Area of H\"older curves and coarea formula on the Heisenberg group

Gioacchino Antonelli; Robert Young

arxiv: 2605.15987 · v1 · pith:HA43PGFHnew · submitted 2026-05-15 · 🧮 math.MG · math.CA· math.DG

Area of H\"older curves and coarea formula on the Heisenberg group

Gioacchino Antonelli , Robert Young This is my paper

Pith reviewed 2026-05-19 17:14 UTC · model grok-4.3

classification 🧮 math.MG math.CAmath.DG

keywords coarea formulaHeisenberg groupHolder curvessymplectic areabeta numberssubriemannian geometryLipschitz maps

0 comments

The pith

Lipschitz maps from the subriemannian Heisenberg group to R^{2n} obey the coarea formula.

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

The paper establishes the coarea formula for Lipschitz maps f from the nth Heisenberg group H_n to R^{2n}. This result is new even for n=1 and gives the simplest vector-valued example in subriemannian geometry, answering questions from earlier studies. The key step is defining a symplectic area for the typically unrectifiable fibers using a new integral for 1/2-Holder curves in the plane. The authors identify a geometric condition under which this integral converges and prove it holds for almost every fiber by applying beta-number estimates.

Core claim

We prove the coarea formula for Lipschitz maps from the subriemannian nth Heisenberg group H_n to R^{2n} by introducing an integral that defines the symplectic area of 1/2-Holder curves in R^{2n} and of projections of vertical curves in H_n, along with a geometric condition for convergence of the integral that is shown to hold almost everywhere via beta-number estimates from the Fassler-Orponen Dorronsoro Theorem.

What carries the argument

The integral defining the symplectic area of 1/2-Holder curves whose convergence is guaranteed by a geometric condition.

If this is right

The coarea formula holds even when n=1.
New results follow on the existence of the signed area of 1/2-Holder planar curves.
The formula applies to the measure of unrectifiable curves in the Heisenberg group.

Where Pith is reading between the lines

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

This method may extend to coarea formulas in other subriemannian spaces or for maps to different dimensions.
It could provide tools for analyzing rectifiability and measures in Carnot groups more broadly.
Similar integrals might be useful for Holder curves in other geometric settings.

Load-bearing premise

The geometric condition for convergence of the area integral holds for almost every fiber of the map.

What would settle it

Finding a Lipschitz map from H_n to R^{2n} where the new integral diverges for a positive measure set of fibers would show the coarea formula does not hold in general.

read the original abstract

We prove the coarea formula for Lipschitz maps from the subriemannian $n$th Heisenberg group $\mathbb H_n$ to $\mathbb R^{2n}$. Our result is new even when $n=1$ and provides the simplest vector-valued instance of the coarea formula in subriemannian geometry. This answers a question left open in the works of Magnani, Kozhevnikov, Magnani--Stepanov--Trevisan, and Julia--Nicolussi Golo--Vittone. The main difficulty of the proof is that a fiber of a $C^1_{\mathrm{H}}$ map $f: \mathbb H_n\to \mathbb R^{2n}$ is typically an unrectifiable curve. Its measure depends on the symplectic area of its projection to $\mathbb R^{2n}$. A bound on this area would imply the coarea formula, but examples of Kozhevnikov show that this area can be infinite or undefined. To overcome this, we introduce an integral that we use to define both the symplectic area of $\frac{1}{2}$--H\"older curves in $\mathbb R^{2n}$ and the symplectic area of projections of vertical curves in $\mathbb H_n$. Then, we give a geometric condition for this integral to converge. This yields, in addition, new results on the existence of the signed area of $\tfrac12$--H\"older planar curves that may be of independent interest. Finally, we use $\beta$--number estimates from the F\"assler--Orponen Dorronsoro Theorem to show that this geometric condition holds for almost every fiber.

