Poincaré-type inequality for Lipschitz continuous vector fields

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Journal

de

Mathématiques

Pures

et

Appliquées

www.elsevier.com/locate/matpur

Poincaré-type

inequality

for

Lipschitz

continuous

vector

fields

Francesco Serra Cassanoc

a_{Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126, Bologna, Italy} b_{Scuola Normale Superiore di Pisa, Piazza dei Cavalieri, 7, 56126, Pisa, Italy}

c_{Dipartimento di Matematica, Università di Trento Via Sommarive 14, 38123, Povo (Trento), Italy}

a r t i cl e i n f o a b s t r a c t

Article history:

Received23June2013

Availableonline8September2015

MSC: 43A80 35R45 42B20 Keywords: Poincaréinequalities Nonlinearvectorfields Liegroups

The scope of this paper is to prove a Poincaré type inequality for a family of nonlinearvectorfields,whosecoefficientsareonlyLipschitzcontinuouswithrespect tothedistanceinducedbythevectorfieldsthemselves.

© 2015ElsevierMassonSAS.All rights reserved. r é s u m é

Dans cet article on démontre une inégalité de type Poincaré pour une famille dechamps devecteurs nonlinéaires dontles coefficientssont seulementcontinus lipschitziensparrapportàladistanceinduiteparleschampseux-mêmes.

© 2015ElsevierMassonSAS.All rights reserved.

1. Introduction andstatementoftheresult

The Poincaré inequality is one of the main tools in the proof of regularity of solutions of PDEs in divergenceform.Indeed,asprovedbySaloff-Costein[1]andGrigor’yanin[2](seealso[3]),itisequivalent to the Harnack inequality and to Hölder continuity for solutions. Thus, to prove regularity of solutions, it sufficestoestablishasuitablePoincaréinequality.

ThePoincaréinequalityforsmoothHörmandervectorfieldsiswellknownandwasprovedbyJerison[4]. We recall that a Hörmander family of vector fields in Rn_{, is defined by m ≤ n smooth vector fields,}

say∇= (∇1,. . . ,∇m),suchthatthegeneratedLie algebrahasmaximumrankateverypoint.

DenotebyBr(x)⊂ Rnthemetricballofcenterx andradiusr > 0 associatedtotheCC-distancedefined

intermsofthefamily∇.ThePoincaréinequalityprovedin[4]is:

* Correspondingauthor.

E-mail addresses:giovanna.citti@unibo.it(G. Citti),maria.manfredini@unibo.it(M. Manfredini), andrea.pinamonti@gmail.com(A. Pinamonti),casssano@science.unitn.it(F. Serra Cassano). http://dx.doi.org/10.1016/j.matpur.2015.09.001



Br(x)

|ψ(y) − ψr(y)| dLn(y)≤ C r



Br(x)

|∇ψ(y)| dLn_{(y) ∀ψ ∈ C}∞_(B r(x)),

where,asusual,ψr(y):= 1/Ln(Br(y))



Br(y)ψ andC > 0.Thepreviousinequalitycanbealsostatedusing

balls defined with respect to different (but equivalent) distances. We mention here the ball box distance (see [5])andthefrozendistancedefinedbyRothschildandSteinin[6].

ThePoincaréinequalityfornon-smoothvectorfieldswasfirstconsideredin[7].Thelaterworksonrelated questions include the papers by Biroli and Mosco [8], Capogna, Danielli and Garofalo [9,10], Chernikov and Vodop’yanov [11], Danielli, Garofalo,Nhieu[12], Franchi, Gallotand Wheeden [13], Franchi, Luand Wheeden[14]andLu[15,16].Morerecently,in[17],theauthorsprovedageneralPoincaréinequalitywhich wasappliedin[18–20]tofamiliesofLipschitzvectorfieldswithdifferentregularityconditions,anddifferent assumptions ontherankofthegeneratedLie algebra.

In [21], the authors studied the relationship between the validity of the Poincaré inequality and the existence of representation formulas for functions as (fractional) integral transforms of first-order vector fields. Theyshowed thatthePoincaré inequalityleadsto (and infactit is oftenequivalent to) asuitable representation formula.Thisapproachwaslaterdevelopedin[9],inwhichanotherproofhasbeengivenof the representationformula relyingon thePoincaréinequalityprovedbyJerison. Finally,in[22], ageneral representation formulaisprovedintermsofthefundamentalsolutionofaHörmandertypesub-Laplacian. Unfortunately,alltheseresultsareexpressedintermsofvectorfieldswithLipschitzcontinuouscoefficients with respect totheEuclidean distance.Ontheother hand,inorder to studypartial differentialequations withnonlinearvectorfields,thisassumptionisnolongernatural.Atypicaldifferentialequationofthistype canbeof theform

m



i,j=1

∇φ

i(aij(φ)∇φjφ) = f, (1.1)

where (aij) is a smooth, symmetric, uniformly positive definite matrix, f is a fixed function and the

coefficients of the vector fields ∇φ depend on the solution φ. Equations of this type naturally arise while studying curvature equations [23], Monge–Ampére equation [24–26], mathematical finance [27–29] or intrinsic minimalgraphsintheHeisenberggroup(seeforinstance [30–35]).

Aparticular,butveryinterestinginstanceof(1.1),istheso-calledminimalsurfaceequationforintrinsic graphsintheHeisenberggroup(seealso[36,35,37]forthecaseofT-graphs).Inthen-dimensionalHeisenberg groupHn,suchgraphsaredescribed asfollows(see [38,39]):

M ={(φ(x1, . . . , x2n), x2, . . . , x2n, x2n+ 2xnφ(x1, . . . , x2n)), (x1, . . . , x2n)∈ ω},

where ω ⊂ R2n

(x1,...,x2n) isan openset andφ: ω → R isacontinuousfunctionsatisfyingsuitable regularity

properties.Intrinsicgraphshavebeenextensivelystudiedinconnectionwiththenotionofrectifiablesetsin Hn _{(seeforinstance[40–43]),theregularityproblemforminimalsurfaces(seeforinstance[31,32,44–46,35])}

andtheBernsteinprobleminHn_{[30,33,47–49].Inparticular,in[38]itisprovedthattheso-calledhorizontal}

perimeter ofM canbe expressedby

PH(M ) = 

ω



1 +|∇φ_φ|2_dL2n_,

where ∇φ= (∇φ₁,. . . ,∇_2nφ ₋₁) isthefamilyofnonlinearvectorfieldsdefinedby:

∇φ i = ∂xi− xi+n∂x2n, ∇ φ n = ∂xn+ 2φ∂x2n,∇ φ i+n= ∂xi+n+ xi∂x2n, (1.2)

wherei= 1,. . . ,n−1.Moreover,aspointedoutin[38,33],onecanshowthattheconditionthattheintrinsic surfaceM beH-minimalisexpressed bythenonlinearequationoftype(1.1):

∇φ_· ∇φφ

1 +|∇φ_φ|2 = 0.

Noticethattheregularityofthesolutionφ of(1.1)canbeobtainedonlyintheHölderspacesdefinedinterms of the distance naturallyassociated to the family∇φ. As aconsequence, the coefficientsof the equation, (whichdependonthesolutionitself),arenotexpectedtobeLipschitzwithrespecttotheEuclideandistance. Moreover,thislackofregularityofthecoefficientsimpliesthattheequivalenceofthedifferentdefinitionsof distancescannotbededucedusing[5].Toovercomethisproblem,adistancemodeledonthefrozendistance ofRothschildandSteinwasdefinedin[23]whilestudyingtheLeviequation.In[38,39],ananalogousfrozen distance dφ hasbeen proposedforthevectorfields in(1.2),as thesymmetrizeddistance associatedto the

frozenvectorfields

∇φ(x0)

i :=∇

φ

i for i = 1, . . . , 2n− 1, i = n, ∇φ(xn 0):= ∂xn+ 2φ(x0)∂x2n, (1.3)

wherex0∈ ω isfixed.Themainadvantageofworkingwiththisfamilyofvectorfieldsreliesonthefactthat

theyhaveC∞ coefficientsandtheycanbe consideredas azeroorderapproximationof thefamily∇φ. We pointoutthattheequivalencebetweenthefrozendistance dφ andtheball boxdistancedefinedin[5]was

provedin[39].Moreover, theequivalencebetween dφ and theCC-distance generated by∇φ canbefound

in[50].

Motivatedbythediscussionabove,inthispaperweproveaPoincarétypeinequalityforthemodelvector fields in(1.2),under theassumption thatthecoefficients areLipschitz continuouswith respect to dφ and

forfunctionswhichbelongtoanintrinsicSobolevspace:

Definition 1.1. Let φ : ω ⊂ R2n −→ R be an intrinsic Lipschitz continuous function, in the sense of Definition 2.1 below.Wesaythatafunctionψ : ω⊂ R2n _{−→ R belongsto thespaceW}

φ(ω) ifthere exist

sequences{ψk}k∈Nand {φk}k∈N inC∞(ω) suchthat

(i) ψk → ψ inL1loc(ω) ask→ +∞;

(ii) φk→ φ uniformlyinω ask→ +∞;

(iii) |∇φk_ψ

k(x)|≤ M ∀x∈ ω andk andforsomepositiveconstantM ;

(iv) ∇φk_ψ

k ∗∇φψ ask→ +∞.

