$\mathbb{E}_{\iota}[\operatorname{TV}(P_{X^{n}_{2},\omega},P_{X^{n}_{2}}P_{\omega})]\leq\epsilon$.

We then recover the achievability of $R_{\text{bro}}(X_{1:m})$ using a
modification of the coding scheme in Section VIII.

###### Proof.

From [24], we know that there exists a solution of (116) such that
$I(X_{1:m};Z)=C(X_{1:m}|Z)$. Let $P_{X_{1:m},Z}$ be the solution. Let
$z^{n}\in\mathcal{T}^{(n)}_{\epsilon}(Z)$ be available for each user, we know
that a transition kernel $P_{X^{n}_{1:m}|z^{n}}$ such that
$\operatorname{TV}(P_{X^{n}_{1:m}|z^{n}},P_{X^{\otimes
n}_{1:m}|z^{n}})\leq\delta$ can be reproduced by transmitting classical
information $\omega_{1,j}$ with the rate $I(X_{1:j-1};X_{j}|Z)+\delta$,
according to Proposition 6 and Definition 4. We first generate the codebook
which is written as a joint probability on three random variables
$(\omega^{\prime}_{2:m},\omega^{\prime\prime}_{2:m},Z^{\otimes n})$. We can
obtain bit strings $\omega^{\prime}_{j}:=\iota_{j}(\omega_{1,j})$ and
$\omega^{\prime\prime}_{j}:=\iota^{\prime}_{j}(\omega_{1,j})$ by random
binning $\omega_{1,j}$ with the rate $I(X_{1:j-1};X_{j}|Z)-\delta$ and
$2\delta$. From Slepian-Wolf theorem, we know that
$(\omega^{\prime}_{j},\omega^{\prime\prime}_{j})$ recovers $\omega_{1,j}$
almost surely, thus further recovering $X^{n}_{1:m}$ for a given $z^{n}$.
Because $\omega_{1,j}$ is randomly binned from $X^{n}_{j}$, every single
$x^{n}_{j}\in\mathcal{T}^{(n)}_{\epsilon}(X_{j})$ is randomly assigned to
$\omega^{\prime}_{j}$ and $\omega^{\prime\prime}_{j}$, then
$\omega^{\prime}_{j}$ and $\omega^{\prime\prime}_{j}$ can be considered as
random bins of $X^{n}_{j}$. Following Lemma 8 and the fact that
$H(X_{j}|Z)\geq I(X_{1:j-1};X_{j}|Z)$, we know that $\omega^{\prime}_{2:m}$
and $Z^{\otimes n}$ can be considered independent with an infinitesimal change
of total variation. Then, we generate the message. We generate
$\omega^{\prime}_{2:m}$ and $i:=(\omega^{\prime}_{2}\oplus
u^{\prime}_{2},\omega^{\prime}_{3}\oplus
u^{\prime}_{3},\cdots,\omega^{\prime}_{m}\oplus u^{\prime}_{m},u^{\prime})$,
where $u^{\prime}_{j}$ is the independent and uniformly distributed bit string
shared by $\alpha_{1}$ and user $j$, and $u^{\prime}$ is another bit string
with the rate of $m\delta$. From the property of XOR, we know that $i$ is
independent with $\omega^{\prime}_{2:m}$ if $u^{\prime}_{2:m}$ are not used
further. The broadcast message is written as
$\omega_{\alpha_{1}}:=(i,\omega^{\prime\prime}_{2:m})$. Because
$u^{\prime}_{j}$ is the shared randomness, $\omega^{\prime}_{j}$ can be
recovered at user $j$. Further map $i$ to randomly sampled
$z^{n}[i]\in\mathcal{T}^{(n)}_{\epsilon}(Z)$ and generate $X^{n}_{1:m}$
following $\sum_{i}P_{X^{n}_{1:m}|z^{n}[i]}P_{z^{n}[i]}$. Using the soft
covering [33, Lemma IV.1] and [26, Lemma 3], we know the recovered
$X^{n}_{1:m}$ satisfying $\operatorname{TV}(P^{\otimes
n}_{X_{1:m}},P_{X^{n}_{1:m}})\leq\epsilon$.

$P_{\omega_{1,2:m}|Z^{n}}$$\omega_{1,2:m}$$\mathcal{B}$$\omega^{\prime\prime}_{2:m}$$\mathcal{B}$$\omega^{\prime}_{2:m}$$Z^{n}$

Figure 11: Construction of codebook.

$\oplus$$u^{\prime}_{2:m}$$P_{\omega^{\prime\prime}_{2:m}|\omega^{\prime}_{2:m},z^{n}[i]}$$\omega^{\prime}_{2,m}$$i$$\omega^{\prime\prime}_{2:m}$

Figure 12: Generation of message
$\omega_{\alpha_{1}}:=(i,\omega^{\prime\prime}_{2:m})$.