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

Antonelli and Young close the open coarea question for maps from H_n to R^{2n} by defining a workable integral for symplectic area on 1/2-Holder curves and checking convergence a.e. via beta estimates.

read the letter

The main takeaway is that this paper proves the coarea formula for Lipschitz maps from the nth Heisenberg group to R^{2n}. It works even for n=1 and fills a gap left open in several earlier papers on subriemannian coarea formulas and area in Carnot groups. The fibers are usually unrectifiable vertical curves, so the usual area can be infinite or undefined, as Kozhevnikov's examples show. The authors get around this by introducing a new integral that defines the symplectic area for 1/2-Holder curves in R^{2n} and for the projections of those vertical curves. They also give a geometric condition for when the integral converges, and they prove that condition holds for almost every fiber using beta-number estimates from the Fässler-Orponen Dorronsoro theorem. This gives a clean coarea formula without reducing everything to a fitted quantity defined inside the paper. The new integral looks like it could be useful on its own for questions about signed area of Holder curves. The argument is direct and leans on standard background plus one external theorem, so there is no obvious circularity. The soft spot is the step that applies the beta estimates to these particular fibers. The projections are only 1/2-Holder, not Lipschitz, and the theorem is typically stated for Euclidean Sobolev or Lipschitz maps. The paper needs to confirm that the almost-everywhere statement survives the change in regularity and the subriemannian geometry, especially since the target measure is 2n-dimensional. If that verification is careful and explicit, the central claim holds up. If it is only sketched, a referee might ask for more detail on the quantifiers. This paper is aimed at people working on geometric measure theory in subriemannian spaces or on coarea formulas in Carnot groups. Readers who follow the literature on Magnani, Kozhevnikov, and related work will find it directly relevant. It deserves a serious referee because it supplies a missing result with a concrete new definition and a proof that can be checked line by line. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript proves the coarea formula for Lipschitz maps from the subriemannian nth Heisenberg group H_n to R^{2n}. It introduces a new integral to define the symplectic area of 1/2-Hölder curves in R^{2n} (and equivalently for projections of vertical curves in H_n), states a geometric condition ensuring convergence of this integral, derives new results on signed area for such curves, and invokes β-number estimates from the Fässler-Orponen Dorronsoro theorem to establish that the condition holds for almost every fiber.

Significance. If correct, the result resolves an open question from Magnani, Kozhevnikov, and related works, supplying the simplest vector-valued coarea formula in subriemannian geometry. The custom integral and accompanying existence results for signed area of 1/2-Hölder curves constitute a technical contribution that may be of independent interest. The strategy of combining a tailored integral with external β-number estimates is a standard and appropriate response to the unrectifiability phenomena illustrated by Kozhevnikov’s examples.

major comments (2)

[Abstract and proof of main theorem] Abstract (final paragraph) and the section applying the Fässler-Orponen Dorronsoro theorem: the argument that β-number estimates guarantee the geometric convergence condition for a.e. fibers relies on transferring L^p-integrability of β-numbers to the projections of vertical curves, which are merely 1/2-Hölder (not Lipschitz). The standard statements of the theorem apply to Sobolev or Lipschitz maps in Euclidean space; an explicit verification is needed that the almost-everywhere quantifier survives with respect to the 2n-dimensional measure on the target (or the appropriate measure on the domain) and that Kozhevnikov-type examples do not produce a null set of bad fibers.
[Definition of the integral and geometric condition] Section introducing the new integral (presumably §3): the geometric condition for convergence of the integral is load-bearing for both the area definition and the subsequent coarea formula. It should be stated with a precise formulation (e.g., an explicit integral test or decay condition on the curve) and shown to be equivalent, for vertical curves in H_n, to the symplectic area appearing in the coarea statement.

minor comments (2)

[Notation] Notation for the Heisenberg group should be uniform (H_n versus mathbb H_n) across the abstract, introduction, and main statements.
[Introduction / references] The abstract refers to “examples of Kozhevnikov” showing infinite or undefined area; a precise citation to the relevant theorem or example in Kozhevnikov’s work should be added for the reader’s convenience.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the thorough review of our manuscript. We appreciate the referee's recognition of the significance of our results and the constructive suggestions for improvement. Below, we provide point-by-point responses to the major comments.

read point-by-point responses

Referee: Abstract (final paragraph) and the section applying the Fässler-Orponen Dorronsoro theorem: the argument that β-number estimates guarantee the geometric convergence condition for a.e. fibers relies on transferring L^p-integrability of β-numbers to the projections of vertical curves, which are merely 1/2-Hölder (not Lipschitz). The standard statements of the theorem apply to Sobolev or Lipschitz maps in Euclidean space; an explicit verification is needed that the almost-everywhere quantifier survives with respect to the 2n-dimensional measure on the target (or the appropriate measure on the domain) and that Kozhevnikov-type examples do not produce a null set of bad fibers.

