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arxiv: 2509.00920 · v2 · submitted 2025-08-31 · 🧮 math.FA

A surprising threshold for the validity of the method of singular projection

Antoine Detaille This is my paper

Pith reviewed 2026-05-18 19:27 UTC · model grok-4.3

classification 🧮 math.FA
keywords singular projectionfractional Sobolev spacesmanifold-valued mapsW^{s,p} regularitycounterexamplesHardt-Lin methodalmost retraction
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The pith

For 0 < s < 1 the singular projection method requires the stricter condition p < ℓ to preserve W^{s,p} regularity, while s ≥ 1 permits the full range sp < ℓ.

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

The paper studies when a projection P onto a manifold N, singular on a set of codimension ℓ, can convert an R^ν-valued W^{s,p} map into an N-valued map of the same class. For s ≥ 1 the method succeeds throughout the expected range sp < ℓ. For fractional exponents 0 < s < 1 the authors construct explicit bounded maps from R^ℓ to R^ℓ whose images under singular projection onto the sphere S^{ℓ-1} leave the space W^{s,p} whenever p ≥ ℓ. This establishes that the previously known threshold p < ℓ is sharp in the fractional case. The same sharp threshold is obtained for the related almost-retraction method.

Core claim

We prove that the method of projection applies in W^{s,p} for s ≥ 1 whenever sp < ℓ. When 0 < s < 1 we construct, for every p ≥ ℓ, a bounded map u in W^{s,p}(R^ℓ, R^ℓ) such that every singular projection of u onto the sphere fails to belong to W^{s,p}. A similar conclusion holds for the method of almost retraction.

What carries the argument

Counterexample maps from R^ℓ to R^ℓ whose fractional seminorm stays finite but whose images under the codimension-ℓ singular projection onto S^{ℓ-1} have infinite Gagliardo seminorm.

If this is right

  • For s ≥ 1 any W^{s,p} map satisfies sp < ℓ implies the projected map stays in W^{s,p}.
  • For 0 < s < 1 the projection method is valid only when p < ℓ.
  • The almost-retraction method likewise requires p < ℓ when 0 < s < 1.
  • Bounded maps exist whose singular projections lose all fractional regularity once p reaches ℓ.

Where Pith is reading between the lines

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

  • The result indicates that fractional regularity is more sensitive to low-dimensional singularities than integer-order Sobolev regularity.
  • Similar thresholds may appear when projecting maps into other manifolds whose singular sets have comparable codimension.
  • The construction technique could be adapted to test sharpness for projections with different codimensions or targets.

Load-bearing premise

The counterexamples for 0 < s < 1 rely on the specific geometry of the sphere and the codimension-ℓ singular set interacting with the fractional seminorm.

What would settle it

An explicit calculation of the Gagliardo seminorm for one of the constructed maps showing that the projected function remains in W^{s,p} for some choice of s < 1 and p ≥ ℓ.

Figures

Figures reproduced from arXiv: 2509.00920 by Antoine Detaille.

Figure 1
Figure 1. Figure 1: Estimate of |𝑃(𝑐 + − 𝑎) − 𝑃(𝑐 − − 𝑎)| in Lemma 4.2 Let us make a comment about the strategy for estimating the energy of 𝑃◦(𝑣𝑛−𝑎)in the case where 𝑝 = ℓ in the above proof. The idea was to take into account the contribution to the energy of not only one "patch" but of many of them. More specifically, we took into account the contribution of every patch whose associated center 𝑐𝑛,𝑘 lies in a cone with verte… view at source ↗
Figure 2
Figure 2. Figure 2: Estimate of |𝑃(𝑐 + − 𝑎) − 𝑃(𝑐 − − 𝑎)| in Lemma 4.3 where we have used the fact that 𝑐 + − 𝑐˜ and 𝑐 − − 𝑐˜ are aligned and have opposite directions. Since 𝑐˜ is an orthogonal projection, we have |𝑐˜ − 𝑎| ≤ |𝑐 ± − 𝑎|. Hence, we find |𝑃(𝑐 + − 𝑎) − 𝑃(𝑐 − − 𝑎)| ≥ 2 |𝑐 + − 𝑐˜||𝑐 − − 𝑐˜| |𝑐 + − 𝑎||𝑐 − − 𝑎| . Finally, since 𝑎 ∈ 𝑄ℓ 2 −𝑛 (𝑐), we find that |𝑐 ± − 𝑐˜| ≳ 2 −𝑛 while |𝑐 ± − 𝑎| ≲ 2 1−𝑛 , whence we conclud… view at source ↗
read the original abstract

Given a compact manifold $ \mathcal{N} $ embedded into $ \mathbb{R}^{\nu} $ and a projection $ P $ that retracts $ \mathbb{R}^{\nu} $ except a singular set of codimension $ \ell $ onto $ \mathcal{N} $, we investigate the maximal range of parameters $ s $ and $ p $ such that the projection $ P $ can be used to turn an $ \mathbb{R}^{\nu} $-valued $ W^{s,p} $ map into an $ \mathcal{N} $-valued $ W^{s,p} $ map. Devised by Hardt and Lin with roots in the work of Federer and Fleming, the method of projection is known to apply in $ W^{1,p} $ if and only if $ p < \ell $, and has been extended in some special cases to more general values of the regularity parameter $ s $. As a first result, we prove in full generality that, when $ s \geq 1 $, the method of projection can be applied in the whole expected range $ sp < \ell $. When $ 0 < s < 1 $, the method of projection was only known to be applicable when $ p < \ell $, a more stringent condition than $ sp < \ell $. As a second result, we show that, somehow surprisingly, the condition $ p < \ell $ is optimal, by constructing, for every $ 0 < s < 1 $ and $ p \geq \ell $, a bounded $ W^{s,p} $ map into $ \mathbb{R}^{\ell} $ whose singular projections onto the sphere $ \mathbb{S}^{\ell-1} $ all fail to belong to $ W^{s,p} $. As a byproduct of our method, a similar conclusion is obtained for the closely related method of almost retraction, devised by Haj\l asz, for which we also prove a more stringent threshold of applicability when $ 0 < s < 1 $.

