Ahlfors-David regularity of intrinsically quasi-symmetric sections in

metric spaces

DANIELA DI DONATO

Department of Mathematics

University of Pavia

Via Adolfo Ferrata, 5, 27100 Pavia,

ITALY

Abstract: - We introduce a deﬁnition of intrinsically quasi-symmetric sections in metric spaces and we prove the

Ahlfors-David regularity for this class of sections. We follow a recent result by Le Donne and the author where

we generalize the notion of intrinsically Lipschitz graphs in the sense of Franchi, Serapioni and Serra Cassano.

We do this by focusing our attention on the graph property instead of the map one.

Key-Words: - Quasi-symmetric sections, Ahlfors-David regularity, quotient maps, intrinsically Lipschitz

sections, quasi-conformal maps, metric spaces

Received: July 13, 2024. Revised: November 15, 2024. Accepted: December 11, 2024. Published: December 31, 2024.

1 Introduction

The notion of Lipschitz maps is a key one for recti-

ﬁability theory in metric spaces [1] that is a key one

in Calculus of Variations and in Geometric Measure

Theory. The reader can see [2, 3, 4, 5, 6, 7, 8, 9,

10, 11, 12]. On the other hand, in [13] and [14] the

authors prove that the classical Lipschitz deﬁnition

not work in the context of SubRiemannian Carnot

groups [15, 16, 17]. Then in a similar way of Eu-

clidean case, Franchi, Serapioni and Serra Cassano

[18, 19, 20, 21, 22, 23] introduce a suitable deﬁnition

of intrinsic cones which is deep different to Euclidean

cones and then they say that a map ϕis intrinsic Lips-

chitz if for any p∈graph(ϕ)it is possible to consider

an intrinsic cone Cwith vertex on psuch that

C∩graph(ϕ) = /0.(1)

In [24], we generalize this concept in general met-

ric spaces. Roughly speaking, in our new approach a

section ψis such that graph(ϕ) = ψ(Y)⊂Xwhere

Xis a metric space and Yis a topological space.

We prove some important properties as the Ahlfors

regularity, the Ascoli-Arzel´

a Theorem, the Exten-

sion theorem for so-called intrinsically Lipschitz sec-

tions. Following this idea, the author introduce other

two natural deﬁnitions: intrinsically H¨

older sections

[25] and intrinsically quasi-isometric sections [26]

in metric spaces. Yet, thanks to the seminal papers

[27] [28, 29] it is possible to found suitable sets of

this class of sections in order to get the convexity

and vector space over Rand C.Finally, in [30] we

study the link between the continuous sections and

the Hamilton-Jacobi equation.

Following [31], the purpose of this note is to give a

natural intrinsically quasi-symmetric notion and then,

following again [24], to prove the Ahlfors-David reg-

ularity result for this class of sections which includes

intrinsically Lipschitz sections. More precisely, the

main result of this paper is Theorem 2.1.

1.1 Quasi-symmetric sections

Before to give a suitable deﬁnition of quasi-

symmetric sections, we recall the classical notion of

quasi-conformal maps [32, 33, 34, 35, 36]. Let X

and Ybe two metric spaces and let f:Y→Xbe an

homeomorphism (i.e., fand its inverse are continu-

ous maps). For ¯y∈Y,r>0 we deﬁne

Lf(¯y,r):=sup{d(f(¯y),f(y)) :d(¯y,y)≤r},(2)

ℓf(¯y,r):=inf{d(f(¯y),f(y)) :d(¯y,y)≥r},(3)

and the ratio Hf(¯y,r):=Lf(¯y,r)/ℓf(¯y,r)which mea-

sures the eccentricity of the image of the ball B(¯y,r)

under f.We say that fis H-quasiconformal if

limsup

r→0

Hf(¯y,r)≤H,∀¯y∈Y.(4)

A good point of our research is that Yis just a topo-

logical space because, in many cases, we just con-

sider the metric on X.On the other hand, we can not

do a automatically choice of ℓfand the reason will be

clear after to present our setting. We have a metric

space X, a topological space Y, and a quotient map

π:X→Y, meaning continuous, open, and surjective.

