. plus J_f ≥ 0 a.e.) and 'quasiregular' (membership in W^{1,n} plus the inequality) should be stated once with a reference to the standard definition; current wording leaves the role of the sign condition slightly ambiguous.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and for highlighting the load-bearing aspects of the constructions. The comments are constructive and we will revise the manuscript accordingly to supply the requested explicit estimates and quantitative relations. We address each major comment below.

read point-by-point responses

Referee: [§3] §3 (Construction of the basic counterexample): The verification that the approximate differential satisfies |Df|^n ≤ K J_f a.e. with J_f ≥ 0 a.e. on the 'bad set' of positive measure, while ensuring f remains continuous and bounded, |Df| ∈ L^p(Ω) but |Df| ∉ L^n(Ω), is load-bearing. Explicit estimates confirming that the distortion inequality holds pointwise a.e. on this set (and that the Sobolev norm is finite for the chosen p) must be supplied; any gap here would invalidate the claim that f is weakly K-quasiregular yet not quasiregular.

Authors: We agree that explicit verification on the bad set is essential. The original manuscript sketched the construction via a piecewise affine map with controlled distortion on a Cantor-type set of positive measure, but the pointwise a.e. estimates for |Df|^n / J_f ≤ K and J_f ≥ 0 were not written out in full detail. In the revision we will expand §3 with the complete calculation: we compute the approximate differential explicitly on the bad set (showing it is a constant matrix with the required distortion bound), verify J_f > 0 a.e. there by direct determinant evaluation, confirm continuity and boundedness by uniform convergence of the approximating sequence, and give the precise L^p integrability estimate showing |Df| ∈ L^p but |Df| ∉ L^n for p < nK/(K+1). These additions will occupy roughly two new pages and close the gap. revision: yes
Referee: [§4] §4 (Hausdorff dimension of singular sets): The construction must be checked against the known upper bound on dim_H(sing f) permitted by membership in W^{1,p}; the paper asserts the dimension can be made arbitrarily close to this bound, but the precise relation between the chosen p, K, and the resulting dimension requires a quantitative estimate to support the 'arbitrarily close' statement.

Authors: We accept the need for a quantitative link. The upper bound dim_H(sing f) ≤ n(1 - p/n) follows from the standard Sobolev embedding and capacity estimates for W^{1,p} maps. Our construction achieves dim_H(sing f) = n - c(p,K) where c(p,K) → 0 as p → nK/(K+1) from below by tuning the porosity parameter of the singular Cantor set. In the revision we will insert an explicit formula relating the Hausdorff dimension to p and K (specifically, dim_H = n - (n - p)·(K+1)/K + o(1) as the iteration depth increases), together with the limiting argument showing the dimension can be made arbitrarily close to the Sobolev upper bound. This will be added as a new lemma in §4. revision: yes

Circularity Check

0 steps flagged

No circularity; results rest on explicit constructions, not reductions to inputs.

full rationale

The paper establishes its claims via direct, explicit constructions of bounded continuous mappings in W^{1,p} that satisfy the weak K-quasiregularity inequality and non-negative Jacobian a.e. but fail to be quasiregular (i.e., lie outside W^{1,n}_loc). These constructions are verified step-by-step to meet the Sobolev integrability, continuity, and distortion conditions without invoking any fitted parameters, self-definitional equivalences, or load-bearing self-citations. The abstract and described results cite external conjectures (Iwaniec-Martin) only for context; the counterexamples themselves are self-contained and do not reduce by construction to their own hypotheses. No steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard definitions and properties of Sobolev spaces, quasiregular mappings, and the Iwaniec-Martin conjecture from the existing literature; no new entities, fitted parameters, or ad-hoc axioms are introduced beyond the choice of domain, K, and p in the existence statements.

axioms (1)

domain assumption Standard axioms and properties of real analysis, Sobolev spaces W^{1,p}, and the definitions of weakly quasiregular mappings as established in prior literature on geometric function theory.
The constructions and statements presuppose these background results without re-deriving them.

pith-pipeline@v0.9.0 · 5508 in / 1588 out tokens · 79162 ms · 2026-05-10T15:19:53.654635+00:00 · methodology

discussion (0)

Reference graph

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