Then,ourmain resultisthefollowing:

Theorem 1.2. Let ω be a bounded and open subset of R2n _{with n ≥ 2, and let p≥ 1. Let φ : ω → R be}

an intrinsic Lipschitz function and ψ ∈ Wφ(ω). Then there exist positive constants C1,C2 with C2 > 1

(dependingcontinuously ontheLipschitzconstant Lφ ofφ) suchthat

 Uφ(¯x,r) |ψ(y) − ψUφ(¯x,r)| p _dL2n_{(y)≤ C} 1rp  Uφ(¯x,C2r) |∇φ_ψ(y)|p _dL2n_{(y), (1.4)}

forevery Uφ(¯x,C2r)⊂ ω, where

Uφ(x, r) :={y ∈ ω : dφ(x, y) < r} . (1.5)

ψUφ(¯x,r):= 1 L2n_(U φ(¯x, r))  Uφ(¯x,r) ψ(y) dL2n(y) . (1.6)

Corollary 1.3. If φ : ω → R is an intrinsic Lipschitz function, then there exist positive constants C1, C2

with C2> 1 (dependingcontinuously ontheLipschitz constant Lφ of φ)suchthat

 Uφ(¯x,r) |φ(y) − φUφ(¯x,r)| dL 2n_{(y)≤ C} 1r  Uφ(¯x,C2r) |∇φ_{φ(y)| dL}2n_{(y), (1.7)} forevery Uφ(¯x,C2r)⊂ ω.

We briefly describe our approach. Since the coefficients of the vector fields ∇φ _{are only Lipschitz}

continuous (with respect to dφ), we cannot consider the Lie algebra generated by the vector fields.

Nevertheless,theexplicitexpressionofthevectorfields ∇φ₁,. . . ,∇φ_2n−1 ensuresthat [∇φ_i,∇φ_n+i] = 2 ∂x2n ∀ i = 1, . . . , n − 1,

sothatthevectorfieldsandtheircommutatorsspanthewholespaceateverypoint,whichcanbeinterpreted as aHörmandercondition for non-regular vectorfields. This approachcanbe consideredaversion ofthe Rothschildand Steinmethodfornon-smoothvectorfields,andhasbeenused in[23]inadifferent setting. In particular,everyψ∈ C∞(ω) can be representedbymeansof asuitablerepresentationformula (proved in[22])intermsof thevectorfields ∇φ(x0)

i ,thefundamentalsolutionΓx0 of theLaplacianoperator

Lφ(x0):= 2n−1 i=1 (∇φ(x0) i ) 2

and thesuperlevelsets Ωφ(x0)(x0,r) ofΓx0,whichareequivalentto theballsUφ(x0,r).

InordertoproveTheorem 1.2,insection3 wewillfirstmodifytheaforementionedrepresentationformula to obtainanotherrepresentation formulaintermsofthefamily∇φ.Subsequently,usinganapproximation result for intrinsicLipschitz functionscontained in[45], (see also[50] for arefinement)we provethatthe representation formulaprovedinSection3 stillholdsforintrinsicLipschitzfunctions.Finally,inSection4 we willprovidetheproofofTheorem 1.2.

2. Preliminaries

2.1. Theintrinsic distance

Fixn≥ 2.Letω⊂ R2n beanopenandbounded setandφ: ω−→ R beacontinuousfunction.TheLie algebrageneratedbythefamily∇φ:= (∇φ₁,. . . ,∇φ_2n−1) definedin(1.2)hasmaximumrankateverypoint, hence it is possible to define on ω the exponential ball box distance and the CC-distance, see [5]. These distances are not explicitly computable, therefore, it is convenient to introduce an equivalent, explicitly computable, quasi-distanceonω. Todothis,we usethefreezing methoddevelopedin[6]andsuccessively refined in[39] (see also [38]). Precisely, let us fix x0 ∈ ω and consider the familyof smooth vector fields

∇φ(x0)_{definedin(1.3)andthenewfamilyoffrozenvectorfields}

ˆ

∇φ(x0)_{= (}∇φ(x0)_{, ∂}

x2n) . (2.8)

LetusnowintroducetheLiealgebraG generatedbythefamilyofvectorfields∇φ(x0)_{.Noticethat,since}

theonlynon-vanishingcommutatoris[∇φ(x0)

i ,∇

φ(x0)

n+i ]= 2∇ φ(x0)

toHn−1×R,asCarnotgroups,whereHn−1≡ R2n−1_{denotesthe(n−1)-dimensionalHeisenberggroup.Any}

elementx˜∈ G canbeidentifiedbyitscoordinateswithrespecttothebasis∇ˆφ(x0)_,thatisx˜≡ (˜x

1,. . . ,x˜2n),

ifx =˜ 2n_i=1x˜i∇ˆiφ(x0).Wecanalso induceanormonG bydefining

(˜x1, . . . , ˜x2n) := max{|(˜x1, . . . , ˜x2n−1)|R2n−1,|˜x2n| 1

2} (2.9)

foreach G.Theexponential map associated to thefamilyof vectorfields ∇ˆφ(x0) _{iswell defined.Precisely,}

foreachx∈ R2n_:

Exp_φ(x0),x:G −→ R2n, Exp_φ(x0),x(˜y) := exp

2n  i=1 ˜ yi∇ˆφ(xi 0) (x) . Incoordinatesweget Exp_φ(x0),x(˜y) =x1+ ˜y1, . . . , x2n−1+ ˜y2n−1, x2n+ ˜y2n+ 2˜ynφ(x0)− σ(˜y, x) , (2.10) where σ(x, y) := n−1  i=1 (yi+nxi− xi+nyi). (2.11)

TheinversemappingofExp_φ(x0),xwill bedenotedbyLog_φ(x0),x:R2n_{−→ G, andtheball-boxexponential}

distanceassociated tothevectorfields ∇ˆφ(x0)_{isdefinedforeveryx,y}∈ R2n

dφ(x0)(x, y) :=Logφ(x0),x(y).

Inparticular,asimplecomputationgives:

Logφ(x0),x(y) =



y1− x1, . . . , y2n−1− x2n−1, y2n− x2n− 2φ(x0)(yn− xn)− σ(x, y)

(2.12)

andthereforeusing (2.9)weget:

dφ(x0)(x, y) = max

|ˆx − ˆy|R2n−1, σφ,x0(x, y)

, (2.13)

whereforeveryx= (x1,. . . ,x2n)∈ R2n wehavedenotedx := (xˆ 1,. . . ,x2n−1)∈ R2n−1 and

σφ,x0(x, y) := |y2n− x2n− 2φ(x0)(yn− xn)− σ(x, y)| 1/2 _x

0∈ ω, x, y ∈ R2n. (2.14)

Moreoverwewillsimplydenote

σφ(x, y) := σφ,x(x, y) x, y∈ ω. (2.15)

Finally,wedefinethefollowing symmetricfunction:

dφ(x, y) = 1 2  dφ(x)(x, y) + dφ(y)(y, x) ∀x, y ∈ ω. (2.16)

Definition 2.1. We say thatφ : ω ⊂ R2n _{−→ R isan intrinsic Lipschitz continuous functionin ω andwe}

writeφ∈ Lip(ω),ifthere isaconstantL> 0 suchthat

The Lipschitz constant of φ in ω is the infimum of the numbers L such that (2.17) holds and we write

Lφ,ω (or simplyLφ)to denoteit. Wealsosaythatφ is alocally intrinsicLipschitz function,andwe write

φ∈ Liploc(ω) ifφ∈ Lip(ω) foreveryω  ω.

Remark2.2.Itimmediatelyfollowsfromtheexplicitexpressionofdφ (seealso[23])that,ifφ∈ Lip(ω) then

dφ isaquasi-distance onω. Precisely,

dφ(x, y) = 0⇐⇒ x = y;

dφ(x, y) = dφ(y, x);

and foreachx,y,z∈ ω:

dφ(x, y)≤

≤ dφ(x, z) + dφ(y, z) +|φ(x) − φ(z)|1/2|xn− zn|1/2+|φ(y) − φ(z)|1/2|yn− zn|1/2 (2.18)

so that

dφ(x, y)≤ (1 + Lφ)1/2(dφ(x, z) + dφ(y, z)) .

Remark2.3. Itiseasy toseethat,if φ∈ Lip(ω), then

σφ(y, x)≤ σφ(x, y) +|φ(x) − φ(y)|1/2|xn− yn|1/2 ∀x, y ∈ ω

whence,by (2.16),

dφ(x, y)≤ |ˆx − ˆy|R2n−1+ σφ(x, y) +|φ(x) − φ(y)|1/2|xn− yn|1/2 ∀x, y ∈ ω. (2.19)

Remark 2.4. Moreover, by a simple calculation, we obtain that there existsC1 > 1 depending only on Lφ

such thatforeach x,y∈ ω

1

C1

dφ(y)(y, x)≤ dφ(x, y)≤ C1dφ(y)(y, x), (2.20)

1

C1

dφ(x)(x, y)≤ dφ(x, y)≤ C1dφ(x)(x, y). (2.21)

Besides,there existsapositiveconstant C2= C2(Lφ) suchthat foreachx,y,z∈ ω

dφ(x)(x, y)≤ C2



dφ(x)(x, z) + dφ(z)(z, y)

. (2.22)

2.2. Lipschitzcontinuous functionswithrespecttononlinearvector fields

A detailed analysis of Lip(ω) canbe found in[50,40],here werecall only those propertiesthatwe will useintheproofofTheorem 1.2.