Authors: We acknowledge that the transfer of the β-number estimates to 1/2-Hölder curves requires careful justification, as the standard Fässler-Orponen Dorronsoro theorem is stated for Lipschitz or Sobolev maps. In our proof, the projections are indeed 1/2-Hölder due to the subriemannian structure. To strengthen the argument, we will revise the section to include an explicit verification. This will involve showing that the L^p integrability holds for the Hölder maps by approximating with Lipschitz functions or using the specific form of the curves as projections of vertical lines in H_n. We will also specify that the almost everywhere is with respect to the 2n-dimensional Hausdorff measure on the image, and demonstrate that any potential bad set arising from Kozhevnikov-type examples has measure zero because the Lipschitz map f ensures that such pathologies are avoided almost everywhere. This revision will clarify the survival of the a.e. quantifier. revision: yes
Referee: Section introducing the new integral (presumably §3): the geometric condition for convergence of the integral is load-bearing for both the area definition and the subsequent coarea formula. It should be stated with a precise formulation (e.g., an explicit integral test or decay condition on the curve) and shown to be equivalent, for vertical curves in H_n, to the symplectic area appearing in the coarea statement.

Authors: We agree with the referee that the geometric condition needs a more precise formulation to make the manuscript self-contained. In the revised version, we will state the condition explicitly in §3 as a decay estimate on the β-numbers of the curve, specifically requiring that the integral of β(x,r)^p r^{-1} dr over scales is finite. Furthermore, we will add a proposition demonstrating the equivalence between this condition for vertical curves in H_n and the symplectic area appearing in the coarea formula. This will be shown by directly relating the new integral to the pullback of the standard symplectic form on R^{2n} under the projection map from the Heisenberg group. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation applies external Dorronsoro theorem to new integral without self-reduction

full rationale

The paper defines a new integral to assign symplectic area to 1/2-Hölder curves and vertical fibers, states a geometric convergence condition for that integral, and then cites the external Fässler-Orponen Dorronsoro theorem to verify the condition holds for a.e. fiber with respect to the appropriate measures. This step is an application of a prior result by different authors rather than a reduction of the target coarea formula to any quantity defined or fitted inside the present work. No equations equate the final statement to an input by construction, no self-citations are load-bearing, and the argument remains self-contained against external benchmarks. The derivation therefore receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claim rests on the new integral definition and the almost-everywhere verification step; background structure of the Heisenberg group is treated as given.

axioms (2)

domain assumption The nth Heisenberg group carries a subriemannian structure with horizontal distribution and vertical complement.
Invoked when discussing fibers of maps and vertical curves.
domain assumption Lipschitz and C^1_H maps from H_n to R^{2n} have fibers that are typically unrectifiable curves.
Stated as the main difficulty in the abstract.

invented entities (1)

Integral defining symplectic area of 1/2-Holder curves no independent evidence
purpose: To assign a finite area to projections of vertical curves when classical area is infinite or undefined.
Newly introduced to bypass the obstacle described in the abstract.

pith-pipeline@v0.9.0 · 5837 in / 1379 out tokens · 67246 ms · 2026-05-19T17:14:29.922400+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

Rectifiable sets in metric and Ba- nach spaces

L. Ambrosio and B. Kirchheim. “Rectifiable sets in metric and Ba- nach spaces”. In:Math. Ann.318.3 (2000), pp. 527–555.ISSN: 0025- 5831,1432-1807

work page 2000
[2]

Antonelli.Rectifiability in Carnot groups

G. Antonelli.Rectifiability in Carnot groups. PhD thesis available on cvGmt. 2022

work page 2022
[3]

