A $C^m$ Lusin Approximation Theorem for Horizontal Curves in the Heisenberg Group

Andrea Pinamonti; Gareth Speight; Marco Capolli

A $C^m$ Lusin Approximation Theorem for Horizontal Curves in the Heisenberg Group

T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Every absolutely continuous horizontal curve in the Heisenberg group whose velocity is $m-1$ times $L^1$-differentiable almost everywhere coincides with a $C^m$ horizontal curve except on a set of arbitrarily small measure.

desk verdict A correct and sharp higher-order Lusin approximation theorem for horizontal curves in H^1, with the main load carried by the authors' earlier Whitney extension theorem; deserves refereeing. read the letter →

arxiv 1908.07624 v2 pith:TVFQMBM5 submitted 2019-08-20 math.MG math.FA

Marco Capolli , Andrea Pinamonti , Gareth Speight This is my paper · ORCID

classification math.MGmath.FA MSC 53C1726A2428A15

keywords HeisenberggrouphorizontalcurvesLusinapproximationL^1differentiabilityapproximateextensiontheoremforjetsCarnotgroupssub-Riemanniangeometry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves a $C^m$ Lusin approximation theorem for horizontal curves in the first Heisenberg group $\mathbb{H}^1$. It shows that every absolutely continuous horizontal curve $\Gamma=(f,g,h)$ for which the horizontal velocity components $f'$ and $g'$ are $m-1$ times $L^1$ differentiable almost everywhere has the $m$-Lusin property: for every $\eta>0$ there is a $C^m$ horizontal curve that agrees with $\Gamma$ except on a set of measure less than $\eta$. The argument works by extracting a large compact set on which the jets of $f,g,h$ satisfy the hypotheses of the $C^m$ extension theorem for horizontal curves, then extending. The paper also constructs a horizontal curve with $f,g,h$ twice $L^p$ differentiable for every $p\ge1$ and $f',g',h'$ once approximately differentiable that nevertheless has no $C^2$ horizontal Lusin approximation, showing the hypothesis cannot be weakened to approximate differentiability. The result matters because it identifies the regularity of the velocity, not of the curve itself, that controls smoothing in sub-Riemannian geometry.

What carries the argument

The carrying object is the extension theorem for $C^m$ horizontal curves in $\mathbb{H}^1$ (Theorem 2.15), quoted from an earlier paper. It says a jet $(F,G,H)$ of order $m$ on a compact set $K$ extends to a $C^m$ horizontal curve exactly when three conditions hold: the jet is a $C^m$ Whitney field (Taylor remainders are $o(|a-b|^{m-k})$); the derivatives satisfy the Leibniz-type relation $H^k = 2\sum_{i=0}^{k-1}\binom{k-1}{i}(F^{k-i}G^i - G^{k-i}F^i)$; and an area discrepancy $A(a,b)/V(a,b)\to 0$ uniformly, where $A$ measures the mismatch between the vertical increment $h(b)-h(a)$ and the increment predicted by the Taylor polynomials of $f$ and $g$, while $V$ controls the scale. The proof of Theorem 4.1 chooses a large compact set $K$ using Lemma 3.1 and Proposition 3.4 (integration of $L^1$ derivatives and a polynomial estimate on density sets), then verifies the three hypotheses, with the $L^1$ differentiability of $f',g'$ giving the uniform estimates needed for the area discrepancy.

What would settle it

Produce a horizontal curve $\Gamma=(f,g,h)$ satisfying the hypotheses of Theorem 4.1 for which every $C^m$ horizontal curve differs from $\Gamma$ on a set of positive measure; equivalently, on the compact set $K$ built in the proof, show that the vertical increment $h(b)-h(a)$ cannot be reproduced from the Taylor polynomials of $f$ and $g$ up to the required order on a positive fraction of pairs $(a,b)$. Either would refute the theorem.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 4.1: if $\Gamma=(f,g,h):I\to\mathbb{H}$ is an absolutely continuous horizontal curve and $f',g'$ are $m-1$ times $L^1$ differentiable at almost every point of $I$, then $\Gamma$ has the $m$-Lusin property; moreover the approximating $C^m$ horizontal curve can be chosen so that its derivatives up to order $m$ agree with the jets of $\Gamma$ on the large set. The converse half, Theorem 5.1, constructs an explicit absolutely continuous horizontal curve for which $f,g,h$ are twice $L^p$ differentiable for all $p\ge1$ and $f',g',h'$ are once approximately differentiable almost everywhere, yet no $C^2$ horizontal curve agrees with it on a set of positive measure. Together these show that the $L^1$ differentiability of the velocity is the right hypothesis and that the Euclidean sufficient condition, approximate differentiability of the curve itself, is not sufficient in the Heisenberg group, because the approximating curve must remain horizontal.