The standard example for us is when Xis a metric Lie

group G(meaning that the Lie group Gis equipped

with a left-invariant distance that induces the mani-

fold topology), for example a subRiemannian Carnot

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group, and Yis the space of left cosets G/H, where

H<Gis a closed subgroup and π:G→G/His the

projection modulo H,g7→ gH.

In [24], we consider a section ϕ:Y→Xof π:

X→Y(i.e., π◦ϕ=idY) such that πproduces a foli-

ation for X,i.e., X=π−1(y)and the Lipschitz prop-

erty of ϕconsists to ask that the distance between two

points ϕ(y1),ϕ(y2)is not comparable with the dis-

tance between y1and y2but between ϕ(y1)and the

ﬁber of y2. Following this idea, the corresponding

notion given in 4 becomes

limsup

r→0

Hϕ(¯y,r)≤H,∀¯y∈Y,(5)

where

Lϕ(¯y,r):=sup{d(ϕ(¯y),ϕ(y)) :d(ϕ(¯y),π−1(y)) ≤r},

ℓϕ(¯y,r):=inf{d(ϕ(¯y),ϕ(y)) :d(ϕ(¯y),π−1(y)) ≥r},

and the intrinsic ratio Hf(¯y,r):=Lϕ(¯y,r)/ℓϕ(¯y,r).

Now we can understand why we can not choice

ℓϕ.Indeed, in this case,

r≤d(ϕ(y1),π−1(y2)) ≤d(ϕ(y1),ϕ(y2))

and so

ℓf(¯y,r) = r,∀¯y∈Y.

Because of this, we follow Pansu in [31], and we

give the following non-trivial deﬁnition.

Deﬁnition 1.1 We say that a map ϕ:Y→X is an

intrinsically η-quasi-symmetric section of π, if it is a

section, i.e.,

π◦ϕ=idY,(6)

and if there exists an homeomorphism η:(0,∞)→

(0,∞)(i.e., ηand its inverse are continuous maps)

measuring the intrinsic quasi-symmetry of ϕ.This

means that for any y1,y2,y3∈Y distinct points of Y

which not belong to the same ﬁber, it holds

d(ϕ(y1),ϕ(y2))

d(ϕ(y1),ϕ(y3)) ≤ηd(ϕ(y2),π−1(y1))

d(ϕ(y3),π−1(y1)).(7)

Here d denotes the distance on X, and, as usual, for

a subset A ⊂X and a point x ∈X, we have d(x,A):=

inf{d(x,a):a∈A}.

Equivalently to equation10aprile, we are request-

ing that

d(x1,x2)

d(x1,x3)≤ηd(x2,π−1(π(x1)))

d(x3,π−1(π(x1))),(8)

for all x1,x2,x3∈ϕ(Y)where we ask that x2,x3/∈

π−1(π(x1)).

We give some examples of this class of maps.

Exemple 1.1 (Intrinsically Lipschitz section of π)

Following [24], we say that a map ϕ:Y→X is an

intrinsically Lipschitz section of πwith constant L,

with L ∈[1,∞), if it is a section and

d(ϕ(y1),ϕ(y2)) ≤Ld(ϕ(y1),π−1(y2)),

for all y1,y2∈Y.

Here, η(x) = Lx for every x ∈(0,∞). In fact,

d(ϕ(y1),ϕ(y2))

d(ϕ(y1),ϕ(y3)) =

d(ϕ(y1),ϕ(y2))

d(ϕ(y2),π−1(y1))

d(ϕ(y3),π−1(y1))

d(ϕ(y1),ϕ(y3))

d(ϕ(y2),π−1(y1))

d(ϕ(y3),π−1(y1))

≤Ld(ϕ(y2),π−1(y1))

d(ϕ(y3),π−1(y1)),

where in the last inequality we used the simple fact

ϕ(y1)∈π−1(y1)and so

d(ϕ(y3),π−1(y1))

d(ϕ(y1),ϕ(y3)) ≤1.

Exemple 1.2 (BiLipschitz embedding) BiLipschitz

embedding are examples of intrinsically η-quasi-

symmetric sections of π.This follows because in

the case πis a Lipschitz quotient or submetry

[37, 38], being intrinsically Lipschitz is equivalent to

biLipschitz embedding, (see Proposition 2.4 in [24]).

