Intrinsically H¨

older sections in metric spaces

DANIELA DI DONATO

Department of Mathematics

University of Pavia

Via Adolfo Ferrata, 5, 27100 Pavia,

ITALY

Abstract: - We introduce the notion of intrinsically H¨

older graphs in metric spaces that generalized the one of

intrinsically Lipschitz sections. This concept is relevant because it has many properties similar to H¨

older maps but

is profoundly different from them. We prove some relevant results as the Ascoli-Arzel`

a compactness Theorem,

Ahlfors-David regularity and the Extension Theorem for this class of sections. In the ﬁrst part of this note, thanks

to Cheeger theory, we deﬁne suitable sets in order to obtain a vector space over Ror C,a convex set and an

equivalence relation for intrinsically H¨

older graphs. These last three properties are new also in the Lipschitz case.

Key-Words: - H¨

older graphs, Ahlfors-David regularity, Extension theorem, Ascoli-Arzel`

a compactness theorem,

vector space, equivalence relation, Metric spaces

Received: April 17, 2024. Revised: September 11, 2024. Accepted: October 4, 2024. Published: October 25, 2024.

1 Introduction

Starting to the seminal papers [1, 2, 3] (see also

[4, 5]), in [6] we generalize the notion of intrinsi-

cally Lipschitz maps introduced in subRiemannian

Carnot groups [7, 8, 9]. This concept was intro-

duced in order to give a good deﬁnition of rectiﬁ-

ability in subRiemannian geometry after the nega-

tive result shown in [10] (see also [11]) regarding

the classical deﬁnition of rectiﬁability using Lips-

chitz maps given by [12]. The notion of rectiﬁ-

able sets is a key one in Calculus of Variations and

in Geometric Measure Theory. The reader can see

[13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

In this paper, we give a new natural deﬁnition

of intrinsically H¨

older sections (see Deﬁnition 2.1)

which includes intrinsically Lipschitz ones. More

precisely, this paper differs from the related ones be-

cause the settings are metric spaces which are more

general than Carnot groups and because intrinsically

H¨

older maps has many properties similar to H¨

older

maps but are profoundly different from them (see [4,

Example 4.58]). There are two main reasons for the

importance of these mappings. First, their deﬁni-

tion is extremely simple and widely applicable, and

so they can be found in abundance on any metric

space without any assumptions of smoothness. Sec-

ond, despite the the simplicity of their deﬁnition, they

often possess many rigidity properties and therefore

their study can yield surprising analytic and geomet-

ric conclusions.

We prove the following results using basic mathe-

matical tools.

1. Theorem 2.1, i.e., Compactness Theorem a l´

Ascoli-Arzel`

a for the intrinsically H¨

older sec-

tions.

2. Theorem 2.2, i.e., Ahlfors-David regularity for

the intrinsically H¨

older sections.

3. Proposition 3.3 states that the class of the intrin-

sically H¨

older sections is a convex set.

4. Theorem 3.1 states that a suitable class of the in-

trinsically H¨

older sections is a vector space over

Ror C.

5. Theorem 4.1 gives an equivalence relation for a

suitable class of the intrinsically H¨

older sections.

6. Theorem 5.1, i.e., Extension Theorem for the in-

trinsically H¨

older sections.

The points (3)−(4)−(5)are new results also in

the context of Lipschitz sections.

2 Intrinsically H¨

older sections:

deﬁnition and basic properties

Deﬁnition 2.1 (Intrinsic H¨

older section) Let (X,d)

be a metric space and let Y be a topological space.

We say that a map ϕ:Y→X is a section of a quotient

map π:X→Y if π◦ϕ=idY.Moreover, we say

that ϕis an intrinsically (L,α)-H¨

older section with

constant L >0and α∈(0,1)if in addition

d(ϕ(y1),ϕ(y2)) ≤Ld(ϕ(y1),π−1(y2))α+d(ϕ(y1),π−1(y2)),

(1)

for all y1,y2∈Y.

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Equivalently, we are requesting that d(x1,x2)≤

Ld(x1,π−1(π(x2)))α+d(x1,π−1(π(x2))),for all

x1,x2∈ϕ(Y).

The standard example for us is when Xis a met-

ric Lie group G(meaning that the Lie group Gis

equipped with a left-invariant distance that induces

the manifold topology), for example a subRieman-

nian Carnot 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.

When α=1,a section ϕis intrinsically Lipschitz

in the sense of [6]. Moreover, we underline that, in

the case α=1 and πis a Lipschitz quotient or sub-

metry [24, 25], the results trivialize, since in this case

being intrinsically Lipschitz is equivalent to biLips-

chitz embedding, see Proposition 2.4 in [6].