Notice that Lip(ω) is not a vector space (see [35, Remark 4.2]). Nevertheless, the intrinsic Lipschitz functionsamounttoathickclassoffunctions.Indeed,itholdsthat

Lip_E(ω) Lip_loc(ω) C_loc1/2(ω) , (2.23) where,Lip_E(ω) andC_loc1/2(ω) denotetheclassesofreal-valuedEuclideanLipschitzandlocally1/2-Euclidean– Höldercontinuousfunctionsonω respectively,see [40,Propositions4.8and4.11].

Theorem2.5. (See[40].)If φ∈ Lip(ω) then φ is∇φ_{-differentiableforL}2n_{-a.e. x∈ ω, inthesense defined}

in [38].Besides,forL2n-a.e.x∈ ω there isaunique vector∇φφ(x)∈ R2n−1 called ∇φ-gradientof φ such that

φ(y) = φ(x) +∇φφ(x), ˜π(y) + o(dφ(x, y)) as y→ x

where ·, · denotes theEuclidean scalar productin R2n−1 _{and π :˜ R}2n _{→ R}2n−1_{, π(x˜}

1,. . . ,x2n−1,x2n):=

(x1,. . . ,x2n−1),∀x∈ R2n−1.

In [50] thefollowing estimates for Lφ are proved.Precisely, for each x¯∈ ω and eachr > 0 sufficiently

smallthere isC1> 0 depending onlyon∇φφL∞(ω) suchthat

Lφ,Uφ(¯x,r)≤ C1∇ φ_φ

L∞(ω),

andthere isC2= C2(n)> 0 such that

∇φ_φ

L∞(ω)≤ C2Lφ(Lφ+ 1)

whereUφ(x,r) isdefinedin(1.5).

Moreover,thefollowingapproximationresultforintrinsicLipschitzfunctionsithasbeenrecentlyproved in[45]:

Theorem 2.6. Letω ⊂ R2n be a bounded open set and let φ ∈ Lip(ω). Then there exists asequence {φk}

with φk∈ C∞(ω) suchthat

(i) φk→ φ uniformlyinω as k→ ∞,

(ii) |∇φk_φ

k(x)|≤ ∇φφL∞(ω) ∀x∈ ω.

Wealsoquotethepaper[50]whereweprovedthateveryφ∈ Lip(ω) canbeapproximatedbyasequence

{φk}k∈Nofsmoothfunctionssatisfying(i),(ii)andalso

∇φk_φ

k(x)→ ∇φφ(x) L2n-a.e. in ω.

2.3. Sub-Laplacianand fundamentalsolution

In order to study the dependence of the vector fields ∇ˆφ(x0) _{(defined in (2.8)) on the variable x} 0 we

recognizethatthemap

Log_φ(x0),x0 :R2n−→ G

changesthefamilies∇φ(x0) _and∇ˆφ(x0)_{into thefamily}∇ and∇ respectively,ˆ _where

∇i:=∇φ(xi 0), ∇ˆj:= ˆ∇φ(xj 0) for i∈ {1, . . . , 2n − 1}, j ∈ {1, . . . , 2n}, i, j = n,

∇n:= ∂xn, ∇ˆn= ∂xn. (2.24)

Precisely,foreachψ∈ C∞(R2n_{),ifwedefine}

˜

ψ(˜x) := ψ(Log−1_φ(x

then

ˆ

∇φ(x0)

i ψ(x) = ˆ∇iψ(Log˜ φ(x0),x0(x)), ∀i ∈ {1, . . . , 2n}.

Wecandefineametric˜d onG associatedtothevectorfields∇,ˆ independentofx0.Namely,givenx∈ R2nlet

Exp_∇,xˆ : G → R2n, Exp_∇,xˆ (˜y) := exp

_2n i=1 ˜ yi∇ˆi  x if ˜y = 2n  i=1 ˜ yi∇ˆi.

WecanalsoidentifyG withR2n_{,byidentifyinganelementofG withitscoordinateswithrespecttobasis ˆ∇.}

Insuchaway,wecandefine

˜

d(˜x, ˜y) :=Exp−1_ˆ

∇,˜x(˜y), (2.26)

where ·  denotesthenormin(2.9).Inparticular itholds that ˜

d(0, ˜x) =˜x, ∀˜x ∈ G ≡ R2n,

dφ(x0)(x, y) = ˜d(Logφ(x0),x0(x), Logφ(x0),x0(y)) ∀x, y, x0∈ ω. (2.27)

Moreoveritfollows thatd turns˜ outto beahomogeneousnormonG ≡ Hn−1× R. Letus callsub-Laplacianthesecondorder differentialoperatordefinedas

Lφ(x0):= 2n_−1

i=1

(∇φ(x0)

i )2. (2.28)

It iswellknownthatLφ(x0) admitsafundamentalsolutionwhichwewill denotebyΓφ(x0)(see for[51]for

thedetails).Thisoperatorischangedbythemap Log_φ(x0),x0 into thesub-Laplacian operator

L :=

2n_−1 i=1

(∇i)2.

Thatis,foreachψ∈ C∞(R2n₎

(Lφ(x0)ψ)(x) = (L ˜ψ)(Logφ(x0),x0(x)) ∀ x ∈ R 2n_,

where ψ is˜ definedin(2.25).

Clearly, the operator L has a fundamental solution Γ of class C∞ far from the pole ˜x = ˜y, which is homogeneousofdegree2− Q withrespect tothedilationfamilynaturallyassociatedtoG,where Q isthe homogeneous dimension of Hn−1× R (see [51, Section 5.3] and the references therein). This means that there existpositiveconstantsC1,C2 suchthatforeveryx and˜ y in˜ R2n,x˜= ˜y

C1 ˜ d(˜x, ˜y)Q−2 ≤ Γ(˜x, ˜y) ≤ C2 ˜ d(˜x, ˜y)Q−2; |∇iΓ(˜x, ˜y)| ≤ C2 ˜ d(˜x, ˜y)Q−1; (2.29) |∇j∇iΓ(˜x, ˜y)| ≤ C2 ˜ d(˜x, ˜y)Q,

for every i,j = 1,. . . ,2n− 1 (see [52] and [51, Section 5.4]). Besides, the fundamental solution Γφ(x0) of Lφ(x0)canbeexplicitlywrittenintermsofΓ as

Γφ(x0)(x, y) = Γ(Logφ(x0),x0(x), Logφ(x0),x0(y)), (2.30)

and

∇φ(x0)

i Γφ(x0)(x, y) =∇iΓ(Logφ(x0),x0(x), Logφ(x0),x0(y)),

fori= 1,. . . ,2n−1.Itfollowsthattheinequalitiesin(2.29)aresatisfiedalsoforΓφ(x0)(x,y) anddφ(x0)(x,y)

with the sameconstants. In particular, it is clear thatthese constants are independent of x0. Using the

estimatesforΓφ(x0)itfollowsthatthespheresinthemetricdφ(x0)areequivalenttothesuperlevelsofthe

fundamentalsolutionΓφ(x0):

Ωφ(x0)(x, r) =



y∈ R2n | Γφ(x0)(x, y) > r

2−Q_{, r > 0. (2.31)}

Moreover,for everyfixedx0∈ ω,theset Ωφ(x0)(x0,r) hasregularboundary(see [22]).Inparticular, from

(2.21), (2.27)and(2.29), thereexistr0,α > 0 withα = α(Lφ) suchthatforany x0∈ ω andr≤ r0

Ωφ(x0)(x0, r/α)⊂ Uφ(x0, r)⊂ Ωφ(x0)(x0, α r), (2.32)

whereUφ(x0,r) istheballdefinedin(1.5).By(2.30) wehavethat

Ωφ(x0)(x, r) =

y∈ R2n | Γ(Logφ(x0),x0(x), Logφ(x0),x0(y)) > r

2−Q_{, (2.33)}

inparticularthesetsΩφ(x0)(x0,r) canbeexpressedintermsofthesuperlevelsofthefundamentalsolution

Γ asfollows: Ωφ(x0)(x0, r) = y∈ R2n | Γ(0, Log_φ(x0),x0(y)) > r2−Q = Exp_φ(x0),x0( ˜Ω(0, r)), (2.34) where ˜ Ω(0, r) :=y˜∈ R2n | Γ(0, ˜y) > r2−Q. (2.35) Wewillalsodenote

K(˜y) := Γ−(Q−2)1 _{(0, ˜y), ˜y}∈ R2n_{, (2.36)}

sothat,wecanrewriteΩ(0,˜ r) as:

˜

Ω(0, r) =y˜∈ R2n | K(˜y) < r. (2.37) 3. A representationformulaintermsoftheintrinsicgradient

Thissectionisorganizedintwosubsections.Inthefirstone,wefixasmoothfunctionφ: ω⊂ R2n_{→ R}

andweintroducethenotionofintegralmeanm of¯ anotherregularfunctionψ : ω⊂ R2n _{→ R onthesuper}

levels Ωφ(x0)(x0,r) of the fundamental solution Γφ(x0). Then, we prove a representation formula for any

regularfunctionψ intermsofitsintegralmeananditsintrinsicgradient.