Exemple 1.3 (Intrinsically H¨

older section of π(in

the discrete case)) Let X be a metric space with the

additional hypothesis that there is ε>0such that

d(ϕ(y1),ϕ(y2)) ≥ε>0for any y1,y2∈Y.Follow-

ing [25], we say that a map ϕ:Y→X is an intrinsi-

cally (L,α)-H¨

older section of π, with L ∈[1,∞)and

α∈(0,1), if it is a section and

d(ϕ(y1),ϕ(y2)) ≤Ld(ϕ(y1),π−1(y2))α,

for all y1,y2∈Y.

Here, η(x) = Lεα−1xαfor any x ∈(0,∞). Indeed,

d(ϕ(y1),ϕ(y2))

d(ϕ(y1),ϕ(y3))

=d(ϕ(y1),ϕ(y2))

d(ϕ(y2),π−1(y1))α

d(ϕ(y3),π−1(y1))α

d(ϕ(y1),ϕ(y3))

d(ϕ(y2),π−1(y1))α

d(ϕ(y3),π−1(y1))α

≤Lεα−1d(ϕ(y2),π−1(y1))α

d(ϕ(y3),π−1(y1))α,

as desired.

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2 Ahlfors-David regularity

Regarding Ahlfors-David regularity in metric setting,

the reader can see [24] for intrinsically Lipschitz sec-

tions; [25] for H¨

older sections; [26] for intrinsically

quasi-isometric sections.

The main result of this paper is the following.

Theorem 2.1 (Ahlfors-David regularity) Let

π:X→Y be a quotient map between a metric space

X and a topological space Y such that there is a

measure µon Y such that for every r0>0and every

x,x′∈X with π(x) = π(x′)there is C >0such that

µ(π(B(x,r))) ≤Cµ(π(B(x′,r))),(9)

for every r ∈(0,r0).

We also assume that ϕ:Y→X is an intrinsically

η-quasi-symmetric section of πsuch that

1. ϕ(Y)is Q-Ahlfors-David regular with respect to

the measure ϕ∗µ, with Q ∈(0,∞)

2. it holds

ℓη:=sup

g,q∈ϕ(Y)π(g)=π(q)

ηd(g,π−1(¯y))

d(q,π−1(¯y))<∞,

(10)

for any ¯y∈Y such that g,q/∈π−1(¯y)

Then, for every intrinsically η-quasi-symmetric

section ψ:Y→X,the set ψ(Y)is Q-Ahlfors-David

regular with respect to the measure ψ∗µ, with Q ∈

(0,∞).

Namely, in Theorem 2.1 Q-Ahlfors-David regularity

means that the measure ϕ∗µis such that for each

point x∈ϕ(Y)there exist r0>0 and C>0 so that

C−1rQ≤ϕ∗µB(x,r)∩ϕ(Y)≤CrQ,(11)

for all r∈(0,r0).

We need to a preliminary result.

Lemma 2.1 Let X be a metric space, Y a topological

space, and π:X→Y a quotient map. If ϕ:Y→X is

an intrinsically η-quasi-symmetric section of πsuch

that equationeta holds, then

π(B(p,r)) ⊂π(B(p,ℓηr)∩ϕ(Y)) ⊂π(B(p,ℓηr)),

(12)

for all p ∈ϕ(Y)and r >0.

Proof 1 Regarding the ﬁrst inclusion, ﬁx p =ϕ(y)∈

ϕ(Y),r>0and q ∈B(p,r)with q =p.

We need to show that π(q)∈π(ϕ(Y)∩B(p,ℓηr)).

Actually, it is enough to prove that

ϕ(π(q)) ∈B(p,ℓηr),(13)

because if we take g :=ϕ(π(q)),then g ∈ϕ(Y)and

π(g) = π(ϕ(π(q))) = π(q)∈π(ϕ(Y)∩B(p,ℓηr)).