We further rephrase the deﬁnition as saying that

ϕ(Y), which we call the graph of ϕ, avoids some par-

ticular sets (which depend on α,Land ϕitself):

Proposition 2.1 Let π:X→Y be a quotient map

between a metric space and a topological space,

ϕ:Y→X be a section of π,α∈(0,1)and L >0.

Then ϕis intrinsically (L,α)-H¨

older if and only if

ϕ(Y)∩Rx,L=/0,for all x ∈ϕ(Y),where Rx,L:=

{x′∈X|

Ld(x′,π−1(π(x)))α+d(x′,π−1(π(x))) <d(x′,x)}.

Proposition 2.1 is a triviality, still its purpose is to

stress the analogy with the intrinsically Lipschitz sec-

tions theory introduced in [6] when α=1. In partic-

ular, the sets Rx,Lare the intrinsic cones in the sense

of Franchi, Serapioni and Serra Cassano when Xis a

subRiemannian Carnot group and α=1.

Deﬁnition 2.1 it is very natural if we think that

what we are studying graphs of appropriate maps.

However, in the following proposition, we introduce

an equivalent condition of (1)when Yis a compact

set.

Proposition 2.2 Let π:X→Y be a quotient map be-

tween a metric space X and a topological and com-

pact space Y and let α∈(0,1).The following are

equivalent:

1. there is L >0such that d(ϕ(y1),ϕ(y2)) ≤

Ld(ϕ(y1),π−1(y2))α+d(ϕ(y1),π−1(y2)),for

all y1,y2∈Y.

2. there is K ≥1such that

d(ϕ(y1),ϕ(y2)) ≤Kd(ϕ(y1),π−1(y2))α,(2)

for all y1,y2∈Y.

(1)⇒(2).This is trivial when

d(ϕ(y1),π−1(y2)) ≤1. On the other hand,

if we consider y1,y2∈Yand ¯x∈Xsuch that

d(ϕ(y1),π−1(y2)) = d(ϕ(y1),¯x)>1,then it is pos-

sible to consider ℓequidistant points x1,...,xℓ∈X

such that

d(ϕ(y1),¯x) = d(ϕ(y1),x1)+

ℓ−1

∑

i=1

d(xi,xi+1)+ d(xℓ,¯x),

with d(ϕ(y1),x1) = d(xi,xi+1) = d(xℓ,¯x)∈(1

2,1).

Here, ℓ≤ ⌊d(ϕ(y1),¯x)⌋+1 depends on y1,y2and ⌊k⌋

denotes the integer part of k.However, it is possible

to choose k∈R+deﬁned as

k:=sup

y1,y2∈Y

d(ϕ(y1),π−1(y2)),(3)

such that knot depends on the points and ℓ≤

k.We notice that this constant is ﬁnite be-

cause, on the contrary, we get the contradiction

∞=d(ϕ(y1),π−1(y2)) ≤d(ϕ(y1),ϕ(y2)).Hence,

d(ϕ(y1),ϕ(y2)) ≤Ld(ϕ(y1),¯x)α+d(ϕ(y1),¯x)

=Ld(ϕ(y1),¯x)α

+d(ϕ(y1),x1) + ∑ℓ−1

i=1d(xi,xi+1) + d(xℓ,¯x)

≤Ld(ϕ(y1),¯x)α

+d(ϕ(y1),x1)α+∑ℓ−1

i=1d(xi,xi+1)α+d(xℓ,¯x)α

≤(L+3(⌊k⌋+1))d(ϕ(y1),¯x)α

=:Kd(ϕ(y1),¯x)α.

(2)⇒(1).This is a trivial implication.

Deﬁnition 2.2 (Intrinsic H¨

older with respect to a section)

Given sections ϕ,ψ:Y→X of π. We say that ϕis

intrinsically (L,α)-H¨

older with respect to ψat point

ˆx, with L >0,α∈(0,1)and ˆx∈X , if

1. ˆx∈ψ(Y)∩ϕ(Y);

2. ϕ(Y)∩Cψ

ˆx,L=/0,

where Cψ

ˆx,L:={x∈X:

d(x,ψ(π(x))) >Ld(ˆx,ψ(π(x)))α+d(ˆx,ψ(π(x)))}.

Remark 1 Deﬁnition 2.2 can be rephrased as fol-

lows. A section ϕis intrinsically (L,α)-H¨

older with

respect to ψat point ˆx if and only if there is ˆy∈Y

such that ˆx=ϕ(ˆy) = ψ(ˆy)and

d(x,ψ(π(x))) ≤Ld(ˆx,ψ(π(x)))α+d(ˆx,ψ(π(x))),

(4)

for all x ∈ϕ(Y), which equivalently means

d(ϕ(y),ψ(y)) ≤Ld(ψ(ˆy),ψ(y))α+d(ψ(ˆy),ψ(y)),

(5)

for all y ∈Y.