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

2 major / 2 minor

Summary. The paper studies the applicability of the method of singular projection (and the related method of almost retraction) for producing manifold-valued maps in the fractional Sobolev space W^{s,p}. For s ≥ 1 it proves in full generality that the method works throughout the expected range sp < ℓ. For 0 < s < 1 it shows that the previously known sufficient condition p < ℓ is in fact optimal, by constructing, for every such pair (s,p) with p ≥ ℓ, an explicit bounded map u : ℝ^ℓ → ℝ^ℓ that lies in W^{s,p} but for which every singular projection onto S^{ℓ-1} fails to lie in W^{s,p}.

Significance. If the central claims hold, the work is significant: it supplies the first sharp threshold for the projection method when 0 < s < 1, demonstrates that the interaction between the Gagliardo seminorm and a codimension-ℓ singular set behaves differently from the integer-order case, and furnishes explicit counterexample maps together with direct seminorm estimates. The full proofs and constructions constitute a clear strength.

major comments (2)
  1. The optimality claim for 0 < s < 1 rests on the explicit family of maps constructed in the second main result. A more detailed verification that the fractional seminorm of each projected map diverges while the original map remains in W^{s,p} would be useful; in particular, the control on the measure of the set where the map hits a neighborhood of the origin should be spelled out with explicit constants.
  2. The general proof for s ≥ 1 (sp < ℓ) is stated to hold for arbitrary compact manifolds N and projections with codimension-ℓ singular sets. It would strengthen the result to include a brief comparison with the known W^{1,p} case of Hardt–Lin, highlighting where the fractional extension uses new arguments.
minor comments (2)
  1. Notation: the sphere is denoted S^{ℓ-1} in the abstract and ℝ^ℓ-valued maps are used for the counterexamples; a short remark clarifying that the construction is tailored to the sphere but the positive result holds for general N would improve readability.
  2. The abstract mentions that a similar conclusion holds for the method of almost retraction; a one-sentence pointer to the precise statement (theorem number) would help readers locate the byproduct result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive assessment, including the recommendation for minor revision. We address each major comment below and will update the manuscript accordingly to improve clarity.

read point-by-point responses

  1. Referee: The optimality claim for 0 < s < 1 rests on the explicit family of maps constructed in the second main result. A more detailed verification that the fractional seminorm of each projected map diverges while the original map remains in W^{s,p} would be useful; in particular, the control on the measure of the set where the map hits a neighborhood of the origin should be spelled out with explicit constants.

    Authors: We agree that expanding the verification would improve readability. In the revised manuscript we will add a more detailed step-by-step estimate of the Gagliardo seminorm for the projected maps, including explicit constants for the measure of the sets on which the original map enters a small neighborhood of the origin. These constants follow directly from the scaling properties of the constructed maps and the choice of the cutoff radii. revision: yes

  2. Referee: The general proof for s ≥ 1 (sp < ℓ) is stated to hold for arbitrary compact manifolds N and projections with codimension-ℓ singular sets. It would strengthen the result to include a brief comparison with the known W^{1,p} case of Hardt–Lin, highlighting where the fractional extension uses new arguments.

    Authors: We thank the referee for this constructive suggestion. In the revised version we will insert a short paragraph, either in the introduction or immediately after the statement of the s ≥ 1 result, that recalls the Hardt–Lin theorem for W^{1,p} and then indicates the new technical ingredients required for the fractional case: the use of the Gagliardo seminorm, the handling of nonlocal interactions across the singular set of codimension ℓ, and the application of fractional potential estimates that are not needed in the first-order setting. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes its main results via direct proofs and explicit constructions rather than any reduction to self-defined quantities or fitted inputs. For s ≥ 1 the applicability in the range sp < ℓ is shown in full generality by estimates that do not presuppose the target conclusion. For 0 < s < 1 the optimality of the stricter threshold p < ℓ is demonstrated by constructing, for each such pair, a bounded map u : ℝ^ℓ → ℝ^ℓ that lies in W^{s,p} while every singular projection P ∘ u fails to lie in W^{s,p}; the argument uses the codimension-ℓ singularity of the radial projection onto S^{ℓ-1} together with controlled estimates on the Gagliardo seminorm performed directly on this geometry. No load-bearing self-citation, imported uniqueness theorem, or ansatz smuggled via prior work appears in the derivation chain, and the constructions are independent of the claimed thresholds. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard properties of fractional Sobolev spaces and on the geometry of the sphere; no free parameters or invented entities are introduced.

axioms (2)
  • standard math Standard embedding and trace theorems for W^{s,p} spaces on R^ν hold.
    Invoked implicitly when discussing preservation of Sobolev regularity under projection.
  • domain assumption The singular set of the projection has codimension exactly ℓ.
    This codimension controls the range sp < ℓ or p < ℓ.

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