Load-bearing premise

The proof assumes, without proving it here, the quoted $C^m$ extension theorem for horizontal curves in the Heisenberg group (Theorem 2.15); if that theorem were false or incomplete, the main positive result would not follow.

Editorial extensions

If this is right

Every horizontal curve in $\mathbb{H}^1$ with $(m-1)$-fold $L^1$-differentiable velocity admits a $C^m$ horizontal approximation, so the $C^1$ result known for all absolutely continuous horizontal curves extends to all orders $m$.
The approximating $C^m$ curve can be arranged to match the original curve and all its derivatives up to order $m$ on the large agreement set.
Approximate differentiability of the curve itself, which is sufficient in Euclidean space, is not sufficient in $\mathbb{H}^1$: the paper's counterexample has $f,g,h$ twice $L^p$-differentiable for every $p\ge1$ and $f',g',h'$ once approximately differentiable, yet no $C^2$ horizontal Lusin approximation exists.
The $L^1$ differentiability hypothesis on the velocity is therefore the correct threshold in the Heisenberg group, at least for $m=2$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the positive proof reduces to verifying the three hypotheses of the extension theorem on a large compact set, any Carnot group with a $C^m$ extension theorem for horizontal curves will inherit the same $m$-Lusin property for curves with $L^1$-differentiable velocity; the group-specific work is concentrated in the extension theorem.
The counterexample's mechanism, vertical increments that concentrate on small intervals and cannot be matched by any horizontal $C^2$ jet, suggests that the area discrepancy is the right quantitative obstruction, and one could try to build counterexamples in other Carnot groups by engineering similar concentration.
A testable open direction is whether the theorem survives if $L^1$ differentiability of $f',g'$ is replaced by $L^p$ differentiability for some fixed $p>1$; the proof's estimates are written for $L^1$ averages, so this is not a direct corollary.
If the quoted $C^m$ extension theorem is available in higher-dimensional Heisenberg groups $\mathbb{H}^n$, the same compact-set construction should give the analogue of Theorem 4.1; the authors expect this, but the paper proves only the first Heisenberg group.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: this paper settles the natural higher-order Lusin approximation question for horizontal curves in the first Heisenberg group, and the counterexample in Section 5 is the right kind of sharpness result. I think it should go to referees.

The positive theorem (Theorem 4.1) is exactly what you'd hope: if the horizontal velocity components f', g' are (m−1)-times L1 differentiable a.e., then the absolutely continuous horizontal curve agrees with a C^m horizontal curve off a set of arbitrarily small measure. The proof is a reduction to the authors' earlier C^m Whitney extension theorem for horizontal curves (Theorem 2.15, published in Trans. AMS 2019). The three hypotheses are checked carefully: Proposition 3.4 builds the Whitney field, Lemma 3.2 gives the jet compatibility condition, and the area-discrepancy estimate in Section 4 follows from uniform L1 differentiability of f' and g' on a large compact set. This is a clean argument.

Section 5 constructs an explicit horizontal curve where f,g,h are twice L^p differentiable a.e. for every p≥1 and f',g',h' are once approximately differentiable a.e., but the curve has no C^2 horizontal Lusin approximation. The construction uses dyadic intervals with small spikes, and the proof that no smooth approximation exists is a nice density argument: if a C^2 horizontal curve agreed with Γ on a large set, then two nearby density points straddling a spike would force H(y)−H(x) to be both large (from the spike height) and small (from the C^2 bound), a contradiction. The parameters are chosen to satisfy the required summability, and I did not find a gap.

Soft spots are minor. The choice of the compact set K in Theorem 4.1 that simultaneously gives Whitney-field control and uniform L1 differentiability is dismissed as 'elementary measure theory'; it is standard (take compact subsets of the full-measure uniform-differentiability sets and intersect with the Whitney-field compact sets), but the paper should spell it out. Lemma 5.9 is terse: from F vanishing on a sequence of points accumulating at x it concludes F'(x)=F''(x)=0; this is correct via Taylor expansion at x, but a sentence would help. The reliance on the authors' own Whitney theorem is the main external input; since that theorem is prior published work, this is not circular, and the burden is appropriately low.