Vertical curves and vertical fibers in the Heisenberg group

G. Antonelli and R. Young. “Vertical curves and vertical fibers in the Heisenberg group”. In: (2024). Accepted for publication in Trans. Amer. Math. Soc. arXiv preprint: https://arxiv.org/abs/2411. 00232

work page 2024
[4]

A T (b) theorem with remarks on analytic capacity and the Cauchy integral

M. Christ. “A T (b) theorem with remarks on analytic capacity and the Cauchy integral”. In:Colloq. Math.60/61.2 (1990), pp. 601–628. ISSN: 0010-1354,1730-6302

work page 1990
[5]

A reverse coarea-type inequality in Carnot groups

F . Corni. “A reverse coarea-type inequality in Carnot groups”. In: Ann. Inst. Fourier (Grenoble)72.1 (2022), pp. 155–185.ISSN: 0373- 0956,1777-5310

work page 2022
[6]

David.Wavelets and singular integrals on curves and surfaces

G. David.Wavelets and singular integrals on curves and surfaces. Vol. 1465. Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1991, pp. x+107.ISBN: 3-540-53902-6. 68 REFERENCES

work page 1991
[7]

Dorronsoro’ s theorem in Heisenberg groups

K. Fässler and T . Orponen. “Dorronsoro’ s theorem in Heisenberg groups”. In:Bull. Lond. Math. Soc.52.3 (2020), pp. 472–488.ISSN: 0024-6093,1469-2120

work page 2020
[8]

Federer.Geometric measure theory

H. Federer.Geometric measure theory. Vol. Band 153. Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York, Inc., New York, 1969, pp. xiv+676

work page 1969
[9]

Rectifiability and perimeter in the Heisenberg group

B. Franchi, R. Serapioni, and F . Serra Cassano. “Rectifiability and perimeter in the Heisenberg group”. In:Math. Ann.321.3 (2001), pp. 479–531.ISSN: 0025-5831,1432-1807

work page 2001
[10]

Intrinsic Lipschitz graphs in Heisenberg groups

B. Franchi, R. Serapioni, and F . Serra Cassano. “Intrinsic Lipschitz graphs in Heisenberg groups”. In:J. Nonlinear Convex Anal.7.3 (2006), pp. 423–441.ISSN: 1345-4773,1880-5221

work page 2006
[11]

Intrinsic Lipschitz graphs within Carnot groups

B. Franchi and R. P . Serapioni. “Intrinsic Lipschitz graphs within Carnot groups”. In:J. Geom. Anal.26.3 (2016), pp. 1946–1994.ISSN: 1050-6926,1559-002X

work page 2016
[12]

P . K. Friz and M. Hairer.A course on rough paths. Universitext. With an introduction to regularity structures. Springer, Cham, 2014

work page 2014
[13]

Area of intrinsic graphs and coarea formula in Carnot groups

A. Julia, S. Nicolussi Golo, and D. Vittone. “Area of intrinsic graphs and coarea formula in Carnot groups”. In:Math. Z.301.2 (2022), pp. 1369–1406.ISSN: 0025-5874,1432-1823

work page 2022
[14]

A coarea formula for smooth contact mappings of Carnot-Carathéodory spaces

M. Karmanova and S. Vodopyanov. “A coarea formula for smooth contact mappings of Carnot-Carathéodory spaces”. In:Acta Appl. Math.128 (2013), pp. 67–111.ISSN: 0167-8019,1572-9036

work page 2013
[15]

Propriétés métriques des ensembles de niveau des applications différentiables sur les groupes de Carnot

A. Kozhevnikov. “Propriétés métriques des ensembles de niveau des applications différentiables sur les groupes de Carnot”. In:Géométrie métrique [math.MG]. Université Paris Sud - Paris XI, (2015)(2015)

work page 2015
[16]

Magnani.Elements of geometric measure theory on subRieman- nian groups

V . Magnani.Elements of geometric measure theory on subRieman- nian groups. Scuola Normale Superiore, Pisa, 2002, pp. viii+195.ISBN: 88-7642-152-1

work page 2002
[17]