Thesecondsubsectionisquitetechnical,andisdevotedtoestablishsomepropertiesoftheintegralmean ofψ.

3.1. Representationformula

Inthis section,wefix ω⊂ R2n _{openand bounded,n≥ 2 andφ,ψ∈ C}∞_{(ω).Weprovearepresentation}

formula forψ onthesuperlevelsΩφ(x0)(x0,r) ofΓφ(x0). Asimilarrepresentation formulahasbeenproved

in[51,22]fortheapproximatedvectorfields ∇φ(x0)_{.From thisformula,we willdeduceanewandintrinsic}

representationformulaforψ,expressedintermsofitsintrinsicgradient∇φ_{ψ.Inthecaseunderconsideration}

theresultof[51,22]canbe statedasfollows:

Proposition3.1.Foreveryx0∈ ω andR > 0 suchthatΩφ(x0)(x0,R)⊂ ω andforeveryψ∈ C∞(ω) wehave

ψ(x0) = Q (Q − 2)(1 −₂1Q)RQ  Ω_φ(x0)(x0,R)\Ω_φ(x0)(x0,R2) |∇φ(x0)_Γ φ(x0)(x0, y)| 2 Γ2(_φ(xQ−1)/(Q−2) 0) (x0, y) ψ(y) dL2n(y) + Q (1− 1 2Q)RQ R  R 2 rQ−1  Ω_φ(x0)(x0,r)  ∇φ(x0)_Γ φ(x0)(x0, y),∇ φ(x0)_ψ(y)  dL2n(y)dr. (3.38)

Here·, · denotesthestandard Euclideanscalar productinR2n−1_.

Remark3.2. Weexplicitlynotethat,if wechoose ψ≡ 1,then from(3.38) weget

1 = C(Q) RQ  Ω_φ(x0)(x0,R)\Ωφ(x0)(x0,R2) |∇φ(x0)_Γ φ(x0)(x0, y)| 2 Γ2(_φ(xQ−1)/(Q−2) 0) (x0, y) dL2n(y), (3.39) where C(Q):=_(Q−2)(1−Q 1 2Q) .

This remark allows tosay that(3.38) representsafunction ψ as thesum ofits meanon asuitable set and thegradient∇φ(x0)_{ψ.Hence,itseemsnaturalto givethefollowing definition}

Definition 3.3. Let ψ ∈ L1_loc(ω).For every x0 ∈ ω and R > 0 such that Ωφ(x0)(x0,R) ⊂ ω we define the

following meanofψ,ontheset Ωφ(x0)(x0,R)\ Ωφ(x0)(x0,

R

2),intermsofthefundamentalsolutionΓφ(x0):

¯ m(ψ, φ, R)(x0) := C(Q) RQ  Ω_φ(x0)(x0,R)\Ωφ(x0)(x0,R2) |∇φ(x0)_Γ φ(x0)(x0, y)| 2 Γ2(_φ(xQ−1)/(Q−2) 0) (x0, y) ψ(y) dL2n(y).

In thesequel wewill need anothermean of ψ on thesameset Ωφ(x0)(x0,R)\ Ωφ(x0)(x0,

R 2). Precisely, we denote: m(ψ, φ, R)(x0) := 2 R R  R 2 ¯ m(ψ, φ, r)(x0) dr. (3.40)

Thefollowingremark,whichwillbeveryusefullateron,providesanintegrationformulabypartsforthe derivative∂2n.

Remark3.4.Letg∈ C1_(R2n_{),r > 0 andc}

1,c2> 0 wedefine:

Ar,c1,c2 :={y ∈ R 2n _{: c}

1r < g(y) < c2r}.

Then, for every f,ψ ∈ C1_(R2n_{) and R}

1,R2 ∈ R with R1 < R2, using the fact that

∂2n= 1₂(∇φ(x1 0)∇ φ n+1− ∇ φ(x0) n+1 ∇ φ

1) andintegrating by partwehave

R2  R1 rQ−1  Ar,c1,c2

f (y)∂2nψ(y) dL2n(y)dr =

=1 2 R2  R1 rQ−1  {y:g(y)/c2=r} f (y)∇φ_n+1ψ(y)∇ φ(x0) 1 g(y) |∇Eg(y)| dH2n−1(y) dr −1 2 R2  R1 rQ−1  {y:g(y)/c1=r} f (y)∇φ_n+1ψ(y)∇ φ(x0) 1 g(y) |∇Eg(y)| dH2n−1(y)dr −1 2 R2  R1 rQ−1  {y:g(y)/c2=r} f (y)∇φ₁ψ(y)∇ φ(x0) n+1 g(y) |∇Eg(y)| dH2n−1(y)dr +1 2 R2  R1 rQ−1  {y:g(y)/c1=r} f (y)∇φ₁ψ(y)∇ φ(x0) n+1 g(y) |∇Eg(y)| dH2n−1(y)dr −1 2 R2  R1 rQ−1  A_r,c1,c2 ∇φ(x0) 1 f (y)∇ φ n+1ψ(y) dL2n(y)dr +1 2 R2  R1 rQ−1  A_r,c1,c2 ∇φ(x0) n+1 f (y)∇ φ 1ψ(y) dL2n(y)dr,

where∇E denotestheEuclidean gradient. Bythecoarea formulawe inferthat R2  R1 rQ−1  A_r,c1,c2

f (y)∂2nψ(y) dL2n(y)dr =

=1 2  A_r,c2R1,c2R2 gQ−1(y) cQ−1₂ f (y)∇ φ n+1ψ(y)∇ φ(x0) 1 g(y) dL2n(y) −1 2  Ar,c1R1,c1R2 gQ−1(y) cQ−1₁ f (y)∇ φ n+1ψ(y)∇ φ(x0) 1 g(y) dL2n(y) −1 2  A_r,c2R1,c2R2 gQ−1(y) cQ−1₂ f (y)∇ φ 1ψ(y)∇ φ(x0) n+1 g(y) dL 2n_(y) +1 2  Ar,c1R1,c1R2 gQ−1(y) cQ−1₁ f (y)∇ φ 1ψ(y)∇ φ(x0) n+1 g(y) dL2n(y)

−1 2 R2  R1 rQ−1  A_r,c1,c2 ∇φ(x0) 1 f (y)∇ φ n+1ψ(y) dL 2n_(y)dr +1 2 R2  R1 rQ−1  Ar,c1,c2 ∇φ(x0) n+1 f (y)∇ φ 1ψ(y) dL2n(y)dr.

If in additionc1= 0 thentheintegralson Ar,c1R1,c1R2 arenot present.

Inthefollowing propositionwewillslightlymodifytherepresentationformulaprovedinProposition 3.1 inorder to obtain a mean representation formula containing only derivatives with respect to the vector fields ∇φ_.

Proposition 3.5. Letφ,ψ∈ C∞(ω). Foreveryx0∈ ω andR > 0 such that Ωφ(x0)(x0,R)⊂ ω wehave ψ(x0)− m(ψ, φ, R)(x0) = IR(x0), where IR(x0) = 2 R R  R 4 f1 _r R  Ω_φ(x0)(x0,r)  K1(x0, y),∇φψ(y) dL2n(y)dr + 2 R R  R 2  Ω_φ(x0)(x0,r)\Ωφ(x0)(x0,r2)  K2(x0, y, r),∇φψ(y) dL2n(y)dr.

Here,f1∈ C0([1₄,1]) andthevectorvaluedfunctionsK1andK2aredefinedin(3.44)and(3.45)respectively.

Moreover, |K1(x0, y)| ≤ ˜C1(Lφ,Ω_φ(x0)(x0,R)+ 1) 2_d1−Q φ(x0)(x0, y) ∀y ∈ Ωφ(x0)(x0, R) (3.41) and |K2(x0, y, r)| ≤ ˜C2(Lφ+ 1)2d1φ(x−Q0)(x0, y), ∀y ∈ Ωφ(x0)(x0, R)\ Ωφ(x0)(x0, R 2), r∈ _R 2, R , (3.42)

where Lφ meansLφ,Ω_φ(x0)(x0,R)\Ω_φ(x0)(x0,R/2) and C˜1,C˜2 > 0 are suitable constants depending only on the homogeneous dimensionQ andon thestructureconstantsC1 andC2 in(2.29).

Proof. We will always denote byC a positiveconstantdepending only onQ whichcan be different from line toline.ByProposition 3.1forallr∈ (0,R)

ψ(x0)− ¯m(ψ, φ, r)(x0) = = C rQ r  r 2 sQ−1  Ω_φ(x0)(x0,s)  ∇φ(x0)_Γ φ(x0)(x0, y),∇ φ(x0)_ψ(y)  dL2n(y)ds = C rQ r  r 2 sQ−1  Ω_φ(x0)(x0,s)  ∇φ(x0)_Γ φ(x0)(x0, y),∇ φ_ψ(y)_dL2n_(y)ds

+ C rQ r  r 2 sQ−1  Ω_φ(x0)(x0,s) ∇φ(x0)

n Γφ(x0)(x0, y)(φ(x0)− φ(y))∂2nψ(y) dL

2n_(y)ds.