∎

## References

  * [1] E. Chitambar and G. Gour, “Quantum resource theories,” _Reviews of modern physics_ , vol. 91, no. 2, p. 025001, 2019.
  * [2] A. Acín, J. I. Cirac, and L. Masanes, “Multipartite bound information exists and can be activated,” _Physical review letters_ , vol. 92, no. 10, p. 107903, 2004.
  * [3] O. Leskovjanová and L. Mišta Jr, “Classical analog of the quantum marginal problem,” _Physical Review A_ , vol. 101, no. 3, p. 032341, 2020.
  * [4] V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” _Reviews of modern physics_ , vol. 81, no. 3, p. 1301, 2009.
  * [5] V. Scarani, N. Gisin, N. Brunner, L. Masanes, S. Pino, and A. Acín, “Secrecy extraction from no-signaling correlations,” _Physical Review A_ , vol. 74, no. 4, p. 042339, 2006.
  * [6] M. Christandl, A. Ekert, M. Horodecki, P. Horodecki, J. Oppenheim, and R. Renner, “Unifying classical and quantum key distillation,” in _Theory of Cryptography: 4th Theory of Cryptography Conference, TCC 2007, Amsterdam, The Netherlands, February 21-24, 2007. Proceedings 4_. Springer, 2007, pp. 456–478.
  * [7] R. Augusiak and P. Horodecki, “Multipartite secret key distillation and bound entanglement,” _Physical Review A_ , vol. 80, no. 4, p. 042307, 2009.
  * [8] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” _Reviews of modern physics_ , vol. 81, no. 2, p. 865, 2009.
  * [9] H. Buhrman, R. Cleve, S. Massar, and R. De Wolf, “Nonlocality and communication complexity,” _Reviews of modern physics_ , vol. 82, no. 1, p. 665, 2010.
  * [10] J. Barrett, N. Linden, S. Massar, S. Pironio, S. Popescu, and D. Roberts, “Nonlocal correlations as an information-theoretic resource,” _Physical review A_ , vol. 71, no. 2, p. 022101, 2005.
  * [11] L. Masanes, A. Acín, and N. Gisin, “General properties of nonsignaling theories,” _Physical Review A_ , vol. 73, no. 1, p. 012112, 2006.
  * [12] A. Acín and N. Gisin, “Quantum correlations and secret bits,” _Physical review letters_ , vol. 94, no. 2, p. 020501, 2005.
  * [13] A. K. Ekert, “Quantum cryptography based on bell’s theorem,” _Phys. Rev. Lett._ , vol. 67, pp. 661–663, Aug 1991.
  * [14] M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” _Physical Review A_ , vol. 59, no. 3, p. 1829, 1999.
  * [15] M. Pivoluska, M. Huber, and M. Malik, “Layered quantum key distribution,” _Physical Review A_ , vol. 97, no. 3, p. 032312, 2018.
  * [16] F. Grasselli, H. Kampermann, and D. Bruß, “Finite-key effects in multipartite quantum key distribution protocols,” _New Journal of Physics_ , vol. 20, no. 11, p. 113014, 2018.
  * [17] M. Proietti, J. Ho, F. Grasselli, P. Barrow, M. Malik, and A. Fedrizzi, “Experimental quantum conference key agreement,” _Science Advances_ , vol. 7, no. 23, p. eabe0395, 2021.
  * [18] M. Epping, H. Kampermann, D. Bruß _et al._ , “Multi-partite entanglement can speed up quantum key distribution in networks,” _New Journal of Physics_ , vol. 19, no. 9, p. 093012, 2017.
  * [19] S. Wengerowsky, S. K. Joshi, F. Steinlechner, H. Hübel, and R. Ursin, “An entanglement-based wavelength-multiplexed quantum communication network,” _Nature_ , vol. 564, no. 7735, pp. 225–228, 2018.
  * [20] S. K. Joshi, D. Aktas, S. Wengerowsky, M. Lončarić, S. P. Neumann, B. Liu, T. Scheidl, G. C. Lorenzo, Ž. Samec, L. Kling _et al._ , “A trusted node–free eight-user metropolitan quantum communication network,” _Science advances_ , vol. 6, no. 36, p. eaba0959, 2020.
  * [21] C. E. Shannon, “A mathematical theory of communication,” _The Bell system technical journal_ , vol. 27, no. 3, pp. 379–423, 1948.
  * [22] N. J. Cerf, S. Massar, and S. Schneider, “Multipartite classical and quantum secrecy monotones,” _Physical Review A_ , vol. 66, no. 4, p. 042309, 2002.
  * [23] S. Watanabe, “Information theoretical analysis of multivariate correlation,” _IBM Journal of research and development_ , vol. 4, no. 1, pp. 66–82, 1960.
  * [24] G. R. Kurri, V. M. Prabhakaran, and A. D. Sarwate, “Coordination through shared randomness,” _IEEE Transactions on Information Theory_ , vol. 67, no. 8, pp. 4948–4974, 2021.
  * [25] A. El Gamal and Y.-H. Kim, _Network information theory_. Cambridge University Press, 2011.
  * [26] M. H. Yassaee, M. R. Aref, and A. Gohari, “Achievability proof via output statistics of random binning,” _IEEE Transactions on Information Theory_ , vol. 60, no. 11, pp. 6760–6786, 2014.
  * [27] M. Winczewski, T. Das, and K. Horodecki, “Limitations on a device-independent key secure against a nonsignaling adversary via squashed nonlocality,” _Physical Review A_ , vol. 106, no. 5, p. 052612, 2022.
  * [28] K. Horodecki, M. Winczewski, and S. Das, “Fundamental limitations on the device-independent quantum conference key agreement,” _Physical Review A_ , vol. 105, no. 2, p. 022604, 2022.
  * [29] A. Philip, E. Kaur, P. Bierhorst, and M. M. Wilde, “Multipartite intrinsic non-locality and device-independent conference key agreement,” _Quantum_ , vol. 7, p. 898, 2023.
  * [30] J. Pearl, _Causality_. Cambridge University Press, 2009.
  * [31] T. S. Han and S. Verdú, “Approximation theory of output statistics,” _IEEE Transactions on Information Theory_ , vol. 39, no. 3, pp. 752–772, 1993.
  * [32] P. W. Cuff, H. H. Permuter, and T. M. Cover, “Coordination capacity,” _IEEE Transactions on Information Theory_ , vol. 56, no. 9, pp. 4181–4206, 2010.
  * [33] P. Cuff, “Distributed channel synthesis,” _IEEE Transactions on Information Theory_ , vol. 59, no. 11, pp. 7071–7096, 2013.
  * [34] A. Winter, ““extrinsic”and “intrinsic”data in quantum measurements: Asymptotic convex decomposition of positive operator valued measures,” _Communications in mathematical physics_ , vol. 244, pp. 157–185, 2004.
  * [35] C. H. Bennett, I. Devetak, A. W. Harrow, P. W. Shor, and A. Winter, “The quantum reverse shannon theorem and resource tradeoffs for simulating quantum channels,” _IEEE Transactions on Information Theory_ , vol. 60, no. 5, pp. 2926–2959, 2014.
  * [36] M. M. Wilde, P. Hayden, F. Buscemi, and M.-H. Hsieh, “The information-theoretic costs of simulating quantum measurements,” _Journal of Physics A: Mathematical and Theoretical_ , vol. 45, no. 45, p. 453001, 2012.
  * [37] T. A. Atif, M. Heidari, and S. S. Pradhan, “Faithful simulation of distributed quantum measurements with applications in distributed rate-distortion theory,” _IEEE Transactions on Information Theory_ , vol. 68, no. 2, pp. 1085–1118, 2021.
  * [38] O. Oreshkov, F. Costa, and Č. Brukner, “Quantum correlations with no causal order,” _Nature communications_ , vol. 3, no. 1, p. 1092, 2012.
  * [39] K. Goswami, C. Giarmatzi, M. Kewming, F. Costa, C. Branciard, J. Romero, and A. G. White, “Indefinite causal order in a quantum switch,” _Physical review letters_ , vol. 121, no. 9, p. 090503, 2018.
  * [40] E.-E. Tselentis and A. Baumeler, “Admissible causal structures and correlations,” _PRX Quantum_ , vol. 4, p. 040307, Oct 2023.
  * [41] O. Oreshkov and C. Giarmatzi, “Causal and causally separable processes,” _New Journal of Physics_ , vol. 18, no. 9, p. 093020, 2016.
  * [42] A. A. Abbott, C. Giarmatzi, F. Costa, and C. Branciard, “Multipartite causal correlations: polytopes and inequalities,” _Physical Review A_ , vol. 94, no. 3, p. 032131, 2016.
  * [43] G. Rubino, L. A. Rozema, A. Feix, M. Araújo, J. M. Zeuner, L. M. Procopio, Č. Brukner, and P. Walther, “Experimental verification of an indefinite causal order,” _Science advances_ , vol. 3, no. 3, p. e1602589, 2017.
  * [44] C. Hitchcock and M. Rédei, “Reichenbach’s Common Cause Principle,” in _The Stanford Encyclopedia of Philosophy_ , Summer 2021 ed., E. N. Zalta, Ed. Metaphysics Research Lab, Stanford University, 2021.
  * [45] G. Murta, F. Grasselli, H. Kampermann, and D. Bruß, “Quantum conference key agreement: A review,” _Advanced Quantum Technologies_ , vol. 3, no. 11, p. 2000025, 2020.
  * [46] L. Salvail, M. Peev, E. Diamanti, R. Alléaume, N. Lütkenhaus, and T. Länger, “Security of trusted repeater quantum key distribution networks,” _Journal of Computer Security_ , vol. 18, no. 1, pp. 61–87, 2010.
  * [47] D. M. Greenberger, M. A. Horne, and A. Zeilinger, “Going beyond bell’s theorem,” _Bell’s theorem, quantum theory and conceptions of the universe_ , pp. 69–72, 1989.
  * [48] R. W. Hamming, “Error detecting and error correcting codes,” _The Bell system technical journal_ , vol. 29, no. 2, pp. 147–160, 1950.
  * [49] B. Groisman, S. Popescu, and A. Winter, “Quantum, classical, and total amount of correlations in a quantum state,” _Physical Review A_ , vol. 72, no. 3, p. 032317, 2005.
  * [50] M. Fannes, “A continuity property of the entropy density for spin lattice systems,” _Communications in Mathematical Physics_ , vol. 31, pp. 291–294, 1973.
  * [51] M. A. Nielsen and I. L. Chuang, “Quantum computation and quantum information,” _Phys. Today_ , vol. 54, no. 2, p. 60, 2001.
  * [52] D. Slepian and J. Wolf, “Noiseless coding of correlated information sources,” _IEEE Transactions on information Theory_ , vol. 19, no. 4, pp. 471–480, 1973.
  * [53] I. Csiszár and J. Körner, _Information theory: coding theorems for discrete memoryless systems_. Cambridge University Press, 2011.
  * [54] D. Yang, K. Horodecki, M. Horodecki, P. Horodecki, J. Oppenheim, and W. Song, “Squashed entanglement for multipartite states and entanglement measures based on the mixed convex roof,” _IEEE Transactions on Information Theory_ , vol. 55, no. 7, pp. 3375–3387, 2009.
  * [55] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” _Physical Review A_ , vol. 53, no. 4, p. 2046, 1996.
  * [56] B. Pass, “On the local structure of optimal measures in the multi-marginal optimal transportation problem,” _Calculus of Variations and Partial Differential Equations_ , vol. 43, no. 3, pp. 529–536, 2012.
  * [57] P. Dupuis and R. S. Ellis, _A weak convergence approach to the theory of large deviations_ , ser. Wiley series in probability and statistics. Hoboken: Wiley-Interscience, 2011.
  * [58] J.-D. Benamou, G. Carlier, M. Cuturi, L. Nenna, and G. Peyré, “Iterative bregman projections for regularized transportation problems,” _SIAM Journal on Scientific Computing_ , vol. 37, no. 2, pp. A1111–A1138, 2015.
  * [59] L. M. Bregman, “The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming,” _USSR computational mathematics and mathematical physics_ , vol. 7, no. 3, pp. 200–217, 1967.
  * [60] M. J. Wainwright, M. I. Jordan _et al._ , “Graphical models, exponential families, and variational inference,” _Foundations and Trends® in Machine Learning_ , vol. 1, no. 1–2, pp. 1–305, 2008.
  * [61] S. Boyd, S. P. Boyd, and L. Vandenberghe, _Convex optimization_. Cambridge university press, 2004.
  * [62] I. Haasler, R. Singh, Q. Zhang, J. Karlsson, and Y. Chen, “Multi-marginal optimal transport and probabilistic graphical models,” _IEEE Transactions on Information Theory_ , vol. 67, no. 7, pp. 4647–4668, 2021.
  * [63] G. Wunder, A. Flinth, D. Becker, and B. Groß, “Mimicking dh key exchange over a full duplex wireless channel via bisparse blind deconvolution,” in _2023 6th International Conference on Advanced Communication Technologies and Networking (CommNet)_ , 2023, pp. 1–8.
  * [64] J. Csima, “Multidimensional stochastic matrices and patterns,” _Journal of algebra_ , vol. 14, no. 2, pp. 194–202, 1970.
  * [65] G. Birkhoff, “Tres observaciones sobre el algebra lineal,” _Univ. Nac. Tucuman, Ser. A_ , vol. 5, pp. 147–154, 1946.
  * [66] J. Von Neumann, “A certain zero-sum two-person game equivalent to the optimal assignment problem,” _Contributions to the Theory of Games_ , vol. 2, no. 0, pp. 5–12, 1953.
  * [67] M. D. Plummer and L. Lovász, _Matching theory_. Elsevier, 1986.
  * [68] G. Friesecke, “A simple counterexample to the monge ansatz in multimarginal optimal transport, convex geometry of the set of kantorovich plans, and the frenkel–kontorova model,” _SIAM Journal on Mathematical Analysis_ , vol. 51, no. 6, pp. 4332–4355, 2019.
  * [69] G. Monge, “Mémoire sur la théorie des déblais et des remblais,” _Histoire de l’Académie Royale des Sciences de Paris_ , 1781.
  * [70] L. V. Kantorovich, “On the translocation of masses,” in _Dokl. Akad. Nauk. USSR (NS)_ , vol. 37, 1942, pp. 199–201.
  * [71] D. Fulkerson, _Hitchcock transportation problem_. Rand Corporation, 1956.
  * [72] Y. Bai, B. Schmitzer, M. Thorpe, and S. Kolouri, “Sliced optimal partial transport,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2023, pp. 13 681–13 690.
  * [73] P. Jouguet, S. Kunz-Jacques, A. Leverrier, P. Grangier, and E. Diamanti, “Experimental demonstration of long-distance continuous-variable quantum key distribution,” _Nature photonics_ , vol. 7, no. 5, pp. 378–381, 2013.
  * [74] T. M. Cover, _Elements of information theory_. John Wiley & Sons, 1999.
  * [75] T. HAN, “Information-spectrum methods in information theory,” _Applications of Mathematics_ , 2003.
  * [76] A. Gamal and T. Cover, “Achievable rates for multiple descriptions,” _IEEE Transactions on Information Theory_ , vol. 28, no. 6, pp. 851–857, 1982.
  * [77] H. Yamamoto, “Source coding theory for cascade and branching communication systems,” _IEEE Transactions on Information Theory_ , vol. 27, no. 3, pp. 299–308, 1981.
  * [78] ——, “Source coding theory for a triangular communication system,” _IEEE Transactions on Information Theory_ , vol. 42, no. 3, pp. 848–853, 1996.
  * [79] C. Léonard, “A survey of the schrödinger problem and some of its connections with optimal transport,” _Discrete and Continuous Dynamical Systems_ , vol. 34, no. 4, p. 1533, 2014.