Hence using the intrinsic η-quasi-symmetric

property of ϕand 10, we have that for any p =

ϕ(y),q,g∈ϕ(Y)with g =ϕ(π(q)),

d(p,g) = d(p,g)

d(p,q)d(p,q)≤ηd(g,π−1(y))

d(q,π−1(y))r≤ℓηr,

(14)

i.e., 13 holds, as desired. Finally, the second inclu-

sion in 12 follows immediately noting that π(ϕ(Y)) =

Y because ϕis a section and the proof is complete.

At this point, we are able to prove Theorem 2.1.

Proof 2 (Proof of Theorem 2.1)Let ϕand ψintrin-

sically η-quasi-symmetric sections. Fix y ∈Y.By

Ahlfors regularity of ϕ(y),we know that there are

c1,c2,r0>0such that

c1rQ≤ϕ∗µB(ϕ(y),r)∩ϕ(Y)≤c2rQ,(15)

for all r ∈(0,r0).We would like to show that there is

c3,c4>0such that

c4rQ≤ψ∗µB(ψ(y),r)∩ψ(Y)≤c4rQ,(16)

for every r ∈(0,r0).We begin noticing that, by sym-

metry and 2.2

C−1µ(π(B(ψ(y),r))) ≤µ(π(B(ϕ(y),r)))

≤Cµ(π(B(ψ(y),r))).

Moreover,

ψ∗µB(ψ(y),r)∩ψ(Y)=µ(ψ−1B(ψ(y),r)∩ψ(Y))

=µ(πB(ψ(y),r)∩ψ(Y)),

and, consequently,

ψ∗µB(ψ(y),r)∩ψ(Y)

≥µ(π(B(ψ(y),r/ℓη))) ≥C−1µ(π(B(ϕ(y),r/ℓη)))

≥C−1µ(π(B(ϕ(y),r/ℓη)∩ϕ(Y)))

=C−1ϕ∗µB(ϕ(y),r/ℓη)∩ϕ(Y)

≥c1C−1ℓ−Q

ηrQ,

where in the ﬁrst inequality we used the ﬁrst inclusion

of 12 with ψin place of ϕ, and in the second one we

used 2. In the third inequality we used the second

inclusion of 12 and in the fourth one we used 2 with

ϕin place of ψ.Moreover, in a similar way we have

that

ψ∗µB(ψ(y),r)∩ψ(Y)≤µ(π(B(ψ(y),r)))

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≤Cµ(π(B(ϕ(y),r)))

≤Cµ(π(B(ϕ(y),ℓηr)) ∩ϕ(Y)))

=Cϕ∗µB(ϕ(y),ℓηr)∩ϕ(Y)

≤c2CℓQ

ηrQ.

Hence, putting together the last two inequalities we

have that 16 holds with c3=c1C−1ℓ−Q

ηand c4=

c2CℓQ

η.

2.1 Quasi-conformal sections

In this section we present the deﬁnition of quasi-

conformal sections. Regarding the classical quasi-

conformal and quasi-symmetric maps the reader can

see [32, 33, 34, 35].

Deﬁnition 2.1 We say that a map ϕ:Y→X is an

intrinsically η-quasi-conformal section of π, if it is a

section, i.e., π◦ϕ=idY,and there exist H ≥0and

an homeomorphism η:(0,∞)→(0,∞)(i.e., ηsuch

that for any y1,y2,y3∈Y distinct points of Y which

not belong to the same ﬁber, it holds

d(ϕ(y1),ϕ(y2))

d(ϕ(y1),ϕ(y3)) ≤

limsup

x,x′∈ϕ(Y),π(x)=π(x′)x→x′

ηd(x,π−1(y1))

d(x′,π−1(y1))<H.

Here d denotes the distance on X, and, as usual, for

a subset A ⊂X and a point x ∈X, we have d(x,A):=

inf{d(x,a):a∈A}.

Finally, this class of section satisﬁes the hypothe-

sis equationeta of Theorem 2.1. Hence, we can con-

clude with the following corollary.

Theorem 2.2 (Ahlfors-David regularity) Let

π:X→Y be a quotient map between a metric

space X and a topological space Y such that there

is a measure µon Y such that for every r0>0and

every x,x′∈X with π(x) = π(x′)there is C >0

such that µ(π(B(x,r))) ≤Cµ(π(B(x′,r))),for every

r∈(0,r0).