Notice that Deﬁnition 2.2 does not induce an

equivalence relation because of lack of symmetry in

the right-hand side of (5). In Section 4 we give a

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stronger deﬁnition in order to obtain an equivalence

relation.

Finally, following Cheeger theory [26] (see also

[27, 28]), we give another equivalent property of

H¨

older section. Here it is fundamental that Yis a

compact set.

Proposition 2.3 Let X be a metric space, Y a topo-

logical and compact space, π:X→Y a quotient

map, L >0and α,β,γ∈(0,1). Assume that every

point x ∈X is contained in the image of an intrinsic

(L,α)-H¨

older section ψxfor π. Then for every sec-

tion ϕ:Y→X of πthe following are equivalent:

1. for all x ∈ϕ(Y)the section ϕis intrinsically

(L1,β)-H¨

older with respect to ψxat x;

2. the section ϕis intrinsically (L2,γ)-H¨

older.

The proof of the last statement is an immediately

consequence of the following result.

Proposition 2.4 Let X be a metric space, Y a topo-

logical and compact space, and π:X→Y a quotient

map. Let L >0and y0∈Y . Assume ϕ0:Y→X is an

intrinsically (L,α)-H¨

older section of π. Let ϕ:Y→

X be a section of πsuch that x0:=ϕ(y0) = ϕ0(y0).

Then the following are equivalent:

1. For some L1>0and β∈(0,1),ϕis intrinsically

(L1,β)-H¨

older with respect to ϕ0at x0;

2. For some L2≥1and γ∈(0,1),ϕsatisﬁes

d(x0,ϕ(y)) ≤L2d(x0,π−1(y))γ,∀y∈Y.(6)

Moreover, the constants L1and L2are quantitatively

related in terms of L.

We begin recall that, by Proposition 2.2, (1)is

equivalent to (2).

(1)⇒(2).For every y∈Y,it follows that

d(ϕ(y),x0)≤d(ϕ(y),ϕ0(y)) + d(ϕ0(y),x0)

≤L1d(ϕ0(y),x0)β+d(ϕ0(y),x0)

≤L1Ld(x0,π−1(y))β α +Ld(x0,π−1(y))α

≤L(L1+1)max{d(x0,π−1(y))β α ,d(x0,π−1(y))α}

where in the ﬁrst inequality we used the triangle

inequality, and in the second one the intrinsic H¨

older

property (1) of ϕ.Then, in the third inequality we

used the intrinsic H¨

older property of ϕ0.Here, notic-

ing that β α <α,we have that if d(x0,π−1(y)) ≤1,

then max{d(x0,π−1(y))β α ,d(x0,π−1(y))α}=

d(x0,π−1(y))β α .On the other hand, if

d(x0,π−1(y)) >1,then using a similar technique

using in Proposition 2.2 we obtain the same maxi-

mum with additional constant K:=L+3(⌊k⌋+1)

where k∈Ris given by costuniversale. Deﬁnitely,

d(ϕ(y),x0)≤LK(L1+1)d(x0,π−1(y))β α .

(2)⇒(1).For every y∈Y,we have that

d(ϕ(y),ϕ0(y)) ≤d(ϕ(y),x0) + d(x0,ϕ0(y))

≤L2d(ϕ0(y),x0)γ+d(ϕ0(y),x0),where in the ﬁrst

equality we used the triangle inequality, and in the

second one we used (6).

It is easy to see that if α=1,then we get β=γ

and so we have the following corollary.

Corollary 2.1 Let X be a metric space, Y a topolog-

ical and compact space, π:X→Y a quotient map,

L≥1and β∈(0,1). Assume that every point x ∈X is

contained in the image of an intrinsically L-Lipschitz

section ψxfor π. Then for every section ϕ:Y→X of

πthe following are equivalent:

1. for all x ∈ϕ(Y)the section ϕis intrinsically

(L1,β)-H¨

older with respect to ψxat x;

2. the section ϕis intrinsically (L2,β)-H¨

older.

2.1 Continuity

An intrinsically (L,α)-H¨

older section ϕ:Y→X

of πis a continuous section. Indeed, ﬁx a point

y∈Y. Since πis open, for every ε∈(0,1)and

every x∈Xsuch that x=ϕ(y)we know that

there is an open neighborhood Uεof π(x) = ysuch

that Uε⊂π(B(x,[ε/(L+1)]1/α)).Hence, if y′∈

Uεthen there is x′∈B(x,[ε/(L+1)]1/α)such that

π(x′) = y′. That means x′∈π−1(y′)and, con-

sequently, d(ϕ(y),ϕ(y′)) ≤Ld(ϕ(y),π−1(y′))α+

d(ϕ(y),π−1(y′))

≤(L+1)d(x,x′)α

≤ε,i.e., ϕ(Uε)⊂B(x,ε).