# On the equivalence of derivatives for maps between Carnot groups

Scott Zimmerman  Department of Mathematics, The Ohio State University, 100
Math Tower (MW) 231 W 18th Ave, Columbus, OH<EMAIL_ADDRESS>
###### Abstract.

This paper gives an alternate, elementary proof of a result of Magnani: maps
between Carnot groups that preserve horizontal curves and are continuously
differential in horizontal directions in the Euclidean sense are continuously
Pansu differentiable. This proof contains primarily Euclidean arguments and
also reproves a version of Magnani’s mean value estimate for continuously
Pansu differentiable maps.

##  1\. Introduction

Suppose $\mathbb{G}$ and $\hat{\mathbb{G}}$ are Carnot groups. Suppose also
that $\Omega\subset\mathbb{G}$ is open and $f:\Omega\to\hat{\mathbb{G}}$ is a
map that preserves horizontal curves. Pansu introduced a natural notion of
differentiability for maps between Carnot groups in [Pan89]. Since Carnot
groups can be viewed as Euclidean spaces with a particular Lie group
structure, it also makes sense to discuss the classical Euclidean
differentiability of maps between Carnot groups. These notions of
differentiability are not the same. In particular, there are continuously
Pansu differentiable maps between Carnot groups that are not Euclidean
differentiable in non-horizontal directions [Mag05].

When $\Omega=\mathbb{G}=\hat{\mathbb{G}}$, Warhurst [War08] proved that $f$ is
continuously Pansu differentiable if and only if it is a Euclidean-$C^{1}$
diffeomorphism. In the more general setting without these added restrictions,
Magnani proved that $f$ is continuously Pansu differentiable if and only if it
is continuously Euclidean differentiable in the horizontal directions of
$\hat{\mathbb{G}}$. See Theorem 1.1 in [Mag13]. The forward implication of
this result is straightforward. It is the reverse implication that is the
focus of this paper; we provide an alternate, elementary proof. Magnani argued
at the level of Lie algebras, while our proof below will be mostly Euclidean
in flavor.

Magnani also provided a mean value estimate on the Pansu difference quotients
of continuously Pansu differentiable maps. See Theorem 1.2 in [Mag13]. In
order to prove our main goal, we also establish Magnani’s estimate (with a
weaker exponent on the modulus of continuity).

Before stating the theorem we aim to prove, let us first establish some
notation. For any open set $\Omega\subset\mathbb{G}$ and compact
$A\subset\Omega$, define
$t_{A}=\frac{\text{dist}(A,\mathbb{G}\setminus\Omega)}{2k_{0}C_{0}\sup_{\|\xi\|=1}|\xi|}$
where $k_{0}$ and $C_{0}$ are the constants depending only on $\mathbb{G}$
from Remark 2.1, and set
$\mathcal{A}=\\{y\in\mathbb{G}:\text{dist}(y,A)\leq\tfrac{1}{2}\text{dist}(A,\mathbb{G}\setminus\Omega)\\}$.
In the following, we will use $\hat{d}$ and $\hat{n}$ to respectively denote
the metric on and horizontal dimension of $\hat{\mathbb{G}}$.

######  Theorem 1.1.

Suppose $\Omega\subset\mathbb{G}$ is open. If $f:\Omega\to\hat{\mathbb{G}}$
preserves horizontal curves and $f_{i}$ is Euclidean-$C^{1}$ on $\Omega$ for
$i\leq\hat{n}$ then, for any compact set $A\subset\Omega$, there is a constant
$C>0$ satisfying the following: for every $x\in A$ and $\xi\in\partial
B(0,1)$, there is some $z(x,\xi)\in\hat{\mathbb{G}}$ such that

(1.1) $\hat{d}(R(x,\xi;t),z(x,\xi))\leq C\omega(t)^{1/s^{k_{0}}}$

for $t\in\left(0,t_{A}\right]$ where $\omega$ is a Euclidean modulus of
continuity on $\mathcal{A}$ for each $\nabla f_{i}$ with $i\leq\hat{n}$. If we
further write $\xi=\xi_{k_{0}}\cdots\xi_{1}$ as in Remark 2.1, then we have

(1.2) $z(x,\xi)=D_{\xi_{k_{0}}}f(x)D_{\xi_{k_{0}-1}}f(x)\cdots
D_{\xi_{1}}f(x),$

where

$\displaystyle D_{\xi_{i}}f(x)$
$\displaystyle=\left(\left.\tfrac{d}{dt}f_{1}(x\delta_{t}(\xi_{i}))\right|_{t=0},\dots,\left.\tfrac{d}{dt}f_{\hat{n}}(x\delta_{t}(\xi_{i}))\right|_{t=0},0,\dots,0\right)$
(1.3) $\displaystyle=\lambda_{i}\left(\nabla f_{1}(x)\cdot
X_{j_{i}}(x),\dots,\nabla f_{\hat{n}}(x)\cdot X_{j_{i}}(x),0,\dots,0\right).$

The bound of $t_{A}$ exists to ensure that $R(x,\xi;t)$ is well-defined and
well controlled (i.e. $x\delta_{t}(\xi)\in\mathcal{A}$). This result can be
interpreted as follows. Inequality (1.1) implies that $f$ is Pansu
differentiable and that its Pansu difference quotients converge uniformly at a
rate controlled by the uniform convergence of the Euclidean derivative. This
is the special case of the mean value estimate (Theorem 1.2 in [Mag13])
mentioned above. Moreover, (1.2) and (1.3) give an explicit expression of the
Pansu derivative in this setting. While this formulation is likely not new, a
relevant reference was not readily found. Since each $D_{\xi_{i}}f$ is
continuous, we may conclude the following.

######  Corollary 1.2 ([Mag13]).

Suppose $\Omega\subset\mathbb{G}$ is open and $f:\Omega\to\hat{\mathbb{G}}$
preserves horizontal curves. If $f_{i}$ is Euclidean-$C^{1}$ on $\Omega$ for
$i\leq\hat{n}$, then $f$ is continuously Pansu differentiable.