We also assume that ϕ:Y→X is an intrinsically

(η,H)-quasi-conformal section of πsuch that ϕ(Y)

is Q-Ahlfors-David regular with respect to the mea-

sure ϕ∗µ, with Q ∈(0,∞)for some ﬁxed ¯y,¯y1∈Y.

Then, for every intrinsically (η,H)-quasi-

conformal section ψ:Y→X,the set ψ(Y)is

Q-Ahlfors-David regular with respect to the measure

ψ∗µ, with Q ∈(0,∞).

References:

[1] H. Federer, Geometric measure theory,

Die Grundlehren der mathematischen Wis-

senschaften, Band 153 (Springer- Verlag New

York Inc., New York, 1969) pp. xiv+676.

[2] S. Pauls, A notion of rectiﬁability modeled on

Carnot groups, Indiana Univ. Math. J. 53, 49

(2004).

[3] D. Cole and S. Pauls, C1hypersurfaces of the

Heisenberg group are N-rectiﬁable, Houston J.

Math. 32, 713 (2006).

[4] D. Bate, Characterising rectiﬁable metric spaces

using tangent spaces, Accepted Invent. math.

(2021).

[5] G. Antonelli and A. Merlo, On rectiﬁable mea-

sures in Carnot groups: existence of density, Ac-

cepted in Journal of Geometric Analysis (2022).

[6] G. Antonelli and A. Merlo, On rectiﬁable mea-

sures in Carnot groups: Marstrand–Mattila recti-

ﬁability criterion, Accepted in Journal of Func-

tional Analysis (2022).

[7] P. Mattila, Geometry of sets and measures in Eu-

clidean spaces, Cambridge Studies in Advanced

Mathematics, vol. 44, p. 343. Cambridge Univer-

sity Press, Cambridge (1995).

[8] J. Marstrand, Hausdorff two-dimensional mea-

sure in 3- space, Proc. London Math. Soc. (3) 11,

91–108 (1961).

[9] P. Mattila, Hausdorff m regular and rectiﬁable

sets in n-space, Trans. Amer. Math. Soc. 205,

263–274 (1975).

[10] A. Naor and R. Young, Vertical perimeter versus

horizontal perimeter, Ann. of Math., (2) 188 , 171

(2018).

[11] G. David and S. Semmes, Singular integrals and

rectiﬁable sets in Rn: Beyond lipschitz graphs,

ast´

erisque, , 171 (1991).

[12] G. David and S. Semmes, Analysis of and on

uniformly rectiﬁable sets, Mathematical Surveys

and Monographs 38 (1993).

[13] L. Ambrosio and B. Kirchheim, Rectiﬁable sets

in metric and Banach spaces, Math. Ann. 318,

527 (2000).

[14] V. Magnani, Unrectiﬁability and rigidity in

stratiﬁed groups, Arch. Math. 83(6), 568 (2004).

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.102

Daniela Di Donato

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995

Volume 23, 2024

[15] A. Agrachev, D. Barilari, and U. Boscain, A

comprehensive introduction to sub-Riemannian

geometry, Cambridge Studies in Advanced Math-

ematics, Cambridge Univ. Press 181, 762 (2019).

[16] A. Bonﬁglioli, E. Lanconelli, and F. Uguzzoni,

Stratiﬁed Lie groups and potential theory for their

sub-Laplacians, Springer Monographs in Mathe-

matics (Springer, Berlin, 2007) pp. xxvi+800.

[17] L. Capogna, D. Danielli, S. D. Pauls, and J. T.

Tyson, An introduction to the Heisenberg group

and the sub- Riemannian isoperimetric problem,

Progress in Mathematics, Vol. 259 (Birkh¨

auser

Verlag, Basel, 2007) pp. xvi+223.

[18] B. Franchi, R. Serapioni, and F. Serra Cassano,

Rectiﬁability and perimeter in the Heisenberg

group, Math. Ann. 321, 479 (2001).

[19] B. Franchi, R. Serapioni, and F. Serra Cassano,

On the structure of ﬁnite perimeter sets in step 2

Carnot groups, The Journal of Geometric Analy-

sis 13, 421 (2003).