2.2 An Ascoli-Arzel`

a compactness theorem

Similar to the Lipschitz case, we have the following

theorem a l´

a Ascoli-Arzel´

Theorem 2.1 (Equicontinuity and Compactness Theorem)

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

space X for which closed balls are compact and a

topological space Y . Then,

(i) For all K′⊂Y compact, L ≥1,α∈(0,1),K⊂X

compact, and y0∈Y the set A0:={ϕ|K′:K′→X|ϕ:

Y→X is intrinsically (L,α)-H¨

older section

of π,ϕ(y0)∈K}is equibounded, equicontinuous,

and closed in the uniform convergence topology.

(ii) For all L ≥1,α∈(0,1),K⊂X compact, and

y0∈Y the set {ϕ:Y→X:ϕis intrinsically

(L,α)-H¨

older section of π,ϕ(y0)∈K}is compact

with respect to the topology of uniform convergence

on compact sets.

The proof is similar to the one of [6, Theorem

1.2].

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2.3 Ahlfors-David regularity

Following again [6], we can prove an Ahlfors-David

regularity for the intrinsically H¨

older sections. Re-

call that in Euclidean case Rs,there are (L,α)-H¨

older

maps such that the (s+1−α)-Hausdorff measure

of their graphs is not zero and the (s+1)-Hausdorff

measure of their graphs is zero, we give the following

result.

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 >0such that

µ(π(B(x,r))) ≤Cµ(π(B(x′,r))),∀r∈(0,r0).

(7)

Let ℓ∈(0,∞).We also assume that there is an in-

trinsically (L,α)-H¨

older section ϕ:Y→X of πsuch

that ϕ(Y)is locally (ℓ+1−α)-Ahlfors-David regu-

lar with respect to the measure ϕ∗µ.

Then, for every intrinsically (L,α)-H¨

older sec-

tion ψ:Y→X of π,the set ψ(Y)is locally Q-

Ahlfors-David regular with respect to the measure

ψ∗µ,where Q =α(ℓ+1−α)when the radius of the

balls is small than 1and Q =ℓ+1−αwhen the ra-

dius of the balls is larger than 1.

Namely, locally 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,f orallr ∈

(0,r0).The same inequality will hold for ψ∗µwith a

possibly different value of Cand Q.

The proof of this statement is similar to the one

of [6, Theorem 1.3]. We notice that in the Lips-

chitz case we use [6, Proposition 2.12 (iii)]; here,

the corresponding result is the following one: Let

Xbe a metric space, Ya topological space, and

π:X→Ya quotient map. If ϕ:Y→Xis an intrin-

sically (L,α)-H¨

older section of πwith α∈(0,1)and

L>0,then π(B(p,r)) ⊂π(B(p,Lrα+r)∩ϕ(Y)) ⊂

π(B(p,Lrα+r)),for all p∈ϕ(Y)and r>0.

3 Properties of linear and quotient

map

In order to give some relevant properties as convexity

and being vector space over Rwe need to ask that π

is also a linear map. Notice that this fact is not too

restrictive because in our idea πis the ’usual’ projec-

tion map. More precisely, throughout the section we

will consider πa linear and quotient map between a

normed space Xand a topological space Y.

3.1 Basic properties

In this section we give two simple results in the par-

ticular case when πis a linear map.

Proposition 3.1 Let π:X→Y be a linear and quo-

tient map between normed spaces X and Y.The set of

all section of πis a convex set.

Fix t∈[0,1]and let ϕ,ψ:Y→Xsections of π.By

the simply fact π(tϕ(y)+(1−t)ψ(y)) = tπ(ϕ(y))+

(1−t)π(ψ(y)) = y,we get the thesis.

Proposition 3.2 Let π:X→Y be a linear and quo-

tient map between normed spaces X and Y.If ϕ:Y→

X is an intrinsically H¨

older section of π,then for any

λ∈R− {0}the section λ ϕ is also intrinsic H¨

older

for 1/λ π with the same Lipschitz constant up to the

constant |λ|1−α.

Fix λ∈R− {0}.The fact that λ ϕ is a sec-

tion is trivial using the similar argument of Propo-

sition 3.1. On the other hand, for any y1,y2∈Y

∥λ ϕ(y1)−λ ϕ (y2)∥≤|λ|Ld(ϕ(y1),π−1(y2))α

=|λ|1−αLd(λ ϕ(y1),(1/λ π)−1(y2))α,i.e., the the-

sis holds. This fact follows by these observations:

1. if d(ϕ(y1),π−1(y2)) = d(ϕ(y1),a)then

|λ|αd(ϕ(y1),π−1(y2))α=∥λ ϕ(y1)−λa∥α.