The main new contribution in this paper and the tool underpinning the proof of
Theorem 1.1 is Lemma 3.2. This lemma shows that smooth horizontal curves are
well approximated by “horizontal rays” in the same way that Euclidean curves
are well approximated by tangent lines.

The paper is outlined as follows. Preliminary information about Carnot groups
and moduli of continuity are given in Section 2. Section 3 is devoted to the
approximation lemma discussed in the previous paragraph. The proof of Theorem
1.1 is the primary content of Section 4.

##  2\. Preliminaries

###  2.1. Carnot groups

Given a positive integer $s$, suppose $\mathbb{G}$ is a connected, simply
connected Lie group and $V_{1},\dots,V_{s}$ are non-zero subspaces of its
associated Lie algebra $\mathfrak{g}$. We say that $\mathbb{G}$ is a step-$s$
Carnot group if $\mathfrak{g}$ is stratified in the following sense:

$\mathfrak{g}=V_{1}\oplus\cdots\oplus V_{s},\quad[V_{1},V_{i}]=V_{i+1}\text{
for }i=1,\dots,s-1,\quad[V_{1},V_{s}]=\\{0\\}.$

The horizontal subbundle $H$ of $T\mathbb{G}$ is defined so that
$H_{0}=V_{1}$, and, for any $p\in\mathbb{G}$, we set
$H_{p}=(L_{p})_{*}(H_{0})$ where $L_{p}(x)=px$ is the left translation
operation. Set $N:=\text{dim }\mathfrak{g}$ and $n:=\text{dim }V_{1}$. We call
$n$ the horizontal dimension of $\mathbb{G}$.

We select a basis $\\{X_{1},\dots,X_{N}\\}$ of $\mathfrak{g}$ adapted to the
stratification in the sense that $\\{X_{(\sum^{i-1}_{j=1}\text{dim
}V_{j})+1},\dots,X_{\sum^{i}_{j=1}\text{dim }V_{j}}\\}$ forms a basis of
$V_{i}$ for each $i\in\\{1,\dots,s\\}$. For any $x\in\mathbb{G}$, then, we can
uniquely write $x=\text{exp}(x_{1}X_{1}+\cdots+x_{N}X_{N})$ for some
$(x_{1},\dots,x_{N})\in\mathbb{R}^{N}$ via the exponential map
$\text{exp}:\mathfrak{g}\to\mathbb{G}$. This allows us to identify
$\mathbb{G}$ with $\mathbb{R}^{N}$ via the relationship
$x\leftrightarrow(x_{1},\dots,x_{N})$. Denote by $|\cdot|$ the Euclidean norm
in $\mathbb{G}=\mathbb{R}^{N}$ determined by our choice of basis above. Note
also that, under this identification, we have $x^{-1}=-x$ for any
$x\in\mathbb{G}$.

Another property that follows from the stratified structure of a Carnot group
is the existence of automorphic dilations. Note that, for any coordinate
$i\in\\{1,\dots,N\\}$, there is a positive integer $d_{i}$ satisfying

$\sum^{d_{i}-1}_{j=1}\text{dim }V_{j}<i\leq\sum^{d_{i}}_{j=1}\text{dim
}V_{j}.$

We call $d_{i}$ the degree of $i$. Intuitively, this describes the “layer” of
the stratification in which the coordinate $i$ lives. Note in particular that,
if $i\leq n$, then $d_{i}=1$. For any $t>0$, define the dilation $\delta_{t}$
as

$\delta_{t}(x)=(t^{d_{1}}x_{1},t^{d_{2}}x_{2},\dots,t^{d_{N}}x_{N}).$

This forms a one parameter family of automorphisms in the sense that
$\delta_{t_{1}}\circ\delta_{t_{2}}=\delta_{t_{1}t_{2}}$ for $t_{1},t_{2}>0$.
We can naturally extend this one parameter family to $t=0$ so that
$\delta_{0}(x)=0$, but this is clearly no longer an automorphism.

The following well known fact follows from the stratified structure of
$\mathbb{G}$. See, for example, Lemma 1.40 in [FS82]. This fact allows us to
“build” any point in $\mathbb{G}$ out of boundedly many points from the first
layer.

######  Remark 2.1.

There is a positive integer $k_{0}$ and a constant $C_{0}>0$ such that, for
each $\xi\in\mathbb{G}$, we may write $\xi=\xi_{k_{0}}\cdots\xi_{1}$ where,
for $k=1,\dots,k_{0}$, we have $\xi_{k}=\lambda_{k}e_{j_{k}}$ for some
$j_{k}\in\\{1,\dots,n\\}$ and $\lambda_{k}\in\mathbb{R}$ with
$|\lambda_{k}|\leq C_{0}|\xi|$.

Since the Lie algebra is nilpotent, one may explicitly compute the group
operation in $\mathbb{G}$ from the Lie bracket combinations in $\mathfrak{g}$
via the Baker–Campbell–Hausdorff formula (see Definition 2.2.11 in [BLU07]).
The following properties of the group law in $\mathbb{G}$ follow from this
computation and are well known. See, for example, Proposition 2.2.22 in
[BLU07].

######  Proposition 2.2.

We may write the group law as $xy=x+y+Q(x,y)$ for some polynomial
$Q=(Q_{1},\dots,Q_{N})$ where

  1. (1)

$Q_{1}=\dots=Q_{n}=0$,

  2. (2)

for $n<i\leq N$, the real valued polynomial $Q_{i}(x,y)$ may be written as a
sum of terms each of which contains a factor of the form
$(x_{j}y_{\ell}-x_{\ell}y_{j})$ for some $1\leq\ell<i$ and $1\leq j<i$,

  3. (3)

$Q_{i}$ is homogeneous of degree $d_{i}$ with respect to dilations i.e.
$Q_{i}(\delta_{t}(x),\delta_{t}(y))=t^{d_{i}}Q_{i}(x,y)$ for all
$x,y\in\mathbb{G}$ and $t>0$,

  4. (4)

if the coordinate $x_{i}$ has degree $d_{i}\geq 2$, then $Q_{i}(x,y)$ depends
only on the coordinates of $x$ and $y$ which have degree strictly less than
$d_{i}$.

Note in particular that the group operation is simply linear in the first
layer and, in any higher layer, depends only on the previous ones. This is
expected due to the stratified structure of $\mathfrak{g}$.

A metric $d$ on $\mathbb{G}$ is called left invariant if $d(xp,xq)=d(p,q)$ and
is called homogeneous if $d(\delta_{t}(p),\delta_{t}(q))=td(p,q)$ for
$x,p,q\in\mathbb{Q}$ and $t>0$. All homogeneous, left invariant metrics on
$\mathbb{G}$ are bi-Lipschitz equivalent to one another due to the compactness
of their unit spheres. For any such metric we also have the following Hölder-
type equivalence (and thus topological equivalence) with the Euclidean norm on
$\mathbb{R}^{N}$. See Proposition 5.15.1 in [BLU07].

######  Proposition 2.3.

For any compact set $K\subset\mathbb{G}$, there is a constant $C_{K}\geq 1$
such that

$C_{K}^{-1}|x-y|\leq d(x,y)\leq C_{K}|x-y|^{1/s}.$

We will choose a particular left invariant, homogeneous metric $d$ on
$\mathbb{G}$ for our purposes below. For $x\in\mathbb{G}$, set

$\|x\|:=\max_{i=1,\dots,N}\mu_{i}|x_{i}|^{1/d_{i}}$

where the constants $\mu_{i}>0$ are chosen so that $\|\cdot\|$ satisfies the
triangle inequality. Such a choice is always possible; see [Gui73]. Define
$d(x,y)=\|x^{-1}y\|$ for $x,y\in\mathbb{G}$. This metric is left invariant and
homogeneous by its definition.

Suppose $\upgamma:[a,b]\to\mathbb{G}$ is absolutely continuous as a curve in
$\mathbb{R}^{N}$. We say that $\upgamma$ is horizontal if
$\upgamma^{\prime}(t)\in H_{\upgamma(t)}$ for almost every $t\in[a,b]$. In
particular, this means

(2.1)
$\upgamma^{\prime}(t)=\sum_{j=1}^{n}\upgamma_{j}^{\prime}(t)X_{j}(\upgamma(t))=\sum_{j=1}^{n}\upgamma_{j}^{\prime}(t)e_{j}+\sum_{j=1}^{n}\upgamma_{j}^{\prime}(t)\left.\frac{\partial
Q(\upgamma(t),y)}{\partial y_{j}}\right|_{y=0}.$

Suppose $\Omega\subset\mathbb{G}$ is open. A map $f:\Omega\to\hat{\mathbb{G}}$
is said to preserve horizontal curves if $f\circ\upgamma$ is a horizontal
curve in $\hat{\mathbb{G}}$ for every horizontal curve $\upgamma$ in $\Omega$.