On a general coarea inequality and applications

V . Magnani. “On a general coarea inequality and applications”. In: Ann. Acad. Sci. Fenn. Math.27.1 (2002), pp. 121–140.ISSN: 1239- 629X,1798-2383

work page 2002
[18]

The coarea formula for real-valued Lipschitz maps on stratified groups

V . Magnani. “The coarea formula for real-valued Lipschitz maps on stratified groups”. In:Math. Nachr .278.14 (2005), pp. 1689–1705. ISSN: 0025-584X,1522-2616

work page 2005
[19]

Non-horizontal submanifolds and coarea formula

V . Magnani. “Non-horizontal submanifolds and coarea formula”. In: J. Anal. Math.106 (2008), pp. 95–127.ISSN: 0021-7670,1565-8538

work page 2008
[20]

Area implies coarea

V . Magnani. “Area implies coarea”. In:Indiana Univ. Math. J.60.1 (2011), pp. 77–100.ISSN: 0022-2518,1943-5258

work page 2011
[21]

A rough calculus ap- proach to level sets in the Heisenberg group

V . Magnani, E. Stepanov, and D. Trevisan. “A rough calculus ap- proach to level sets in the Heisenberg group”. In:J. Lond. Math. Soc. (2)97.3 (2018), pp. 495–522.ISSN: 0024-6107,1469-7750. REFERENCES 69

work page 2018
[22]

Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un

P . Pansu. “Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un”. In:Ann. of Math. (2)129.1 (1989), pp. 1–60.ISSN: 0003-486X

work page 1989
[23]

An inequality of the Hölder type, connected with Stielt- jes integration

L. C. Young. “An inequality of the Hölder type, connected with Stielt- jes integration”. In:Acta Math.67.1 (1936), pp. 251–282.ISSN: 0001- 5962,1871-2509

work page 1936

[1] [1]

Rectifiable sets in metric and Ba- nach spaces

L. Ambrosio and B. Kirchheim. “Rectifiable sets in metric and Ba- nach spaces”. In:Math. Ann.318.3 (2000), pp. 527–555.ISSN: 0025- 5831,1432-1807

work page 2000

[2] [2]

Antonelli.Rectifiability in Carnot groups

G. Antonelli.Rectifiability in Carnot groups. PhD thesis available on cvGmt. 2022

work page 2022

[3] [3]

Vertical curves and vertical fibers in the Heisenberg group

G. Antonelli and R. Young. “Vertical curves and vertical fibers in the Heisenberg group”. In: (2024). Accepted for publication in Trans. Amer. Math. Soc. arXiv preprint: https://arxiv.org/abs/2411. 00232

work page 2024

[4] [4]

A T (b) theorem with remarks on analytic capacity and the Cauchy integral

M. Christ. “A T (b) theorem with remarks on analytic capacity and the Cauchy integral”. In:Colloq. Math.60/61.2 (1990), pp. 601–628. ISSN: 0010-1354,1730-6302

work page 1990

[5] [5]

A reverse coarea-type inequality in Carnot groups

F . Corni. “A reverse coarea-type inequality in Carnot groups”. In: Ann. Inst. Fourier (Grenoble)72.1 (2022), pp. 155–185.ISSN: 0373- 0956,1777-5310

work page 2022

[6] [6]

David.Wavelets and singular integrals on curves and surfaces

G. David.Wavelets and singular integrals on curves and surfaces. Vol. 1465. Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1991, pp. x+107.ISBN: 3-540-53902-6. 68 REFERENCES

work page 1991

[7] [7]

Dorronsoro’ s theorem in Heisenberg groups

K. Fässler and T . Orponen. “Dorronsoro’ s theorem in Heisenberg groups”. In:Bull. Lond. Math. Soc.52.3 (2020), pp. 472–488.ISSN: 0024-6093,1469-2120

work page 2020

[8] [8]

Federer.Geometric measure theory

H. Federer.Geometric measure theory. Vol. Band 153. Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York, Inc., New York, 1969, pp. xiv+676

work page 1969

[9] [9]

Rectifiability and perimeter in the Heisenberg group

B. Franchi, R. Serapioni, and F . Serra Cassano. “Rectifiability and perimeter in the Heisenberg group”. In:Math. Ann.321.3 (2001), pp. 479–531.ISSN: 0025-5831,1432-1807

work page 2001

[10] [10]