2. λa∈π−1(λy).

3. π−1(λy) = (1/λ π)−1(y).

The second point is true because using the linear-

ity of πwe have that π(λa) = λ π(a) = λy.Finally,

the third point holds because π−1(λy) = {x∈X:

π(x) = λy}

={x∈X: 1/λ π(x) = y}

= (1/λ π)−1(y),as desired.

3.2 Convex set

In this section we show that the set of all intrinsically

H¨

older sections is a convex set. We underline that the

hypothesis on boundness of Yis not necessary.

Deﬁnition 3.1 (Intrinsic H¨

older set with respect to ψ)

Let α∈(0,1]and ψ:Y→X a section of π. We

deﬁne the set of all intrinsically H¨

older sections of π

with respect to ψat point ˆx as H ψ,ˆx,α:={ϕ:Y→

X a section of π:ϕis intrinsically

(˜

L,α)-H¨

older w.r.t. ψat point ˆx for some ˜

L>0}.

Proposition 3.3 Let π:X→Y be a linear and quo-

tient map between normed spaces X and Y.Assume

also that α∈(0,1],ψ:Y→X a section of πand

ˆx∈ψ(Y).Then, the set Hψ,ˆx,αis a convex set.

Let ϕ,η∈Hψ,ˆx,αand let t∈[0,1].We want to

show that

w:=tϕ+ (1−t)η∈Hψ,ˆx,α.

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Notice that, by Proposition 3.1, wis a section of πand

w(¯y) = ϕ(¯y) = η(¯y) = ˆxfor some ¯y∈Y.On the other

hand, for every y∈Ywe have ∥w(y)−ψ(y)∥=

∥t(ϕ(y)−ψ(y)) + (1−t)(η(y)−ψ(y))∥,and so

∥w(y)−ψ(y)∥ ≤ t∥ϕ(y)−ψ(y)∥+ (1−t)∥η(y)−

ψ(y)∥.Hence, d(w (y),ψ(y)) ≤tLϕd(ψ(¯y),ψ(y))α

+ (1−t)Lψd(ψ(¯y),ψ(y))α+d(ψ(¯y),ψ(y))

= [t(Lϕ−Lψ) + Lψ]d(ψ(¯y),ψ(y))α

+d(ψ(¯y),ψ(y)),

for every y∈Y,as desired.

3.3 Vector space

In this section we show that a ’large’ class of intrin-

sically H¨

older sections is a vector space over Ror C.

Notice that it is no possible to obtain the statement for

Hψ,ˆx,αsince the simply observation that if ψ(ˆy) = ˆx

then ψ(ˆy) + ψ(ˆy)=ˆx.

Theorem 3.1 Let π:X→Y is a linear and quotient

map between normed spaces X and Y.Assume also

that ψ:Y→X is a section of π.

Then, for any α∈(0,1],the set Sλ∈R+Hλ ψ,λˆx,α∪

{0}is a vector space over Ror C.

Let ϕ,η∈Sλ∈R+Hλ ψ ,λˆx,αand β∈R− {0}.We

want to show that

1. w=ϕ+η∈Sλ∈R+Hλ ψ ,λˆx,α.

2. β ϕ ∈Sλ∈R+Hλ ψ,λˆx,α.

(1). If ϕ∈Hδ1ψ,δ1ˆx,αand η∈Hδ2ψ,δ2ˆx,αfor some

δ1,δ2∈R+it holds

w∈H(δ1+δ2)ψ,(δ1+δ2)ˆx,α.

For simplicity, we choose ϕ,η∈Hψ,ˆx,αand so it re-

mains to prove

w∈H2ψ,2 ˆx,α.

By linear property of π,wis a section of 1/2π.

On the other hand, if ψ(¯y) = ˆx,then w(¯y) = ϕ(¯y) +

η(¯y) = 2ψ(¯y)∈X.Moreover, using (5), we deduce

∥w(y)−2ψ(y)∥=∥ϕ(y) + η(y)−2ψ(y)∥

≤ ∥ϕ(y)−ψ(y)∥+∥η(y)−ψ(y)∥

≤2max{Lϕ,Lη}∥ψ(¯y)−ψ(y)∥α

+2∥ψ(¯y)−ψ(y)∥

=21−αmax{Lϕ,Lη}∥2ψ(¯y)−2ψ(y)∥α

+∥2ψ(¯y)−2ψ(y)∥,for any y∈Y,as desired.