###  2.2. Bounding right translation

While the metric $d$ is not invariant under right multiplication, we do have
the following estimate.

######  Proposition 2.4.

Suppose $M>0$ and $a,b,c\in\mathbb{G}$ satisfy $\max\\{|a|,|b|,|c|\\}\leq M$.
There is a constant $C>0$ depending only on $M$ and $\mathbb{G}$ so that

$d(ac,bc)\leq Cd(a,b)^{1/s}.$

###### Proof.

In this proof and the next, we will write $x\lesssim y$ for quantities
$x,y\geq 0$ to indicate that there is a constant $C\geq 1$ depending only on
$M$ and $\mathbb{G}$ such that $x\leq Cy$.

Write $p=a^{-1}b$. We have that

(2.2) $\displaystyle d(ac,bc)=\|c^{-1}pc\|=\max_{1\leq i\leq
N}\mu_{i}\left|(c^{-1}pc)_{i}\right|^{1/d_{i}}.$

Let us establish the structure of $c^{-1}pc$. First, note that

$\displaystyle
pc=\sum_{i=1}^{n}(p_{i}+c_{i})e_{i}+\sum_{i=n+1}^{N}\left(p_{i}+c_{i}+Q_{i}(p,c)\right)e_{i}$

so that

$\displaystyle
c^{-1}pc=\sum_{i=1}^{n}p_{i}e_{i}+\sum_{i=n+1}^{N}\left(p_{i}+Q_{i}(p,c)+Q_{i}(c^{-1},pc)\right)e_{i}.$

According to Proposition 2.2(2), every polynomial $Q_{i}(p,c)$ can be written
as a sum of terms each of which contains a factor of the form
$(p_{j}c_{\ell}-p_{\ell}c_{j})$ for some $1\leq j,\ell<i$. In particular,
$Q_{i}(p,c)$ can be rewritten as a sum of terms each of which has a factor of
$p_{j}$ for some $1\leq j<i$.

Similarly, we can write $Q_{i}(c^{-1},pc)$ as a sum of terms each of which
contains a factor of the form

$\displaystyle-c_{j}(pc)_{\ell}+c_{\ell}(pc)_{j}$
$\displaystyle=-c_{j}\left(p_{\ell}+c_{\ell}+Q_{\ell}(p,c)\right)+c_{\ell}\left(p_{j}+c_{j}+Q_{j}(p,c)\right)$
$\displaystyle=-c_{j}p_{\ell}-c_{j}Q_{\ell}(p,c)+c_{\ell}p_{j}+c_{\ell}Q_{j}(p,c).$

Again, then, $Q_{i}(c^{-1},pc)$ can be rewritten as a sum of terms each of
which has a factor of $p_{j}$ for some $1\leq j<i$. Therefore, we may conclude
that

(2.3) $\displaystyle\left|(c^{-1}pc)_{i}\right|^{1/d_{i}}\lesssim\max_{1\leq
j\leq i}\left(|p_{j}|^{1/d_{j}}\right)^{d_{j}/d_{i}}$
$\displaystyle\lesssim\|p\|^{1/s}$

since $d_{j}/d_{i}\geq 1/s$. Combining (2.2) and (2.3) completes the proof. ∎

######  Corollary 2.5.

Given $\alpha>0$ and
$a_{1},\dots,a_{k_{0}},b_{1},\dots,b_{k_{0}}\in\mathbb{G}$ satisfying
$d(a_{k},b_{k})\leq\alpha$ for $k=1,\dots,{k_{0}}$ and
$\max_{i=1,\dots,k_{0}}\\{|a_{i}|,|b_{i}|\\}\leq M$ for some $M>0$, there is a
constant $C>0$ depending only on $M$ and $\mathbb{G}$ such that

$d(a_{k_{0}}\cdots a_{1},b_{k_{0}}\cdots b_{1})\leq C\alpha^{1/s^{k_{0}-1}}.$

###### Proof.

Note first that $d(a_{1},b_{1})\leq\alpha$. Suppose for some
$k\in\\{1,\dots,k_{0}-1\\}$ that we have proven

$d(a_{k}\cdots a_{1},b_{k}\cdots b_{1})\lesssim\alpha^{1/s^{k-1}}.$

We then have

$\displaystyle d(a_{k+1}$ $\displaystyle a_{k}\cdots a_{1},b_{k+1}b_{k}\cdots
b_{1})$ $\displaystyle\leq d(a_{k+1}a_{k}\cdots a_{1},a_{k+1}b_{k}\cdots
b_{1})+d(a_{k+1}b_{k}\cdots b_{1},b_{k+1}b_{k}\cdots
b_{1})=:\mathbf{I}+\mathbf{II}.$

The left invariance of the metric provides the bound
$\mathbf{I}\lesssim\alpha^{1/s^{k-1}}\lesssim\alpha^{1/s^{k}}$. Applying the
previous proposition $k$ times gives $\mathbf{II}\lesssim
d(a_{k+1},b_{k+1})^{1/s^{k}}\lesssim\alpha^{1/s^{k}}.$ Repeating this process
until it terminates completes the proof. ∎

###  2.3. Pansu Differentiability

Suppose $\Omega\subset\mathbb{G}$ is open, and consider
$f:\Omega\to\hat{\mathbb{G}}$. The Pansu difference quotient of $f$ at
$x\in\mathbb{G}$ in the direction of $\xi\in\partial B(0,1)\subset\mathbb{G}$
is defined for all $t>0$ as

$R(x,\xi;t)=\hat{\delta}_{1/t}\left(f(x)^{-1}f\left(x\delta_{t}(\xi)\right)\right).$

We say that $f$ is Pansu differentiable at $x\in\Omega$ if there is some
$z(x,\xi)\in\mathbb{G}$ such that, for all $\xi\in\partial B(0,1)$, the
difference quotient $R(x,\xi;t)$ converges to $z(x,\xi)$ as $t\to 0$ and this
convergence is uniform with respect to $\xi$. When this limit exists, the map
$\xi\mapsto z(x,\xi)$ is known to be a homeomorphism. See Proposition 3.2 in
[Pan89]. We say that $f$ is continuously Pansu differentiable if the map
$x\mapsto z(x,\cdot)$ is continuous.

###  2.4. Modulus of continuity

######  Definition 2.6.

Suppose $\mathcal{A}\subset\mathbb{G}$ is compact and fix
$f:A\to\hat{\mathbb{G}}$. We say that $\omega:[0,\infty)\to[0,\infty)$ is a
Euclidean modulus of continuity on $\mathcal{A}$ of $f$ if it is non-constant,
continuous, concave, increasing, $\omega(0)=0$, and

$|f(x)-f(y)|\leq\omega(d(x,y))\quad\text{ for all }x,y\in\mathcal{A}.$

Any continuous function on a compact set yields a concave modulus of
continuity. It follows from the concavity of $\omega$ that
$t\mapsto\omega(t)/t$ is a decreasing function. See, for example, Lemma 2.5 in
[SZ23]. From this, it follows that
$\frac{\omega(t_{A})}{t_{A}}\leq\frac{\omega(t)}{t}$. That is, there is a
constant $c>0$ depending only on $A$ and $\mathbb{G}$ such that $t\leq
c\omega(t)$ for any $t\in[0,t_{A}]$. Moreover, if $C\geq 1$ is a constant,
then, for any $t>0$, $\frac{\omega(Ct)}{Ct}\leq\frac{\omega(t)}{t}$ so that
$\omega(Ct)\leq C\omega(t)$. These facts will be used throughout the proofs
below.

##  3\. Approximating smooth, horizontal curves with horizontal lines

We make the following observation about horizontal curves. Given a curve
$\upgamma:[a,b]\to\mathbb{G}$ that is horizontal at $t\in[a,b]$, write
$h(t):=(\upgamma_{1}^{\prime}(t),\dots,\upgamma_{n}^{\prime}(t),0,\dots,0)$.
Note that $\alpha h(t)=\delta_{\alpha}(h(t))$.

######  Lemma 3.1.

Suppose $\upgamma:[a,b]\to\mathbb{G}$ is a curve which is horizontal at
$t\in[a,b]$. Then, for each $i\in\\{n+1,\dots,N\\}$ and $\alpha>0$, we have
that $\alpha^{d_{i}}\upgamma_{i}^{\prime}(t)$ is a finite sum of terms each of
which possesses

$\alpha^{d_{j}+1}\upgamma_{j}(t)h_{\ell}(t)-\alpha^{d_{\ell}+1}\upgamma_{\ell}(t)h_{j}(t)$

as a factor for some $1\leq j,\ell<i$.