UsingRemark 3.4withg(y):= Γ

1 2−Q φ(x0)(x0,y) weobtain ψ(x0)− ¯m(ψ, φ, r)(x0) = 1 rQ r  r 2 sQ−1  Ω_φ(x0)(x0,s)  K1(x0, y),∇φψ(y) dL2n(y)ds +  Ω_φ(x0)(x0,r)\Ωφ(x0)(x0,r2)  K2(x0, y, r)∇φψ(y) dL2n(y), (3.43) where K1(x0, y) := C∇φ(x0)Γφ(x0)(x0, y) − C∇φ(x0) 1 ∇φ(xn 0)Γφ(x0)(x0, y)(φ(x0)− φ(y))en+1 + C∇φ(x0) n+1 ∇φ(xn 0)Γφ(x0)(x0, y)(φ(x0)− φ(y))e1, (3.44) and K2(x0, y, r) := C rQ ∇φ(x0) n Γφ(x0)(x0, y) Γ2(_φ(xQ−1)/(Q−2) 0) (x0, y) (φ(x0)− φ(y))∇φ(xn+10)Γφ(x0)(x0, y)e1 − C rQ ∇φ(x0) n Γφ(x0)(x0, y) Γ2(_φ(xQ−1)/(Q−2) 0) (x0, y) (φ(x0)− φ(y))∇φ(x1 0)Γφ(x0)(x0, y)en+1, (3.45)

whereei isthei-th elementofthecanonicalbasisofR2n−1.

Integrating(3.43)from R₂ toR weget

ψ(x0)− m(ψ, φ, R)(x0) = = 2 R R  R 2 1 ρQ ρ  ρ 2 rQ−1  Ω_φ(x0)(x0,r)  K1(x0, y),∇φψ(y) dL2n(y)drdρ + 2 R R  R 2  Ω_φ(x0)(x0,ρ)\Ωφ(x0)(x0,ρ2)  K2(x0, y, ρ),∇φψ(y) dL2n(y)dρ.

Exchangingtheorderofintegration inthefirstintegralandsetting:

f1(t) :=

21−Q_{− (2t)}Q−1

1− Q if t∈ [1/4, 1/2], f1(t) :=

tQ−1− 1

1− Q if t∈ [1/2, 1], wegetthethesis.Finally,theestimatesonK1 andK2 aredirectconsequencesof(2.29). 2

3.2. Somepropertiesof theintegralmean

Inthissection wecollectsomepropertiesof theintegralmeanm(ψ,¯ φ,r) tobeused inthenextsection. Wewill seethatinorderto concludetheproof ofthePoincaréinequalitywewillneedadetailedestimate of thedifference

¯

m(ψ, φ, r)(x)− ¯m(ψ, φ, r)(y) (3.46) at twodifferentpointsx,y∈ ω withx= y.This differenceisestimated inProposition 3.11,anditisbased onsometechnicallemmas.

Lemma 3.6. Let ψ ∈ C∞(ω). For each x0 ∈ ω and each R > 0 such that Ωφ(x0)(x0,R) ⊂ ω the mean,

¯

m(ψ,φ,R)(x0),of asmoothfunctionφ canbeexpressed asfollows:

¯ m(ψ, φ, R)(x0) = 1 RQ  ˜ Ω(0,R)\˜Ω(0,R 2) K3(0, ˜y)ψ(Expφ(x0),x0(˜y)) dL 2n_(˜y),

where Ω(0,˜ R) isdefined in(2.35) and

K3(0, ˜y) := Q

(Q − 2)(1 − 1 2Q)

|∇Γ(0, ˜y)|2

Γ(0, ˜y)2(Q−1)/(Q−2).

Moreover, there exist constants C˜3,C˜4 depending only Q and on the structure constants C1 and C2 in

(2.29) suchthat |K3(0, ˜y)| ≤ ˜C3, |∇K3(0, ˜y)| ≤ ˜ C4 ˜y ∀˜y ∈ ˜Ω(0, R) \ ˜Ω  0,R 2 . (3.47)

Proof. By(2.34) wehavethat

Ωφ(x0)(x0, R) = Expφ(x0),x0( ˜Ω(0, R)).

Sothat,byDefinition 3.3and(2.30)wehave:

¯ m(ψ, φ, R)(x0) = = C(Q) (Q − 2)RQ  Ω_φ(x0)(x0,R)\Ω_φ(x0)(x0,R2) |∇φ(x0)_Γ φ(x0)(x0, y)| 2 Γ2(_φ(xQ−1)/(Q−2) 0) (x0, y) ψ(y) dL2n(y) = C(Q) (Q − 2)RQ  ˜ Ω(0,R)\˜Ω(0,R 2) |∇Γ(0, ˜y)|2 Γ2(Q−1)/(Q−2)_{(0, ˜y)}ψ(Expφ(x0),x0(˜y)) dL 2n_(˜y),

where in the last equality we have applied a change of variables and the fact that the determinant of the Jacobian matrixof Expφ(x0),x0 is equalto 1.Finally, we observe that(3.47) follows directly from the

estimatesonΓ in(2.29). 2

In the next proposition we will start studying properties of the difference

ψ(Exp_φ(x),x(˜y))− ψ(Exp_φ(x0),x0(˜y)), which, thanks to the previous lemma, can be considered the first stepintheproofof(3.46).

Proposition 3.7. For every x¯ ∈ ω there exist a constant C0 > 0, such that for every R > 0 with

Ωφ(¯x)(¯x,C0R)  ω, every x,x0 ∈ Ωφ(¯x)(¯x,R) and every y˜ ∈ ˜Ω(0,R) there is an integral curve of the

vector fields ∇ˆφ(x0)_,γ ˜

y : [0,1]→ ω joiningExpφ(x),x(˜y) and Expφ(x0),x0(˜y). Moreover, γy˜ can be explicitly written as γy˜(t) := exp  t˜h ˆ∇φ(x0) _{exp(˜y ˆ}∇φ(x0)_)(x 0) t∈ [0, 1], (3.48) where ˜ h = Logφ(x0),Exp_φ(x0),x0(˜y)  Expφ(x),x(˜y) . (3.49)

Proof. Letusfix¯x∈ ω,andasphereΩφ(¯x)(¯x,CR) subsetofω.TheconstantC = C(Lφ,Ωφ(¯x)(¯x,R))> 0 will

bechosenat theend.Wefirstnotethat,thereexists

C = C(L_{φ,Ωφ(¯x)(¯x, ¯R)}) > 0 (3.50)

such thatfor everyx,x0 ∈ Ωφ(¯x)(¯x,R), y˜∈ ˜Ω(0,R),the points Expφ(x),x(˜y) and Expφ(x0),x0(˜y) belong to

Ωφ(¯x)(¯x,CR).By(2.10)weget Exp_φ(x),x(˜y) =x1+ ˜y1, . . . , x2n−1+ ˜y2n−1, x2n+ ˜y2n+ 2˜ynφ(x)− σ(˜y, x) , Exp_φ(x0),x0(˜y) =x0,1+ ˜y1, . . . , x0,2n−1+ ˜y2n−1, x0,2n+ ˜y2n+ 2˜ynφ(x0)− σ(˜y, x0) .

Then,using (3.49)and(2.12), weobtain

˜ hi = (x− x0)i i = 1, . . . , 2n− 1, ˜ h2n = (x− x0)2n− 2φ(x0)(x− x0)n+ 2˜yn(φ(x)− φ(x0))− 2σ(˜y, x − x0) + σ(x, x0) (3.51) andcalling ˜ x := Log_φ(x0),x0(x), (3.52) werealizethat ˜ h = ˜x +2˜yn(φ(x)− φ(x0))− 2σ(˜y, ˜x) e2n. (3.53)

By(3.48)andtheBaker–Campbell–Hausdorffformulawehave

γy˜(t) = exp  t˜h ˆ∇φ(x0)  expy ˆ˜∇φ(x0) (x0) = exp2t˜yn(φ(x)− φ(x0))∂2n+ 2tσ(˜y, ˜x)∂2n− tσ(˜y, ˜x)∂2n+ (t˜x + ˜y) ˆ∇φ(x0) (x0).