[20] B. Franchi, R. Serapioni, and F. Serra Cas-

sano, Regular hypersurfaces, intrinsic perimeter

and implicit function theorem in Carnot groups,

Comm. Anal. Geom. 11, 909 (2003).

[21] F. Serra Cassano, Some topics of geometric

measure theory in Carnot groups, in Geometry,

analysis and dynamics on sub-Riemannian man-

ifolds. Vol. 1, EMS Ser. Lect. Math. (Eur. Math.

Soc., Z¨

urich, 2016) pp. 1–121.

[22] B. Franchi and R. P. Serapioni, Intrinsic Lip-

schitz graphs within Carnot groups, J. Geom.

Anal. 26, 1946 (2016).

[23] B. Franchi, R. Serapioni, and F. Serra Cassano,

Intrinsic Lipschitz graphs in Heisenberg groups,

J. Nonlinear Convex Anal. 7, 423 (2006).

[24] D. Di Donato and E. Le Donne, Intrinsically

Lipschitz sections and applications to metric

groups, accepted to Communications in Contem-

porary Mathematics (2024).

[25] Di Donato, Daniela, Intrinsically H¨

older sec-

tions in metric spaces, to appear WSEAS Trans-

actions on Mathematics, (2024).

[26] Di Donato, Daniela, Intrinsically quasi-

isometric sections in metric spaces, preprint

(2022).

[27] J. Cheeger, Differentiability of Lipschitz func-

tions on metric measure spaces, Geom. Funct.

Anal. 9 , 428 (1999).

[28] S. Keith, A differentiable structure for metric

measure spaces, Adv. Math. 183 , 271 (2004).

[29] B. Kleiner and J. Mackay, Differentiable struc-

tures on metric measure spaces: a primer, Ann.

Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XVI , 41

(2016).

[30] Di Donato D., The intrinsic Hopf-Lax semi-

group vs. the intrinsic slope, Journal of Mathe-

matical Analysis and Applications 523 (2023) Is-

sue 2, 127051

[31] P. Pansu, On the quasisymmetric H¨

older equiv-

alence problem for Carnot groups. Annales de la

facolt´

e des sciences de Toulouse Math´

ematiques,

2016.

[32] L.V. Ahlfors, Lectures on quasiconformal map-

pings, University Lecture Series. Amer. Math.

Soc., Providence, RI, 38, 2006.

[33] J. Heinonen. Lectures on Analysis on Metric

Spaces, Springer-Verlag. 2001.

[34] J. Heinonen and P. Koskela. Quasiconformal

maps in metric spaces with controlled geometry.

Acta Mathematica, 181(1):1 – 61, 1998.

[35] T. Iwaniec and G. Martin. Geometric function

theory and non-linear analysis. Oxford Mathe-

matical Monographs, Clarendon Press, Oxford

University Press, 2001.

[36] P. Tukia and J. Va¨

ıs¨

al¨

a, Quasisymmetric embed-

dings of metric spaces. Ann. Acad. Sci. Fenn. Ser.

A I Math. 5, pages 97–114, 1980.

[37] S. Bates, W. B. Johnson, J. Lindenstrauss, D.

Preiss, and G. Schechtman, Afﬁne approximation

of Lipschitz functions and nonlinear quotients,

Geom. Funct. Anal. 9, 1092 (1999).

[38] V. N. Berestovskii, Homogeneous manifolds

with intrinsic metric, Sib Math J I, 887 (1988).

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Sources of Funding for Research Presented in a

Scientiﬁc Article or Scientiﬁc Article Itself

D.D.D. is supported by the Italian MUR through

the PRIN 2022 project “Inverse problems in PDE:

theoretical and numerical analysis”, project code

2022B32J5C, under the National Recovery and Re-

silience Plan (PNRR), Italy, funded by the European

Union - Next Generation EU, Mission 4 Component

1 CUP F53D23002710006.

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The author contributed in the present research, at all

stages from the formulation of the problem to the

final findings and solution.

Conflict of Interest

The author has no conflict of interest to declare that

is relevant to the content of this article.

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