###### Proof.

Fix $i\in\\{n+1,\dots,N\\}$. According to (2.1), we have

(3.1)
$\upgamma_{i}^{\prime}(t)=\sum_{j=1}^{n}\upgamma_{j}^{\prime}(t)\left.\frac{\partial
Q_{i}(\upgamma(t),y)}{\partial
y_{j}}\right|_{y=0}=\bar{Q}_{i}(\upgamma(t),h(t))$

where $\bar{Q}_{i}$ is the finite sum of the terms in $Q_{i}(x,y)$ in which
$y$ appears linearly. Proposition 2.2(2) implies that $\bar{Q}_{i}(x,y)$ is a
finite sum of terms of the form

$p_{j,\ell}(x)(x_{j}y_{\ell}-x_{\ell}y_{j})$

for $1\leq j,\ell<i$ where $p_{j,\ell}$ is a polynomial. The $d_{i}$-degree
homogeneity of $Q_{i}$ (and thus of $\bar{Q}_{i}$) gives

$\displaystyle\alpha^{d_{i}}\upgamma_{i}^{\prime}(t)=\bar{Q}_{i}(\delta_{\alpha}(\upgamma(t)),\delta_{\alpha}(h(t)))=\bar{Q}_{i}(\delta_{\alpha}(\upgamma(t)),\alpha
h(t)),$

and this is then a finite sum of terms of the form

$p_{j,\ell}\left(\delta_{\alpha}(\upgamma(t))\right)\left(\alpha^{d_{j}+1}\upgamma_{j}(t)h_{\ell}(t)-\alpha^{d_{\ell}+1}\upgamma_{\ell}(t)h_{j}(t)\right).$

∎

Suppose $f:[0,b]\to\mathbb{G}$ is a horizontal curve such that $f_{i}$ is
Euclidean-differentiable at 0 for $i\leq n$. Define the “horizontal ray”

$L_{f}(t):=f(0)*\left(tf_{1}^{\prime}(0),\dots,tf_{n}^{\prime}(0),0\dots,0\right)\quad\text{for
}t\geq 0.$

The following is a generalization of Lemmas 3 and 4 and Corollary 5 from
[War08]. This is the primary new tool in the proof of Theorem 1.1 and is
similar to approximations proven in Proposition 21 from [CLZ22] and Lemma 3.9
from [Zim22]. However, unlike in these cited results, we now no longer assume
regularity of every component of the curve but only its horizontal ones. It
turns out that this lower level of regularity is still enough to guarantee
that the curve is well approximated by a horizontal ray.

######  Lemma 3.2.

Suppose $t_{0}\geq 0$ and $f:[0,t_{0}]\to\mathbb{G}$ is a horizontal curve
such that $f_{i}$ is Euclidean-$C^{1}$ for $i\leq n$, and suppose that
$|f_{i}^{\prime}(t)-f_{i}^{\prime}(0)|\leq\omega(t)$ for some Euclidean
modulus of continuity $\omega$, all $t\in[0,t_{0}]$, and $i\leq n$. Then there
is a constant $C_{1}>0$ depending only on $\omega$, $\mathbb{G}$, $t_{0}$, and
$|f_{i}^{\prime}(0)|$ for $i\leq n$ such that, for every $t\in[0,t_{0}]$,

(3.2) $d(f(t),L_{f}(t))\leq C_{1}t\,\omega(t)^{1/s}.$

###### Proof.

Throughout this proof, we will write $x\lesssim y$ for quantities $x,y\geq 0$
to indicate that there is some constant $C\geq 1$ depending only on $\omega$,
$\mathbb{G}$, $t_{0}$, and $|f_{i}^{\prime}(0)|$ for $i\leq n$ such that
$x\leq Cy$.

Using the left invariance of the metric on $\mathbb{G}$, we may assume without
loss of generality that $f(0)=0$. Notice first that, for $i\in\\{1,\dots,n\\}$
and all $t\in[0,t_{0}]$,

(3.3)
$\displaystyle|f_{i}(t)-tf_{i}^{\prime}(0)|\leq\int_{0}^{t}|f_{i}^{\prime}(\tau)-f_{i}^{\prime}(0)|\,d\tau\leq
t\,\omega(t).$

Since $d(f(t),L_{f}(t))=N_{\infty}(L_{f}(t)^{-1}*f(t))$, it remains to verify
that

(3.4) $|f_{i}(t)|+|Q_{i}(L_{f}(t)^{-1},f(t))|\lesssim t^{d_{i}}\,\omega(t)$

for $i\in\\{n+1,\dots,N\\}$.

Fix such an index $i$, and choose $t\in[0,t_{0}]$ at which $f$ is horizontal.
Suppose by way of induction that we have proven for all $n+1\leq j<i$ that

(3.5) $\left|f_{j}(t)\right|\lesssim t^{d_{j}}\,\omega(t).$

According to Lemma 3.1, we can write
$t^{1-d_{i}}f_{i}^{\prime}(t)=t\left((t^{-1})^{d_{i}}f_{i}^{\prime}(t)\right)$
as a finite sum of terms each of which is a multiple of

(3.6) $t^{-d_{j}}f_{j}(t)h_{\ell}(t)-t^{-d_{\ell}}f_{\ell}(t)h_{j}(t)$

with $1\leq j,\ell<i$. If $j>n$, we have $h_{j}(t)=0$. In this case (or when
$\ell>n$), we may use (3.5) and the fact that
$|f_{k}^{\prime}(t)|\leq|f_{k}^{\prime}(0)|+\omega(t_{0})$ for $k\leq n$ to
bound (3.6) by a constant multiple of $\omega(t)$ with the constant depending
only on $\omega$, $\mathbb{G}$, $t_{0}$, and $|f_{k}^{\prime}(0)|$ for $k\leq
n$. Otherwise, if $j,\ell\leq n$, we have $d_{j}=d_{\ell}=1$, so (3.6) is
bounded by

$\displaystyle\tfrac{1}{t}\left|f_{j}(t)-tf_{j}^{\prime}(0)\right|\left|f_{\ell}^{\prime}(t)\right|+\tfrac{1}{t}\left|f_{\ell}(t)-tf_{\ell}^{\prime}(0)\right|\left|f_{j}^{\prime}(t)\right|+\left|f_{j}^{\prime}(0)f_{\ell}^{\prime}(t)-f_{\ell}^{\prime}(0)f_{j}^{\prime}(t)\right|.$

This last term is bounded by

$\displaystyle\left|f_{j}^{\prime}(0)\right|\left|f_{\ell}^{\prime}(t)-f_{\ell}^{\prime}(0)\right|+\left|f_{\ell}^{\prime}(0)\right|\left|f_{j}^{\prime}(t)-f_{j}^{\prime}(0)\right|\lesssim\omega(t).$

This, (3.3), and the argument above imply that

$\left|t^{-d_{j}}f_{j}(t)h_{\ell}(t)-t^{-d_{\ell}}f_{\ell}(t)h_{j}(t)\right|\lesssim\omega(t)$

for any $1\leq j,\ell<i$, and hence
$\left|t^{1-d_{i}}f_{i}^{\prime}(t)\right|\lesssim\omega(t)$ for almost all
$t\in[0,t_{0}]$. Therefore, for all $t\in[0,t_{0}]$ and
$i\in\\{n+1,\dots,N\\}$,

(3.7) $\displaystyle\left|f_{i}(t)\right|\leq
t^{d_{i}-1}\int_{0}^{t}\left|t^{1-d_{i}}f_{i}^{\prime}(t)\right|\,dt\lesssim
t^{d_{i}}\,\omega(t).$

It remains to bound the second term in (3.4), and the arguments here are
similar. Again, choose $t\in[0,t_{0}]$ at which $f^{\prime}(t)$ is horizontal
and fix $i\in\\{n+1,\dots,N\\}$. Proposition 2.2 implies that

$t^{-d_{i}}Q_{i}\left(L_{f}(t)^{-1},f(t)\right)=Q_{i}\left(\delta_{1/t}(L_{f}(t)^{-1}),\delta_{1/t}(f(t))\right),$

and this polynomial is a sum of terms each of which contains a factor of the
form

$t^{-d_{j}}f_{j}(t)h_{\ell}(0)-t^{-d_{\ell}}f_{\ell}(t)h_{j}(0)$

for $1\leq j,\ell<i$. This quantity is bounded from above by

$\displaystyle
t^{-d_{j}}\left|f_{j}(t)-t^{d_{j}}h_{j}(0)\right|\left|t^{-d_{\ell}}f_{\ell}(t)\right|+t^{-d_{\ell}}\left|f_{\ell}(t)-t^{d_{\ell}}h_{\ell}(0)\right|\left|t^{-d_{j}}f_{j}(t)\right|$

Note that (3.3) implies $\left|f_{j}(t)\right|\leq
t\left(\omega(1)+\left|f_{j}^{\prime}(0)\right|\right)$ when $j\leq n$. This
together with (3.7) gives

$\left|Q_{i}\left(L_{f}(t)^{-1},f(t)\right)\right|\lesssim
t^{d_{i}}\,\omega(t)$

which completes estimate (3.4) and hence the proof. ∎

##  4\. Proof of the main theorem

As in Remark 2.1, there are constants $k_{0}$ and $C_{0}$ depending only on
$\mathbb{G}$ such that, for any $\xi\in K:=\partial B(0,1)$, we can write
$\xi=\xi_{k_{0}}\cdots\xi_{1}$ where each $\xi_{k}=\lambda_{k}e_{j_{k}}$ for
some $j_{k}\in\\{1,\dots,n\\}$ and $\lambda_{k}\in\mathbb{R}$ with
$|\lambda_{k}|\leq C_{0}|\xi|$.