Who should read this: people working on Lusin properties, Whitney extension problems, or approximation of horizontal curves in Carnot groups. It is a solid contribution that deserves a serious referee. I would recommend accept after minor revisions, mainly to expand the two terse measure-theoretic passages.

Referee Report

0 major / 4 minor

Summary. The paper proves a C^m Lusin approximation theorem for horizontal curves in the first Heisenberg group. The main positive result, Theorem 4.1, states that if Γ=(f,g,h):I→H is an absolutely continuous horizontal curve and f′,g′ are (m−1)-times L1 differentiable at almost every point, then Γ has the m-Lusin property: it agrees, together with its Taylor coefficients up to order m, with a C^m horizontal curve except on a set of arbitrarily small measure. The proof uses Lemma 3.1 and Lemma 3.2 to upgrade L1 differentiability of the velocity components to m-times L1 differentiability of f,g,h, Proposition 3.4 to produce Whitney fields on large compact sets, and then verifies the three hypotheses of the horizontal Whitney extension theorem (Theorem 2.15, quoted from Pinamonti–Speight–Zimmerman). The second main result, Theorem 5.1, constructs an explicit absolutely continuous horizontal curve for which f,g,h are twice L^p differentiable almost everywhere and f′,g′,h′ are once approximately differentiable almost everywhere, yet the curve does not admit a C^2 horizontal Lusin approximation. This counterexample shows that the L1 differentiability hypothesis on the velocity in Theorem 4.1 cannot be weakened to approximate differentiability, in contrast to the Euclidean situation.

Significance. If the results are correct, they give a natural and apparently optimal higher-order version of Lusin approximation for horizontal curves in the Heisenberg group, and the counterexample cleanly demonstrates a genuine difference between the Heisenberg and Euclidean settings. The proof of the positive theorem is well structured: the reduction to the published horizontal Whitney extension theorem is explicit, the verification of all three hypotheses of that theorem is carried out in detail, and the main estimate in the verification of hypothesis (3) is careful and quantitative. The counterexample is explicit and the obstruction is concrete, based on a density-point argument. The paper is readable and the internal logic is coherent; the main external input, Theorem 2.15, is prior published work and is used transparently as a black box, so I do not regard the dependency as circular.

minor comments (4)

[Throughout] The text contains several spelling and typesetting errors, such as 'positon' in the introduction and 'APPROXIMA TION' and 'HORIZONT AL' in the running title; these should be corrected before publication.
[Section 4, proof of Theorem 4.1] The simultaneous choice of the compact set K satisfying the Whitney-field condition, the uniform estimate (4.1), and the measure bound L1(I\K)<η is compressed into the phrase 'elementary measure theory'; since this is the one step where three conditions are obtained together, I suggest adding a short explanation, for instance by taking compact subsets of the almost-everywhere L1-differentiability sets with uniform averaged Taylor error, intersecting them with the compact sets from Proposition 3.4, and then passing to a further compact subset of large measure.
[Section 4, verification of Theorem 2.15(2)] In the verification of condition (2), the passage from the identity for (TH)′ to the identities for (TH)^k would be easier to follow if the polynomials S_a and S_a^k were defined explicitly; in particular, stating that S_a^k is the k-th derivative contribution of the error term would clarify the divisibility argument and the conclusion S_a^k(a)=0.
[Section 2, Theorem 2.15] Since the proof of the paper depends essentially on the horizontal Whitney extension theorem from [24], a sentence in Section 2 explicitly noting that this theorem is used as a black box and not reproved here would help orient readers who expect the paper to be fully self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 4.1 is a legitimate reduction to an independent published Whitney extension theorem with all hypotheses verified in the paper; the counterexample is self-contained.

full rationale

The derivation chain in Theorem 4.1 is a direct and non-circular reduction. The only load-bearing external input is Theorem 2.15, the C^m horizontal Whitney extension theorem of Pinamonti--Speight--Zimmerman. Although two of the present authors are among the authors of that cited theorem, it is a prior published result whose stated assumptions (Whitney fields on a compact set, algebraic compatibility of the jets, and the area/velocity decay condition) do not include the m-Lusin property. The paper does not define the Lusin conclusion into the hypotheses; instead it proves Proposition 3.4, Lemmas 3.1 and 3.2, and then verifies conditions (1), (2), and (3) of Theorem 2.15 explicitly on a carefully chosen compact set K. The verification of condition (3) is a direct estimate using horizontality equation (2.2) and the L^1 differentiability of f' and g', not an appeal to the desired conclusion. The counterexample in Theorem 5.1 is constructed independently and does not presuppose Theorem 4.1. There are no fitted parameters, no quantity is predicted from itself, and no uniqueness claim is imported to force the argument. The self-citation is load-bearing in the sense of being an important theorem, but it is independent support rather than circularity, so the appropriate finding is no significant circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No new entities are introduced. The result is a theorem in established mathematics; the only tunable objects are the auxiliary sequences in the counterexample construction, which are determined up to inequalities and play no role in the main positive theorem.