(2). Let β∈R− {0}and ϕ∈Hψ,ˆx,α.By lin-

ear property of π,β ϕ is a section of 1/β π.On the

other hand, β ϕ(¯y) = β ψ(¯y) = βˆx∈X.Moreover, us-

ing (5), we deduce ∥β ϕ (y)−β ψ(y)∥=|β|∥ϕ(y)−

ψ(y)∥

≤ |β|Lϕ∥ψ(¯y)−ψ(y)∥α

+|β|∥ψ(¯y)−ψ(y)∥

=Lϕ∥β ψ(¯y)−β ψ (y)∥α

+∥β ψ(¯y)−β ψ (y)∥,

for any y∈Y.Hence β ϕ ∈Hβ ψ,βˆx,αand the proof is

complete.

Remark 2 Theorem 3.1 holds also if we consider

λ∈R−instead of R+.

3.4 Examples

In this section, πis a linear map. Here, we present

some examples of linear sections and intrinsically

Lipschitz sections.

1. Let the general linear group X=GL(n,R)or

X=GL(n,C)of degree nwhich is the set of

n×ninvertible matrices, together with the op-

eration of ordinary matrix multiplication. We

consider Y=R∗=GL(n,R)/SL(n,R)or Y=

C∗=GL(n,C)/SL(n,C)where the special lin-

ear group SL(n,R)(or SL(n,C)) is the subgroup

of GL(n,R)(or GL(n,C)) consisting of matri-

ces with determinant of 1. Here the linear map

π=det :GL(n,R)→R∗is a surjective homo-

morphism where Ker(π) = SL(n,R).

2. Let X=GL(n,R)as above and Y=

GL(n,R)/O(n,R)where O(n,R)is the or-

thogonal group in dimension n.Recall that Yis

diffeomorphic to the space of upper-triangular

matrices with positive entries on the diagonal,

the natural map π:X→Yis linear.

3. Let X=R2,Y=Rand π:R2→Rdeﬁned as

π((x1,x2)) :=x1+x2for any (x1,x2)∈R2.An

easy example of sections of πis the following

one: let ϕ:R→R2given by ϕ(y)=(by +

a f (y),(1−b)y−a f (y)),∀y∈R,where a,b∈

Rand f:R→Ris a continuous map.

4. Let X=R2κ,Y=Rand π:R2κ→Rde-

ﬁned as π((x1,...,x2κ)) :=x1+.. . +x2κfor any

(x1,...,x2κ)∈R2κ.An easy example of sections

of πis the following one: let ϕ:R→R2κgiven

by ϕ(y) =

(y+a1f1(y),−a1f1(y),a2f2(y),−a2f2(y),

.. .,aκfκ,−aκfκ),for all y∈R, where ai∈R

and fi:R→Rare continuous maps for any

i=1,...,κ.

5. Regarding examples of intrinsically Lipschitz

sections the reader can see [4, Example 4.58].

4 An equivalence relation

In this section Xis a metric space, Ya topological

space and π:X→Ya quotient map (we do not ask

that πis a linear map). We stress that Deﬁnition 2.2

does not induce an equivalence relation, because of

lack of symmetry in the right-hand side of (5). As

a consequence we must ask a stronger condition in

order to obtain an equivalence relation.

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Deﬁnition 4.1 [Intrinsic H¨

older with respect to a

section in strong sense] Given sections ϕ,ψ:Y→X

of π. We say that ϕis intrinsically (L,α)-H¨

older

with respect to ψat point ˆxin strong sense, with

L>0,α∈(0,1]and ˆx∈X, if

1. ˆx∈ψ(Y)∩ϕ(Y);

2. it holds d(ϕ(y),ψ(y)) ≤

min{Ld(ψ(ˆy),ψ(y))α+d(ψ(ˆy),ψ(y)),

Ld(ψ(ˆy),ϕ(y))α+d(ψ(ˆy),ϕ(y))},

for every y ∈Y.

Now we are able to give the main theorem.

Theorem 4.1 Let α∈(0,1]and π:X→Y be a quo-

tient map from a metric space X to a topological

space Y.Assume also that ψ:Y→X is a section of

πand ˆx∈X.Then, being intrinsically H¨

older with

respect to ψat point ˆx in strong sense induces an

equivalence relation. We will write the class of equiv-

alence of ψat point ˆx as [Hψ,ˆx,α]:={ϕ:Y→

X a section of π:

ϕis intrinsically (˜

L,α)-H¨

older with respect to

ψat point ˆx in strong sense, for some ˜

L>0}.

An interesting observation is that, considering

Hψ,ˆx,α,the intrinsic constants Lcan be change but

it is fundamental that the point ˆxis a common one for

the every section.