For any such $\xi$ and any $x\in\mathbb{G}$, we define

$\displaystyle
D_{\xi_{k}}f(x)=\left(\left.\tfrac{d}{d\tau}f_{1}(x\delta_{\tau}(\xi_{k}))\right|_{\tau=0},\dots,\left.\tfrac{d}{d\tau}f_{\hat{n}}(x\delta_{\tau}(\xi_{k}))\right|_{\tau=0},0,\dots,0\right)$

and

$z(x,\xi)=D_{\xi_{k_{0}}}f(x)D_{\xi_{k_{0}-1}}f(x)\cdots D_{\xi_{1}}f(x).$

Recall also that, given $A\subset\Omega\subset\mathbb{G}$, we define
$t_{A}=\frac{\text{dist}(A,\mathbb{G}\setminus\Omega)}{2k_{0}C_{0}\sup_{\|\xi\|=1}|\xi|}$.

We restate the quantitative version of Theorem 1.1 here for convenience. In
the following, we will use $d$, $\delta$, $N$, and $n$ and $\hat{d}$,
$\hat{\delta}$, $\hat{N}$, and $\hat{n}$ to denote the metrics on, dilations
on, topological dimensions of, and horizontal dimensions of $\mathbb{G}$ and
$\hat{\mathbb{G}}$ respectively.

######  Theorem 4.1.

Suppose $\Omega\subset\mathbb{G}$ is open. If $f:\Omega\to\hat{\mathbb{G}}$
preserves horizontal curves and $f_{i}$ is Euclidean-$C^{1}$ for
$i\leq\hat{n}$, then, for any compact set $A\subset\Omega$, there is a
constant $C>0$ satisfying the following: for every $x\in A$ and
$\xi\in\partial B(0,1)$,

(4.1) $\hat{d}(R(x,\xi;t),z(x,\xi))\leq C\omega(t)^{1/s^{k_{0}}}$

for $t\in\left(0,t_{A}\right]$.

###### Proof.

Throughout this proof, we will write $a\lesssim b$ for quantities $a,b\geq 0$
to indicate that there is some constant $C\geq 1$ depending only on
$\mathbb{G}$, $A$, $\Omega$, and $f$ such that $a\leq Cb$.

We begin by verifying the equality in (1.3). Note first that, for any
$x\in\mathbb{G}$,

$\displaystyle\left.\tfrac{d}{d\tau}x\delta_{\tau}(\xi_{k})\right|_{\tau=0}=\left.\tfrac{d}{d\tau}L_{x}(\tau\lambda_{k}e_{j_{k}})\right|_{\tau=0}=dL_{x}\left(\lambda_{k}\partial_{j_{k}}\right)=\lambda_{k}X_{j_{k}}(x)$

where $L_{x}(p):=xp$. Thus, given
$f:\mathbb{G}\supset\Omega\to\hat{\mathbb{G}}$ such that $f_{i}$ is Euclidean-
differentiable at $x\in\Omega$ for $i\leq\hat{n}$, we have

$\displaystyle D_{\xi_{i}}f(x)=\lambda_{i}\left(\nabla f_{1}(x)\cdot
X_{j_{i}}(x),\dots,\nabla f_{\hat{n}}(x)\cdot X_{j_{i}}(x),0,\dots,0\right).$

Now fix $f:\mathbb{G}\supset\Omega\to\hat{\mathbb{G}}$ and assume that $f_{i}$
is Euclidean-$C^{1}$ for $i\leq\hat{n}$. Fix a compact set $A\subset\Omega$
and choose a function $\omega$ which is a Euclidean modulus of continuity on
$\mathcal{A}$ for each $\nabla f_{i}$ with $i\leq\hat{n}$. To justify our
definition of $z(x,\xi)$, we first recall “Pansu’s trick” for the difference
quotient [Pan89]: given $x\in A$, $p,q\in K$, and $t\in[0,t_{A}]$, we have

$\displaystyle R(x,pq;t)$
$\displaystyle=\hat{\delta}_{1/t}\left(f(x)^{-1}f\left(x\delta_{t}(pq)\right)\right)$
$\displaystyle=\hat{\delta}_{1/t}\left(f(x)^{-1}f(x\delta_{t}(p))\right)\hat{\delta}_{1/t}\left(f(x\delta_{t}(p))^{-1}f\left(x\delta_{t}(p)\delta_{t}(q)\right)\right)$
$\displaystyle=R(x,p;t)R(x\delta_{t}(p),q;t).$

Applying this trick $k_{0}-1$ times, we have for any $x\in A$ that

$\displaystyle R(x,\xi;t)=R(x,\xi_{k_{0}}\cdots\xi_{1};t)$
$\displaystyle=R(x,\xi_{k_{0}};t)R(x\delta_{t}(\xi_{k_{0}}),\xi_{k_{0}-1}\cdots\xi_{1};t)$
$\displaystyle=\dots$ (4.2)
$\displaystyle=R(x_{k_{0}},\xi_{k_{0}};t)R(x_{k_{0}-1}(t),\xi_{k_{0}-1};t)\cdots
R(x_{1}(t),\xi_{1};t)$

where $x_{k_{0}}(t)\equiv x$ and

$\displaystyle
x_{k}(t):=x_{k+1}\delta_{t}(\xi_{k+1})=x\delta_{t}(\xi_{k_{0}}\xi_{k_{0}-1}\cdots\xi_{k+1})$

for $k=1,\dots,{k_{0}}-1$. The left invariance and homogeneity of $d$ together
with the triangle inequality imply that $x_{k}(t)\in\mathcal{A}\subset\Omega$
for each $t\in[0,t_{A}]$. Indeed,

$\displaystyle d(x,x_{k}(t))=td(0,\xi_{k_{0}}\cdots\xi_{k+1})$
$\displaystyle\leq t(d(0,\xi_{k_{0}})+\cdots+d(0,\xi_{k+1}))$ (4.3)
$\displaystyle\leq t(|\lambda_{k_{0}}|+\cdots|\lambda_{k+1}|)\leq
t(k_{0}-1)C_{0}|\xi|.$

For each $t\in[0,t_{A}]$, define

$z_{0}(x,\xi;t):=D_{k_{0}}f(x)*D_{k_{0}-1}f(x_{k_{0}-1}(t))*\cdots*D_{1}f(x_{1}(t)).$

We will complete the proof by verifying that

(4.4) $\hat{d}(z(x,\xi),z_{0}(x,\xi;t))\lesssim\omega(t)^{1/s^{k_{0}}}$

and

(4.5) $\hat{d}(R(x,\xi;t),z_{0}(x,\xi;t))\lesssim\omega(t)^{1/s^{k_{0}}}$

for all $x\in A$, $\xi\in K$, and $t\in[0,t_{A}]$.

We will first attack estimate (4.4). In view of Corollary 2.5, we will prove
the following claim.

Claim: There is a constant $C_{1}>0$ such that, for all $x\in A$ and
$k\in\\{1,\dots,k_{0}\\}$,

(4.6) $\displaystyle\hat{d}(D_{k}f(x),D_{k}f(x_{k}(t)))\leq
C_{1}\omega(t)^{1/s}$

for $t\in[0,t_{A}]$.

###### Proof of Claim.

By (4.3), we have $x_{k}(t)\in\mathcal{A}$, and so for $i\leq\hat{n}$ we get

$\displaystyle\left|\left.\tfrac{d}{d\tau}f_{i}(x\delta_{\tau}(\xi_{k}))\right|_{\tau=0}-\left.\tfrac{d}{d\tau}f_{i}(x_{k}(t)\delta_{\tau}(\xi_{k}))\right|_{\tau=0}\right|$
$\displaystyle\qquad\qquad\qquad=|\nabla
f_{i}(x)\cdot\lambda_{k}X_{j_{k}}(x)-\nabla
f_{i}(x_{k}(t))\cdot\lambda_{k}X_{j_{k}}(x_{k}(t))|$
$\displaystyle\qquad\qquad\qquad\lesssim|\nabla f_{i}(x)-\nabla
f_{i}(x_{k}(t))||X_{j_{k}}(x)|+|\nabla
f_{i}(x)||X_{j_{k}}(x)-X_{j_{k}}(x_{k}(t))|$
$\displaystyle\qquad\qquad\qquad\lesssim\omega(t)|X_{j_{k}}(x)|+t|\nabla
f_{i}(x)|.$

In the last inequality, we applied Proposition 2.3 and (4.3), and we utilized
the smoothness of $X_{j_{k}}$. Therefore, according to Proposition 2.3,

$\hat{d}(D_{k}f(x),D_{k}f(x_{k}(t)))\lesssim|D_{k}f(x)-D_{k}f(x_{k}(t))|^{1/s}\lesssim\omega(t)^{1/s}.$

∎

Now, for $k\in\\{1,\dots,k_{0}\\}$, write $a_{k}=D_{k}f(x)$ and
$b_{k}=D_{k}f(x_{k}(t))$. Note that each $|a_{k}|$ and each $|b_{k}|$ is
bounded by a constant depending only on $A$, $f$, $\Omega$, and $\mathbb{G}$
according to $\eqref{e-otherform}$ and the fact that $x_{k}(t)\in\mathcal{A}$.
Hence, the above claim applied to Corollary 2.5 gives the bound

$\displaystyle\hat{d}\left(z(x,\xi),z_{0}(x,\xi;t)\right)=\hat{d}(a_{k_{0}}\cdots
a_{1},b_{k_{0}}\cdots b_{1})\lesssim\omega(t)^{1/s^{k_{0}}}.$

This is estimate (4.4).