Fromthis andusing(2.10)we get

(γy˜(t))i= t(x− x0)i+ (˜y + x0)i i = 1, . . . , 2n− 1

(γy˜(t))2n= t(x− x0)2n+ (˜y + x0)2n+ 2t˜yn(φ(x)− φ(x0))

+ 2φ(x0)˜yn+ σ(t(x− x0) + x0, ˜y). (3.54)

dφ(x0)(x0, γy˜(t))≤ ˜y + ˜x +



˜y˜x +˜y|φ(x) − φ(x0)|, (3.55)

where x is˜ asin(3.52).Indeed,using(2.13)and (3.54)weget

dφ(x0)(x0, γy˜(t))≤  (t˜x1+ ˜y1, . . . , t˜x2n−1+ ˜y2n−1) R2n−1+ +t˜x2n+ ˜y2n+ 2t˜yn(φ(x)− φ(x0)) + tσ(˜x, ˜y) 1 2 ,

and (3.55)follows usingthetriangleinequality.Since, x,x0∈ Ωφ(¯x)(¯x,R) and˜y≤ R thenby(3.52) and

(2.22) weget

˜x ≤ CR, and dφ(x, x0)≤ CR (3.56)

forsomeconstantC = C(L_{φ,Ωφ(¯x)(¯x, ¯R)})> 0.Finally,by(3.55),(3.56)and(2.29) weconcludethat

γy˜(t)∈ Ωφ(x0)(x0, C1R) ∀t ∈ [0, 1]

forsomeC1= C1(Lφ,Ωφ(¯x)(¯x,R))> 0 andthethesisfollowswith C0= max(C,C1). 2

Proposition3.8.LetC0beasinProposition 3.7.Foreveryx¯∈ ω suchthatΩφ(¯x)(¯x,C0R) ω andforevery

x,x0∈ Ωφ(¯x)(¯x,R) and forevery y˜∈ ˜Ω(0,R) wehave

ψ(Expφ(x),x(˜y))− ψ(Expφ(x0),x0(˜y)) =

= 1  0 2n  i=1 (Log_φ(x0),x0(x))i∇ˆiφ(x0)ψ(γy˜(t))dt + K4(x, x0, ˜y) 1  0 ∂2nψ(γy˜(t))dt, (3.57) where K4(x, x0, ˜y) := 2(φ(x)− φ(x0))˜yn− 2σ(˜y, x − x0). (3.58)

The kernel K4 isofclassC∞ with respecttoy and˜ thefollowingestimateshold:

|K4(x, x0, ˜y)| ≤ 2(Lφ,Ωφ(¯x)(¯x,R)+ 1)dφ(x, x0)˜y, (3.59) |∇K4(x, x0, ˜y)| ≤ 2(Lφ,Ωφ(¯x)(¯x,R)+ 1)dφ(x, x0). (3.60)

Proof. Sinceψ∈ C∞(ω) andγy˜ishorizontalwithrespecttothefamilyofvectorfields{ ˆ∇φ(x0)},weobtain

ψ(Expφ(x),x(˜y))− ψ(Expφ(x0),x0(˜y)) = 1  0 (ψ◦ γy˜)(t)dt = 2n  i=1 1  0 ˜ hi∇ˆφ(xi 0)ψ(γy˜(t))dt,

so that(3.57) immediately follows using (3.51). In order to prove(3.59) it suffices to observe that σ(x− x0,y)˜ ≤ dφ(x,x0)˜y. Moreover, since ∂y˜2nK4(x,x0,y)˜ = 0 itfollows that to prove(3.60) it is enough to

estimate theEuclidean gradientofK4 (withrespectto thevariable y).˜ Byadirectcomputationand using

∂y˜iK4(x, x0, ˜y) =−2(x − x0)n+i if i = 1, . . . , n− 1, ∂y˜nK4(x, x0, ˜y) = 2(φ(x)− φ(x0)),

∂y˜iK4(x, x0, ˜y) = 2(x− x0)i if i = n + 1, . . . , 2n− 1.

Hence|∇K4(x,x0,y)˜|≤ 2(Lφ,Ωφ(¯x)(¯x,R)+ 1)dφ(x,x0),whichisthethesis. 2

There is anaturalchange of variables, naturallyassociated to the curve γy˜ definedin Proposition 3.7.

Indeedthefollowinglemmaholds:

Lemma3.9. Letx¯∈ ω,C0> 0 andγy˜beasProposition 3.7.Foreach R > 0,suchthatΩφ(¯x)(¯x,C0R)⊂ ω,

x,x0∈ Ωφ(¯x)(¯x,R) and y˜∈ ˜Ω(0,R).Thenthefunction

H : [0, 1]× ˜Ω(0, R) −→ [0, 1] × ω

(t, ˜y)→ (t, γy˜(t))

has inverse function(t,F (z,˜ t)), themap z → (t,F (z,˜ t)) is C∞ andits Jacobian matrix has determinant equalto1.

Proof. Using(3.54),(2.10)and setting(t,z):= (t,γy˜(t)),F can˜ beexpressedas

˜

Fi(z, t) = (z− x0)i− t(x − x0)i i = 1, . . . , 2n− 1,

˜

F2n(z, t) = (z− x0)2n− t(x − x0)2n− 2t((z − x0)n− t(x − x0)n)(φ(x)− φ(x0))

− 2φ(x0)((z− x0)n− t(x − x0)n) + σ(z, t(x− x0) + x0). (3.61)

In particular it is clear from (3.61) that F is˜ of class C∞ as a function of the variable z and that the Jacobiandeterminantof z→ ˜F (z,t) is equalto 1 foreacht∈ [0,1]. 2

Lemma3.10. Letg∈ C∞(Rn) andF (z,˜ t) as inLemma 3.9 then

∇φ(x0) zi (g( ˜F (z, t)) = ⎧ ⎪ ⎨ ⎪ ⎩ (∇y˜ig)( ˜F (z, t)) i = 1, . . . , n− 1, (∇y˜ng)( ˜F (z, t))− 2t(φ(x) − φ(x0))(∂y˜2ng)( ˜F (z, t)) i = n, (∇y˜ig)( ˜F (z, t)) i = n + 1, . . . , 2n− 1, (3.62)

where(∇1,. . . ,∇2n−1) isthefamily ofvector fieldsdefined in(2.24).

Proof. Letusstartcomputing ∇φ(x0)

zi (g( ˜F (z,t))) withi= 1,. . . ,n− 1,thatis



∂zi− zi+n∂z2n

(g( ˜F (z, t))). (3.63)

Tothisend,wecalculate

∂zi(g( ˜F (z, t))) and ∂z2n(g( ˜F (z, t))).

Bytheexplicitexpression ofF (z,˜ t) weobtain:

∂zi(g( ˜F (z, t))) = (∂y˜ig)( ˜F (z, t)) + (∂y˜2ng)( ˜F (z, t))∂ziF˜2n(z, t), (3.64)

henceby(3.63), (3.64)and(3.65)weget ∇φ(x0) zi (g( ˜F (z, t))) =  ∂y˜ig− ˜Fi+n(z, t)∂˜y2ng ( ˜F (z, t))+ +  ˜ Fi+n(z, t)− zi+n+ ∂ziF˜2n(z, t) ∂y˜2ng( ˜F (z, t)). Since ˜ Fi(z, t) = (z− x0)i− t(x − x0)i i = 1, . . . , 2n− 1 ˜ F2n(z, t) = (z− x0)2n− t(x − x0)2n− 2t((z − x0)n− t(x − x0)n)(φ(x)− φ(x0)) + − 2φ(x0)((z− x0)n− t(x − x0)n) + σ(z, t(x− x0) + x0) (3.66) this implies ∇φ(x0) zi (g( ˜F (z, t))) = (∇y˜ig)( ˜F (z, t)).

Thecomputations for∇φ(x0)

zi (g( ˜F (z,t))) wheni= n+ 1,. . . ,2n− 1 aresimilar.

Finally,letuscompute∇φ(x0)

zn (g( ˜F (z,t))).Bydefinition: ∇φ(x0) zn (g( ˜F (z, t))) =  ∂zn+ 2φ(x0)∂z2n (g( ˜F (z, t))) (3.67) and since ∂zn(g( ˜F (z, t))) = (∂y˜ng)( ˜F (z, t))− 2[t(φ(x) − φ(x0)) + φ(x0)](∂y˜2ng)( ˜F (z, t)) (3.68) by(3.67),(3.65)and(3.68) weget ∇φ(x0) zn g( ˜F (z, t)) = (∇y˜ng)( ˜F (z, t))− 2t(φ(x) − φ(x0))(∂y˜2ng)( ˜F (z, t)). 2

Proposition 3.11.Forevery t∈ [0,1],c1,c2> 0 and r > 0 letus define

Dt,c1,c2,r :=

z∈ R2n: (z, t)∈ ˜F−1Ω(0, c˜ 2r)− ˜Ω(0, c1r)

. (3.69)

Let x¯ ∈ ω and C0 > 0 be as in Proposition 3.7, then for every 0< R such that Ωφ(¯x)(¯x,C0R) ⊂ ω and

x,x0∈ Ωφ(¯x)(¯x,R) with x= x0 itholds m(ψ, φ, R)(x)− m(ψ, φ, R)(x0) = 2 R 1  0 R  R 2  D_{t, 1} 2,1,r < K5(x, x0, t, z, r),∇φψ(z) > dL2n(z)drdt + 1  0  D_{t, 1} 2,1,R < K6(x, x0, t, z, R),∇φψ(z) > dL2n(z)dt − 1  0  D_{t, 1} 4, 12,R < K7(x, x0, t, z, R),∇φψ(z) > dL2n(z)dt

forsuitablekernelsK5,K6,K7definedin(3.74),(3.75)and(3.76)respectively.Moreover,therearepositive

|K5(x, x0, t, z, r)| ≤ ˜C5(Lφ+ 1)2 dφ(x0)(x0, x) rQ on Dt,12,1,r,∀t ∈ [0, 1], (3.70) |K6(x, x0, t, z, R)| ≤ ˜C6(Lφ+ 1)2 dφ(x0)(x0, x) R ˜F (z, t)Q−1 on Dt,12,1,R,∀t ∈ [0, 1], (3.71) |K7(x, x0, t, z, R)| ≤ ˜C6(Lφ+ 1)2 dφ(x0)(x0, x) R ˜F (z, t)Q−1 on Dt,14, 1 2,R,∀t ∈ [0, 1]. (3.72)

Proof. ByLemma 3.6forevery0< r < R0 suchthatΩφ(x)(x,r),Ωφ(x0)(x0,r) ω,wehave

¯ m(ψ, φ, r)(x)− ¯m(ψ, φ, r)(x0) = = 1 rQ  ˜ Ω(0,r)\˜Ω(0,r 2) K3(0, ˜y) 

ψ(Expφ(x),x(˜y))− ψ(Expφ(x0),x0(˜y))

dL2n(˜y) byProposition 3.7 = 1 rQ  ˜ Ω(0,r)\˜Ω(0,r 2) K3(0, ˜y) 1  0 < Log_φ(x0),x0(x), ˆ∇φ(x0)_ψ(γ ˜ y(t)) > dtdL2n(˜y) + 1 rQ  ˜ Ω(0,r)\˜Ω(0,r 2) K3(0, ˜y) 1  0 K4(x, x0, ˜y)∂2nψ(γy˜(t))dtdL2n(˜y).