free parameters (1)

Sequences h_n, lambda_n, w_n in Theorem 5.1 construction = h_n = 3^{-n}, lambda_n = (2/5)^n, w_n <= 2^{-6n} (one possible choice)
Chosen by hand to satisfy the growth conditions (5.1)-(5.4). The theorem only needs existence of such sequences; the specific values do not enter the central claims and are not fitted to data.

assumptions (4)

domain assumption C^m Whitney extension theorem for horizontal curves in H (Theorem 2.15)
Quoted from [24] and used as the central tool; the proof of Theorem 4.1 verifies its three hypotheses on a large compact set.
domain assumption Measurability of approximate derivative coefficients (Liu-Tai [20])
Used in Proposition 3.4 and Theorem 4.1 to define the jets F,G,H from pointwise derivatives; cited to [20].
standard math De Giorgi's lemma (Lemma 3.3)
Used in Proposition 3.4 to bound polynomial derivatives by L1 averages over high-density sets.
standard math Borel-Cantelli lemma and Lebesgue density theorem
Used in Section 5 for the measure-zero set A and for selecting density points in the contradiction argument.

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Cite this review

Pith. "Pith review of A $C^m$ Lusin Approximation Theorem for Horizontal Curves in the Heisenberg Group." pith.science (2026). https://pith.science/paper/TVFQMBM5

@misc{pith2026190807624,
  author       = {Pith},
  title        = {Pith review of: A $C^m$ Lusin Approximation Theorem for Horizontal Curves in the Heisenberg Group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TVFQMBM5}},
  note         = {Machine review of arXiv:1908.07624}
}

abstract

We prove a $C^m$ Lusin approximation theorem for horizontal curves in the Heisenberg group. This states that every absolutely continuous horizontal curve whose horizontal velocity is $m-1$ times $L^1$ differentiable almost everywhere coincides with a $C^m$ horizontal curve except on a set of small measure. Conversely, we show that the result no longer holds if $L^1$ differentiability is replaced by approximate differentiability. This shows our result is optimal and highlights differences between the Heisenberg and Euclidean settings.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Alberti, G., Bianchini, S., Crippa, G.: On the Lp-Diﬀerentiability of Certain Classes of Functions , Revista Matematica Iberoamericana 30(1) (2014), 349– 367

work page 2014

[2] [2]

Ambrosio, L., Tilli, P.: Topics on Analysis in Metric Spaces , Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press , 2004

work page 2004

[3] [3]

Bierstone, E.: Diﬀerentiable Functions, Bulletin of the Brazilian Mathematical Society 11(2) (1980) 139–190

work page 1980

[4] [4]

Bonﬁglioli, A., Lanconelli, E., Uguzzoni, F.: Stratiﬁed Lie groups and Poten- tial Theory for their Sub-Laplacians , Springer Monographs in Mathematics, Springer, Berlin, 2007

work page 2007

[5] [5]

Campanato, S.: Propriet´ a di una Famiglia di Spazi Funzionali , Annali della Scuola Normale Superiore di Pisa 18 (1964) 137–160

work page 1964

[6] [6]

LUSIN APPROXIMATION IN THE HEISENBERG GROUP 21

Cheeger, J.: Diﬀerentiability of Lipschitz Functions on Metric Measure Spaces, Geometric and Functional Analysis 9(3) (1999), 428–517. LUSIN APPROXIMATION IN THE HEISENBERG GROUP 21

work page 1999

[7] [7]

Capogna, L., Danielli, D., Pauls, S., Tyson, J.: An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Pro blem, Birkhauser, Progress in Mathematics 259, 2007

work page 2007

[8] [8]

Evans, L., Gariepy, R.: Measure Theory and Fine Properties of Functions , Studies in Advanced Mathematics, CRC Press, 1991

work page 1991