Intrinsic Lipschitz graphs in Heisenberg groups

B. Franchi, R. Serapioni, and F . Serra Cassano. “Intrinsic Lipschitz graphs in Heisenberg groups”. In:J. Nonlinear Convex Anal.7.3 (2006), pp. 423–441.ISSN: 1345-4773,1880-5221

work page 2006

[11] [11]

Intrinsic Lipschitz graphs within Carnot groups

B. Franchi and R. P . Serapioni. “Intrinsic Lipschitz graphs within Carnot groups”. In:J. Geom. Anal.26.3 (2016), pp. 1946–1994.ISSN: 1050-6926,1559-002X

work page 2016

[12] [12]

P . K. Friz and M. Hairer.A course on rough paths. Universitext. With an introduction to regularity structures. Springer, Cham, 2014

work page 2014

[13] [13]

Area of intrinsic graphs and coarea formula in Carnot groups

A. Julia, S. Nicolussi Golo, and D. Vittone. “Area of intrinsic graphs and coarea formula in Carnot groups”. In:Math. Z.301.2 (2022), pp. 1369–1406.ISSN: 0025-5874,1432-1823

work page 2022

[14] [14]

A coarea formula for smooth contact mappings of Carnot-Carathéodory spaces

M. Karmanova and S. Vodopyanov. “A coarea formula for smooth contact mappings of Carnot-Carathéodory spaces”. In:Acta Appl. Math.128 (2013), pp. 67–111.ISSN: 0167-8019,1572-9036

work page 2013

[15] [15]

Propriétés métriques des ensembles de niveau des applications différentiables sur les groupes de Carnot

A. Kozhevnikov. “Propriétés métriques des ensembles de niveau des applications différentiables sur les groupes de Carnot”. In:Géométrie métrique [math.MG]. Université Paris Sud - Paris XI, (2015)(2015)

work page 2015

[16] [16]

Magnani.Elements of geometric measure theory on subRieman- nian groups

V . Magnani.Elements of geometric measure theory on subRieman- nian groups. Scuola Normale Superiore, Pisa, 2002, pp. viii+195.ISBN: 88-7642-152-1

work page 2002

[17] [17]

On a general coarea inequality and applications

V . Magnani. “On a general coarea inequality and applications”. In: Ann. Acad. Sci. Fenn. Math.27.1 (2002), pp. 121–140.ISSN: 1239- 629X,1798-2383

work page 2002

[18] [18]

The coarea formula for real-valued Lipschitz maps on stratified groups

V . Magnani. “The coarea formula for real-valued Lipschitz maps on stratified groups”. In:Math. Nachr .278.14 (2005), pp. 1689–1705. ISSN: 0025-584X,1522-2616

work page 2005

[19] [19]

Non-horizontal submanifolds and coarea formula

V . Magnani. “Non-horizontal submanifolds and coarea formula”. In: J. Anal. Math.106 (2008), pp. 95–127.ISSN: 0021-7670,1565-8538

work page 2008

[20] [20]

Area implies coarea

V . Magnani. “Area implies coarea”. In:Indiana Univ. Math. J.60.1 (2011), pp. 77–100.ISSN: 0022-2518,1943-5258

work page 2011

[21] [21]

A rough calculus ap- proach to level sets in the Heisenberg group

V . Magnani, E. Stepanov, and D. Trevisan. “A rough calculus ap- proach to level sets in the Heisenberg group”. In:J. Lond. Math. Soc. (2)97.3 (2018), pp. 495–522.ISSN: 0024-6107,1469-7750. REFERENCES 69

work page 2018

[22] [22]

Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un

P . Pansu. “Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un”. In:Ann. of Math. (2)129.1 (1989), pp. 1–60.ISSN: 0003-486X

work page 1989

[23] [23]

An inequality of the Hölder type, connected with Stielt- jes integration

L. C. Young. “An inequality of the Hölder type, connected with Stielt- jes integration”. In:Acta Math.67.1 (1936), pp. 251–282.ISSN: 0001- 5962,1871-2509

work page 1936