Thechangeofvariablesz = γ˜y(t),changesΩ(0,˜ r)\ ˜Ω(0,r₂) inthesetDt,1

2,1,r andtheinversemappinghas

Jacobiandeterminantequalto1.Hence

m(ψ, φ, R)(x)− m(ψ, φ, R)(x0) = = 2 R 1  0 R  R 2 1 rQ  D_{t, 1} 2,1,r K3(0, ˜F (z, t)) < Logφ(x0),x0(x), ˆ∇ φ(x0)_{ψ(z) > d}L2n_{(z)drdt +} + 2 R 1  0 R  R 2 1 rQ  D_{t, 1} 2,1,r K3(0, ˜F (z, t))K4(x, x0, ˜F (z, t))∂2nψ(z)dL2n(z)drdt. (3.73)

NowapplyingRemark 3.4wegetthethesiscalling:

K5(x, x0, t, z, r) := 1 rQK3(0, ˜F (z, t))Logφ(x0),x0(x) + + 1 2rQ∇ φ(x0) 1  K3(0, ˜F (z, t))K4(x, x0, ˜F (z, t)) en+1+ + 1 2rQ∇ φ(x0) n+1  K3(0, ˜F (z, t))K4(x, x0, ˜F (z, t)) e1; (3.74) K6(x, x0, t, z, R) := 1 R ∇φ(x0) 1 K( ˜F (z, t)) KQ( ˜F (z, t)) K3(0, ˜F (z, t))K4(x, x0, ˜F (z, t))en+1+ + 1 R ∇φ(x0) n+1 K( ˜F (z, t)) KQ( ˜F (z, t)) K3(0, ˜F (z, t))K4(x, x0, ˜F (z, t))e1; (3.75)

K7(x, x0, t, z, R) :=− 1 R ∇φ(x0) 1 K( ˜F (z, t)) 2Q_KQ_{( ˜F (z, t))} K3(0, ˜F (z, t))K4(x, x0, ˜F (z, t))en+1 + 1 R ∇φ(x0) n+1 K( ˜F (z, t)) 2QKQ( ˜F (z, t)) K3(0, ˜F (z, t))K4(x, x0, ˜F (z, t))e1, (3.76)

where asusualei denotesthei-thelementofthecanonicalbasisofR2n−1.Toprove(3.70)weobservethat

byLemma 3.10 K5(x, x0, t, z, r) = 1 rQK3(0, ˜F (z, t))Logφ(x0),x0(x) + 1 2rQ  (∇1K3)(0, ˜F (z, t))K4(x, x0, ˜F (z, t)) + (∇1K4)(x, x0, ˜F (z, t))K3(0, ˜F (z, t)) en+1 + 1 2rQ  (∇n+1K3)(0, ˜F (z, t))K4(x, x0, ˜F (z, t)) + (∇n+1K4)(x, x0, ˜F (z, t))K3(0, ˜F (z, t)) e1

henceusing(3.47),(3.59)and(3.60) weget

|K5(x, x0, t, z, r)| ≤ ˜ C3 rQdφ(x0)(x0, x) + 2 ˜ C4(Lφ+ 1)dφ(x, x0) rQ + 2 ˜ C3(Lφ+ 1)dφ(x, x0) rQ

and theconclusionfollows using(2.20).Finally,(3.71) and(3.72)aredirectconsequencesof(2.29),(3.47), (3.59) andLemma 3.10. 2

4. Poincaréinequality

Thescope ofthissectionis toproveTheorem 1.2.ThePoincarétypeinequalityprovedhereispartially inspired to the Sobolev type inequality for vector fields with non-regular coefficients contained in [23] and successively extended to a more general class of vector fields in [53]. The key point in our strategy, is to establish a representation formula for intrinsic Lipschitz continuous functions. To this end we use Theorem 2.6andtherepresentationformulaprovedinProposition 3.11forC∞functions.

Throughout this sectionwe denote by ω an open and bounded subset of R2n _{with n ≥ 2 andbyφ an}

intrinsic Lipschitzfunctiondefinedonω withLipschitzconstantequaltoLφ.

Let ψ ∈ Wφ and let {ψk}k∈N, {φk}k∈N smooth functions on ω which satisfy conditions (i)− (iv) in

Definition 1.1.Wedenotebydφk(x0)thedistanceintroducedin(2.13),byΓφk(x0)thefundamentalsolution

oftheoperatorLφk(x0)definedin(2.30)andbyΩφk(x0)(x0,r) thesuperlevelsetofΓφk(x0)definedin(2.31).

Westartprovingthattheaveragem(ψ,¯ φ,R)(x0) canbeapproximatedbymeansoftheregularsequence

¯

m(ψk,φk,R)(x0).Precisely:

Lemma 4.1.Letx0∈ ω andR > 0 such that Ωφ(x0)(x0,R)⊂ ω.Then (i) χΩ_φk(x0)(x0,R)→ χΩ_φ(x0)(x0,R) uniformlyin ω ask→ +∞;

(ii) ¯m(ψk,φk,R)(x0)→ ¯m(ψ,φ,R)(x0) uniformly inR > 0 as k→ +∞.

HereχA denotesthecharacteristicfunctionof A.

Proof. Werecallthat

˜

Ω(0, R) =y˜∈ R2n | Γ(0, ˜y) > R2−Q,

Ωφk(x0)(x0, R) = Expφk(x0),x0( ˜Ω(0, R)).

Using the explicit form of Expφk(x0),x0 and Expφ(x0),x0 stated in (2.10) we easily conclude that

(Exp_φk(x0),x0)k∈NuniformlyconvergestoExpφ(x0),x0 inω ask→ +∞.Inordertoprove(i) weobservethat

itissufficienttoprovethatforall > 0 thereexists ¯k = ¯k( )> 0 suchthatforallk > ¯k

Ωφ(x0)(x0, R)⊆ (Ωφk(x0)(x0, R)) (4.77)

where

(Ωφk(x0)(x0, R)):={y ∈ ω | dφk(x0)(∂Ωφk(x0)(x0, R), y) < }. (4.78)

Forsimplifyingthenotationwedefine

Ek( ˜Ω(0, R)) := Expφk(x0),x0( ˜Ω(0, R)), E( ˜Ω(0, R)) := Expφ(x0),x0( ˜Ω(0, R)).

Supposebycontradictionthatthereexists > 0 suchthatforeveryk there¯ arek > ¯k andyk∈ E(˜Ω(0,R))

suchthatyk∈ E/ k( ˜Ω(0,R)).Then,thereexist(kj)j,kj→ +∞ asj→ +∞ and(xkj)j inΩ(0,˜ R) suchthat E(xkj) ∈ E/ kj( ˜Ω(0,R)). Sothat, the distance between E(xkj) and Ekj(xkj) isgreater than and this is

absurdbeing Ek uniformly convergenttoE.Then,(4.77) followsandhence(i).

Toprove(ii) weobservethatbyDefinition 3.3:

lim k→+∞m(ψ¯ k, φk, R)(x0) = = lim k→+∞ C(Q) Q − 2 1 RQ  ω |∇φk(x0)_Γ φk(x0)(x0, y)| 2 Γφk(x0)(x0, y)2(Q−1)/(Q−2) ψk(y) χΩ_φk(x0)(x0,R)\Ωφk(x0)(x0,R2)(y) dL 2n_(y). By(2.29)and(2.30) lim k→+∞ |∇φk(x0)_Γ φk(x0)(x0, y)| 2 Γφk(x0)(x0, y) 2(Q−1)/(Q−2) = |∇φ(x0)_Γ φ(x0)(x0, y)| 2 Γφ(x0)(x0, y) 2(Q−1)/(Q−2) ≤ C ∀y = x0 (4.79)

therefore,(ii) followsfrom (i) andthefactthatψk→ ψ inL1loc(ω). 2

Inwhat follows weprove thattherepresentation formulas obtainedinProposition 3.5 and in Proposi-tion 3.11forC∞ functionsstillholdifφ isintrinsicLipschitzandψ∈ Wφ(ω).