We need to show:

1. reﬂexive property;

2. symmetric property;

3. transitive property.

(1). It is trivial that ϕ∽ϕ.

(2). If ϕ∽ψ,then ψ∽ϕ.This follows from Def-

inition 4.1.

(3). We know that ϕ∽ψand ψ∽η.Hence,

ˆx=ϕ(ˆy) = ψ(ˆy) = η(ˆy).Moreover, by Deﬁnition

4.1, it holds d(ϕ(y),ψ(y)) ≤min{L1d(ψ(ˆy),ψ(y))α

+d(ψ(ˆy),ψ(y)),L1d(ψ(ˆy),ϕ(y))α

+d(ψ(ˆy),ϕ(y))},

d(ψ(y),η(y)) ≤min{L2d(η(ˆy),η(y))α

+d(η(ˆy),η(y)),L2d(η(ˆy),ψ(y))α

+d(η(ˆy),ψ(y))},

for any y∈Yand, consequently, if ˜

2max{L1,L2},then d(ϕ(y),ψ(y)) ≤d(ϕ(y),ψ(y))+

d(ψ(y),η(y))

≤min{˜

Ld(η(ˆy),η(y))α+d(η(ˆy),η(y)),

Ld(ψ(ˆy),ϕ(y))α+d(ψ(ˆy),ϕ(y))},

=min{˜

Ld(η(ˆy),η(y))α+d(η(ˆy),η(y)),

Ld(η(ˆy),ϕ(y))α+d(η(ˆy),ϕ(y))},

f orany y∈Y.This means that ϕ∽η,as desired.

5 Level sets and extensions

A crucial property of H¨

older sections is that under

suitable assumptions they can be extended. This

property is much studied in the context of metric

spaces if we consider the H¨

older maps; the reader

can see [29, 30, 31] and their references. We need to

mention several earlier partial results on extensions

of Lipschitz graphs in the context of Carnot groups,

as for example in [32, 33], [34, Proposition 4.8],

[35, Theorem 1.5]), [36, Proposition 3.4], [5, The-

orem 4.1].

Our proof follows using the link between H¨

older

sections and level sets of suitable maps. This idea is

widespread in the context of subRiemannian Carnot

groups (see, for instance, [37, 38, 39, 35]). In next

result, we say that a map fon Xis L-biLipschitz on

ﬁbers (of π) if on each ﬁber of πit restricts to an L-

biLipschitz map.

Theorem 5.1 (Extensions as level sets) Let π:X→

Y be a quotient map between a metric space X and a

topological space Y .

(5.1.i) If Z is a metric space, z0∈Z and f :X→Z

is (λ,β)-H¨

older and λ-biLipschitz on ﬁbers, with

λ>0and β∈(0,1), then there exists an intrinsi-

cally (λ2,β)-H¨

older section ϕ:Y→X of πsuch that

ϕ(Y) = f−1(z0).

(5.1.ii) Vice versa, assume that X is geodesic and

that there exist k ≥1,α∈(0,1),ρ:X×X→R

k-biLipschitz equivalent to the distance of X,and

τ:X→Ris (k,α)-H¨

older and k-biLipschitz on ﬁbers

such that

1. for all τ0∈Rthe set τ−1(τ0)is an intrinsically

(k,α)-H¨

older graph of a section ϕτ0:Y→X;

2. for all x0∈τ−1(τ0)the map X →R,x7→

δτ0(x):=ρ(x0,ϕτ0(π(x))) is k-Lipschitz on the

set {|τ| ≤ δτ0}.

Let Y ′⊂Y a set and L ≥1. Then for every in-

trinsically (L,α)-H¨

older section ϕ:Y′→π−1(Y′)of

π|π−1(Y′):π−1(Y′)→Y′, there exists a map f :X→R

that is (K,α)-H¨

older and K-biLipschitz on ﬁbers,

with K ≥1,such that ϕ(Y′)⊆f−1(0).In particular,

each ‘partially deﬁned’ intrinsically H¨

older graph

ϕ(Y′)is a subset of a ‘globally deﬁned’ intrinsically

H¨

older graph f −1(0).

We underline that an important point is that the

constant βin (5.1.i) does not change.

[Proof of Theorem 5.1] The proof is the same

to [6, Theorem 1.4]. The only difference that we

want to notice is the ”good” map in the second

point is deﬁned as follows: Fix x0∈τ−1(τ0).

We consider the map fx0:X→Rdeﬁned as

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(2(Γ−γ(δτ0(x)α+δτ0(x)) if |Γ| ≤ 2γ[δτ0(x)α+δτ0(x)]

Γif Γ>2γ[δτ0(x)α+δτ0(x)]

3Γif Γ<−2γ[δτ0(x)α+δτ0(x)]

where Γ:=τ(x)−τ(x0)and γ:=2kL +1.