It remains to bound $\hat{d}(R(x,\xi;t),z_{0}(x,\xi;t))$ for any $x\in A$ and
$\xi\in K$. Recall that $t\mapsto x\delta_{t}(\xi_{k})$ is a horizontal
$C^{\infty}$ curve in $\mathbb{G}$ for each fixed $x\in\mathbb{G}$. Hence our
assumptions on $f$ imply that the curve $\eta:[0,t_{A}]\to\hat{\mathbb{G}}$
defined as $\eta(t)=f(x\delta_{t}(\xi_{k}))$ is a horizontal curve in
$\hat{\mathbb{G}}$, and $\eta_{i}$ is Euclidean-$C^{1}$ for $i\leq\hat{n}$. We
would like to apply Lemma 3.2 to this curve. Following our prior notation,
denote the “horizontal ray” in $\hat{\mathbb{G}}$ approximating $\eta$ by

$\displaystyle L_{x,k}(t):=L_{\eta}(t)$
$\displaystyle=\eta(0)*\left(t\eta_{1}^{\prime}(0),\dots,t\eta_{\hat{n}}^{\prime}(0),0,\dots,0\right)$
$\displaystyle=f(x)*\left(t\left.\tfrac{d}{d\tau}f_{1}(x\delta_{\tau}(\xi_{k}))\right|_{\tau=0},\dots,t\left.\tfrac{d}{d\tau}f_{\hat{n}}(x\delta_{\tau}(\xi_{k}))\right|_{\tau=0},0,\dots,0\right)$
$\displaystyle=f(x)*\hat{\delta}_{t}(D_{k}f(x)).$

We then have for any $x\in A$ that

(4.7) $\displaystyle\hat{d}\left(R(x,\xi_{k};t),D_{k}f(x)\right)$
$\displaystyle=\hat{d}\left(\hat{\delta}_{1/t}\left(f(x)^{-1}f\left(x\delta_{t}(\xi_{k})\right)\right),\hat{\delta}_{1/t}\left(f(x)^{-1}L_{x,k}(t)\right)\right)$
$\displaystyle=\tfrac{1}{t}\hat{d}\left(f\left(x\delta_{t}(\xi_{k})\right),L_{x,k}(t)\right).$

Claim: There is a constant $C_{2}>0$ depending only on $f$, $A$, $\Omega$, and
$\mathbb{G}$ such that, for all $x\in A$, $k\in\\{1,\dots,k_{0}\\}$, and
$t\in[0,t_{A}]$,

(4.8)
$\displaystyle\hat{d}(f(x_{k}(t)\delta_{t}(\xi_{k})),L_{x_{k}(t),k}(t))\leq
C_{2}t\,\omega(t)^{1/s}.$

Before proving this claim, let us see how it can be used to complete the proof
of the theorem. Estimates (4.7) and (4.8) together give

(4.9)
$\displaystyle\hat{d}\left(R(x_{k}(t),\xi_{k};t),D_{k}f(x_{k}(t))\right)\leq
C_{2}\omega(t)^{1/s}$

for all $x\in A$, $k\in\\{1,\dots,k_{0}\\}$, and $t\in[0,t_{A}]$. We will now
use Corollary 2.5 to conclude (4.5). Fix $x\in A$, and, for each
$k\in\\{1,\dots,k_{0}\\}$, write $a_{k}=R(x_{k}(t),\xi_{k};t)$ and
$b_{k}=D_{k}f(x_{k}(t))$. We may bound $|b_{k}|$ as before, and (4.9) then
provides a bound on $|a_{k}|$ depending only on $f$, $A$, $\Omega$, and
$\mathbb{G}$. Therefore, we can apply (4.9) and Corollary 2.5 to conclude that

$\displaystyle\hat{d}\left(R(x,\xi;t),z_{0}(x,\xi;t)\right)=\hat{d}(a_{k_{0}}\cdots
a_{1},b_{k_{0}}\cdots b_{1})\lesssim\omega(t)^{1/s^{k_{0}}}.$

This is the required bound to complete the proof.

###### Proof of Claim.

Fix $t\in[0,t_{A}]$. Write $z=x_{k}(t)$, and set
$\eta(\kappa)=f(z\delta_{\kappa}(\xi_{k}))$ for $\kappa\in[0,t_{A}]$. We want
to apply Lemma 3.2. In order to do so, it remains to prove that

(4.10)
$|\eta_{i}^{\prime}(\kappa)-\eta_{i}^{\prime}(0)|\lesssim\omega(\kappa)$

for all $\kappa\in[0,t_{A}]$ and $i\leq\hat{n}$. As before, for such $i$,

$\displaystyle\eta_{i}^{\prime}(\kappa)=\left.\tfrac{d}{d\tau}f_{i}(z\delta_{\tau}(\xi_{k}))\right|_{\tau=\kappa}=\nabla
f_{i}(z\delta_{\kappa}(\xi_{k}))\cdot X_{j_{k}}(z)$

so that

$\displaystyle|\eta_{i}^{\prime}(\kappa)-\eta_{i}^{\prime}(0)|$
$\displaystyle=\left|\nabla f_{i}(z\delta_{\kappa}(\xi_{k}))\cdot
X_{j_{k}}(z)-\nabla f_{i}(z)\cdot X_{j_{k}}(z)\right|$
$\displaystyle\leq\left|\nabla f_{i}(z\delta_{\kappa}(\xi_{k}))-\nabla
f_{i}(z)\right||X_{j_{k}}(z)|.$

Note that $z\delta_{\kappa}(\xi_{k})\in\mathcal{A}$, so we have (4.10) by the
definition of $\omega$. Applying Lemma 3.2 proves

$\hat{d}(f(x_{k}(t)\delta_{\kappa}(\xi_{k})),L_{x_{k}(t),k}(\kappa))\lesssim\kappa\,\omega(\kappa)^{1/s}$

for all $\kappa\in[0,t_{A}]$. Choosing specifically $\kappa=t$ completes the
proof of the claim. ∎

This establishes (4.5) and proves the theorem. ∎

## References

  * [BLU07] A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni.  Stratified Lie groups and potential theory for their sub-Laplacians.  Springer Monographs in Mathematics. Springer, Berlin, 2007.
  * [CLZ22] Vasileios Chousionis, Sean Li, and Scott Zimmerman.  Singular integrals on $C^{1,\alpha}$ regular curves in Carnot groups.  J. Anal. Math., 146(1):299–326, 2022.
  * [FS82] G. B. Folland and Elias M. Stein.  Hardy spaces on homogeneous groups, volume 28 of Mathematical Notes.  Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1982.
  * [Gui73] Yves Guivarc’h.  Croissance polynomiale et périodes des fonctions harmoniques.  Bull. Soc. Math. France, 101:333–379, 1973.
  * [Mag05] Valentino Magnani.  The coarea formula for real-valued Lipschitz maps on stratified groups.  Math. Nachr., 278(14):1689–1705, 2005.
  * [Mag13] Valentino Magnani.  Towards differential calculus in stratified groups.  J. Aust. Math. Soc., 95(1):76–128, 2013.
  * [Pan89] Pierre Pansu.  Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un.  Ann. of Math. (2), 129(1):1–60, 1989.
  * [SZ23] Gareth Speight and Scott Zimmerman.  A $C^{m,\omega}$ Whitney extension theorem for horizontal curves in the Heisenberg group.  J. Geom. Anal., 33(6):Paper No. 182, 24, 2023.
  * [War08] Ben Warhurst.  Contact and Pansu differentiable maps on Carnot groups.  Bull. Aust. Math. Soc., 77(3):495–507, 2008.
  * [Zim22] Scott Zimmerman.  Singular integrals on $C_{w^{*}}^{1,\alpha}$ regular curves in Banach duals.  Ann. Funct. Anal., 13(2):Paper No. 32, 24, 2022.