Lemma4.2.Letφ beaLipschitzcontinuousfunctionandψ∈ Wφ(ω).Foreachx0∈ ω andeachR > 0 such

that Ωφ(x0)(x0,R)⊂ ω, thefollowingformula holds:

ψ(x0) = m(ψ, φ, R)(x0) + IR(x0), wherem(ψ,φ,R)(x0) is asin(3.40) and IR(x0) := 2 R R  R 4 f1 _r R  Ω_φ(x0)(x0,r)  K1(x0, y),∇φψ(y) dL2n(y)dr +2 R R  R 2  Ω_φ(x0)(x0,r)\Ωφ(x0)(x0,r2)  K2(x0, y, r),∇φψ(y) dL2n(y)dr, (4.80)

where K1, K2 are as in Proposition 3.5. Letx¯ ∈ ω and C0 > 0 be as in Proposition 3.7, then for every

0< R such that Ωφ(¯x)(¯x,C0R)⊂ ω andx,x0∈ Ωφ(¯x)(¯x,R) withx= x0 itholds:

m(ψ, φ, R)(x)− m(ψ, φ, R)(x0) = = 2 R 1  0 R  R 2  D_{t, 1} 2,1,r < K5(x, x0, t, z, r),∇φψ(z) > dL2n(z)drdt + 1  0  D_{t, 1} 2,1,R < K6(x, x0, t, z, R),∇φψ(z) > dL2n(z)dt − 1  0  D_{t, 1} 4, 12,R < K7(x, x0, t, z, R),∇φψ(z) > dL2n(z)dt, (4.81)

whereDt,c1,c2,r isasin(3.69)andK5,K6,K7areasin(3.74),(3.75)and(3.76)respectivelyandtheysatisfy thesame estimatesprovedinProposition 3.11 withpossibly different constants.

Proof. Bydefinitionof1.1thereare{ψk}k∈N,{φk}k∈Nsequencesofsmoothfunctionsdefinedonω satisfying

conditions(i)–(iv) ofDefinition 1.1.ByPropositions 3.5 and3.11,thethesisistrueforeveryφk,ψkasabove.

Passing tothelimitasinthepreviousproposition,itholdstruealsoforthelimitfunctionsφ and ψ. 2

It is well known (see for example [14,21]) that the key step in the proof of the Poincaré inequality is a representation formula as the one proved in Lemma 4.2, which is indeed equivalent to the Poincaré inequality itself. For further applications,we note that we can obtainthe representation formula on any family of balls, equivalent to the superlevels Ωφ(¯x)(¯x,R),which canbe Ωφ(¯x)(¯x,R) or Uφ(¯x,R), defined

respectivelyin(2.31)and(1.5).

Let us denote by Bφ(¯x,R) a family of spheres centered at x and¯ radius R, equivalent to the family

Ωφ(¯x)(¯x,R).Let us denote by ψBφ(¯x,R) the mean of ψ on the set Bφ(¯x,R) with respect to the Lebesgue

measure,i.e. ψBφ(¯x,R):= 1 L2n_(B φ(¯x, R))  Bφ(¯x,R) ψ(x) dL2n(x) (4.82)

we willprovethefollowing result:

Weuseourrepresentationformulatoproveanupperboundforthequantity|ψ(x0)−ψBφ(¯x,R)|,precisely:

Proposition 4.3. Let φ ∈ Lip(ω) and ψ ∈ Wφ(ω). Let x¯ ∈ ω and C0 > 0 be as in Proposition 3.7.

There are C˜0 > C0, C˜1,C˜2 > 0, depending only on Lφ,ω,Q and the structure constants in (2.29), such

that if Bφ(¯x,C˜0R) ω andx0∈ Bφ(¯x,R) then

|ψ(x0)− ψBφ(¯x,R)| ≤ ≤ ˜C1  Bφ(¯x, ˜C0R) d1_φ−Q(x0, y)|∇φψ(y)|dL2n(y) + + C˜2 L2n_(B φ(¯x, R))  Bφ(¯x, ˜C0R)  Bφ(¯x, ˜C0R) d1_φ−Q(x, y)|∇φψ(y)|dL2n(y)dL2n(x). (4.83)

Proof. Let R > 0 such that Bφ(¯x,R) ⊂ Ωφ(¯x)(¯x,C0R) ⊂ Bφ(¯x,C˜0R)  ω. By Lemma 4.2, for each x,x0∈ Bφ(¯x,R) we have: ψ(x0) = m(ψ, φ, R)(x0) + IR(x0), ψ(x) = m(ψ, φ, R)(x) + IR(x), hence ψ(x0)− ψ(x) = m(ψ, φ, R)(x0)− m(ψ, φ, R)(x) + IR(x0)− IR(x). (4.84)

Integratingequation(4.84)withrespecttothevariablex onthesphereBφ(¯x,R) andrecallingthedefinition

ofψBφ(¯x,R) weget ψ(x0)− ψBφ(¯x,R)= 1 L2n_(B φ(¯x, R))  Bφ(¯x,R) m(ψ, φ, R)(x0)− m(ψ, φ, R)(x) dL2n(x) + 1 L2n_(B φ(¯x, R))  Bφ(¯x,R) IR(x0)− IR(x) dL2n(x). Hence: |ψ(x0)− ψBφ(¯x,R)| ≤ 1 L2n_(B φ(¯x, R))  Bφ(¯x,R)  m(ψ, φ, R)(x0)− m(ψ, φ, R)(x) dL2n(x) +|IR(x0)| + 1 L2n_(B φ(¯x, R))  Bφ(¯x,R) |IR(x)| dL2n(x). (4.85)

Now,byLemma 4.2,wehave:

|m(ψ, φ, R)(x0)−m(ψ, φ, R)(x)| ≤ 2 R 1  0 R  R 2  D_{t, 1} 2,1,r | < K5(x, x0, t, z, r),∇φψ(z) >|dL2n(z)drdt + 1  0  D_{t, 1} 2,1,R | < K6(x, x0, t, z, R),∇φψ(z) >|dL2n(z)dt + 1  0  D_{t, 1} 4, 12,R | < K7(x, x0, t, z, R),∇φψ(z) >|dL2n(z)dt.

Weclaimthatthere existsC = C(Lφ)> 0 suchthatforallr∈ (R/2,R),t∈ [0,1] itholds

Dt,1

2,1,r⊆ Ωφ(x0)(x0, CR)⊂ Bφ(¯x, ˜C0R). (4.86)

Tothisend letus fixt∈ [0,1] andr∈ (R/2,R) thenforeachy˜∈ ˜Ω(0,r)\ ˜Ω(0,r/2) wehave

r

and, by(3.55),italsoholds

dφ(x0)(x0, γy˜(t))≤ ˜y + ˜x +



˜y˜x +˜y|φ(x) − φ(x0)|. (4.88)

Since x,x0∈ Bφ(¯x,R) by(2.20)and (2.29)wehave

dφ(x, x0)≤ CR and ˜x ≤ CR (4.89)

forsomeC = C(Lφ)> 0.Using(4.87),(4.88)and(4.89)weimmediatelyget(4.86)withpossiblysmaller R.

By Lemma 4.2 we know that the estimates for K5, K6, K7 proved in Proposition 3.11 also hold for

φ∈ Lip(ω).Hence,by(3.70)and (4.87)foreachz∈ D_t,1

2,1,r andt∈ [0,1] wehave |K5(x, x0, t, z, r)| ≤ ˜C dφ(x0)(x0, x) rQ ≤ ˜C dφ(x0)(x0, x)  ˜F (z, t)Q

forsomeC = ˜˜ C(Lφ)> 0.Using(4.86) and(4.89)weget

dφ(x0)(x0, x) =˜x ≤ CR ≤ 2Cr ≤ C ˜F (z, t)

forasuitableconstantC = C(Lφ)> 0.Then

|K5(x, x0, t, z, r)| ≤ C 1  ˜F (z, t)Q−1. Moreover, by(4.86),z∈ Ωφ(x0)(x0,CR) and 0 < dφ(x0)(x0, z)≤ CR ≤ 2Cr ≤ C ˜F (z, t). Sothat |K5(x, x0, t, z, r)| ≤ Cdφ(x0)(x0, z) 1−Q_{. (4.90)}

Analogously, wecanprovethatthere existsC = C(Lφ)> 0 such that

|K6(x, x0, t, z, r)|, |K7(x, x0, t, z, r)| ≤ Cdφ(x0)(x0, z)

1−Q_{. (4.91)}

Inconclusionweprovedthat

|m(ψ, φ, R)(x0)− m(ψ, φ, R)(x)| ≤ ≤ ˜C1  Ω_φ(x0)x0,CR  d1_φ(x−Q 0)(x0, z)|∇ φ_ψ(z)|dL2n_{(z). (4.92)}

Furthermore, byLemma 4.2,(3.41) and(3.42)wehave

|IR(x0)| ≤ ˜C2Lφ  Ω_φ(x0)(x0,R) d1_φ(x−Q 0)(x0, y)|∇ φ_ψ(y)|dL2n_{(y), (4.93)} |IR(x)| ≤ ˜C2Lφ  Ωφ(x)(x,R) d1_φ(x)−Q(x, y)|∇φψ(y)|dL2n(y). (4.94)