References:

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

Rectiﬁability and perimeter in the Heisenberg

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

[2] 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).

[3] 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).

[4] F. Serra Cassano, Some topics of geometric mea-

sure theory in Carnot groups, in Geometry, analy-

sis and dynamics on sub-Riemannian manifolds.

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

Z¨

urich, 2016) pp. 1–121.

[5] B. Franchi and R. P. Serapioni, Intrinsic Lipschitz

graphs within Carnot groups, J. Geom. Anal. 26,

1946 (2016).

[6] D. Di Donato and E. Le Donne, Intrinsically Lip-

schitz sections and applications to metric groups,

accepted to Communications in Contemporary

Mathematics (2024).

[7] A. Agrachev, D. Barilari, and U. Boscain, A com-

prehensive introduction to sub-Riemannian ge-

ometry, Cambridge Studies in Advanced Math-

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

[8] 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.

[9] 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.

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

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

527 (2000).

[11] V. Magnani, Unrectiﬁability and rigidity in

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

[12] H. Federer, Geometric measure theory,

Die Grundlehren der mathematischen Wis-

senschaften, Band 153 (Springer- Verlag New

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

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

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

(2004).

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

Heisenberg group are N-rectiﬁable, Houston J.

Math. 32, 713 (2006).

[15] D. Bate, Characterising rectiﬁable metric spaces

using tangent spaces, Accepted Invent. math.

(2021).

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

sures in Carnot groups: existence of density, Ac-

cepted in Journal of Geometric Analysis (2022).

[17] 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).

[18] 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).

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

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

91–108 (1961).

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

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

263–274 (1975).

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

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

(2018).

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

rectiﬁable sets in Rn: Beyond lipschitz graphs,

ast´

erisque, , 171 (1991).

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

uniformly rectiﬁable sets, Mathematical Surveys

and Monographs 38 (1993).

[24] 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).

[25] V. N. Berestovskii, Homogeneous manifolds

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

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

tions on metric measure spaces, Geom. Funct.

Anal. 9 , 428 (1999).

Acknowledgment:

We would like to thank Davide Vittone for helpful

suggestions and Giorgio Stefani for the reference

[28].

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.74

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E-ISSN: 2224-2880

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Volume 23, 2024

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

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

[28] 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).

[29] L. Ambrosio and D. Puglisi, Linear extension

operators between spaces of Lipschitz maps and

optimal transport, Journal f¨

ur die reine und ange-

wandte Mathematik (Crelles Journal) 2020, 1

(2020).

[30] J. Lee and A. Naor, Extending Lipschitz func-

tions via random metric partitions, Invent. Math.

160 1, 59 (2005).

[31] S. Ohta, Extending Lipschitz and H¨

older maps

between metric spaces, Positivity 13 2, 407

(2009).

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

Intrinsic Lipschitz graphs in Heisenberg groups,

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

[33] D. Di Donato and K. F´

assler, Extensions and

corona decompositions of low-dimensional in-

trinsic Lipschitz graphs in Heisenberg groups,

Annali di Matematica Pura ed Applicata 201,

453 (2022).

[34] R. Monti, Lipschitz approximation of H-

perimeter minimizing boundaries, Calc. Var. Par-

tial Differential Equations 50, 171 (2014).

[35] D. Vittone, Lipschitz graphs and currents

in Heisenberg groups, Forum of Mathematics,

Sigma E6, 10 (2022).

[36] D. Vittone, Lipschitz surfaces, perimeter and

trace theorems for BV functions in Carnot-

Carath´

eodory spaces, Ann. Sc. Norm. Super. Pisa

Cl. Sci. (5) 11, 939 (2012).

[37] L. Ambrosio, F. Serra Cassano, and D. Vit-

tone, Intrinsic regular hypersurfaces in Heisen-

berg groups, J. Geom. Anal. 16, 187 (2006).

[38] G. Antonelli, D. Di Donato, S. Don, and

E. Le Donne, Characterizations of uni-

formly differentiable co-horizontal intrinsic

graphs in Carnot groups, (2020), accepted

to Annales de l’Institut Fourier, available at

https://arxiv.org/abs/2005.11390.

[39] D. Di Donato, Intrinsic differentiability and in-

trinsic regular surfaces in Carnot groups, Poten-

tial Anal. 54, 1 (2021).

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.

Conﬂicts of Interest The authors declare no conﬂict

of interest.

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The author contributed in the present research, at all

stages from the formulation of the problem to the

final findings and solution.

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