New characterizations of Sobolev spaces on the Heisenberg group

New characterizations of Sobolev spaces on the Heisenberg group

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New characterizations of Sobolev spaces on the

Heisenberg group✩

Xiaoyue Cuia, Nguyen Lama, Guozhen Lub,a,∗

a Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

b School of Mathematical Sciences, Beijing Normal University, Beijing, 100875,

character-iff ∈ L p(H),p >1, thenf ∈ W 1,p(H) if and only if

Bour-in the setting of Heisenberg group Second, corresponding to the casep= 1, we give a characterizations of BV functions on the Heisenberg group ( Theorems 4.1 and 4.2 ) Third, we give some more generalized characterizations of Sobolev spaces on the Heisenberg groups ( Theorems 5.1 and 5.2 ).

It is worth to note that the underlying geometry of the clidean spaces, such as that any two points in RN can be connected by a line-segment, plays an important role in the proof of the main theorems in [29] Thus, one of the main techniques in [29] is to use the uniformity in every directions

Eu-of the unit sphere in the Euclidean spaces More precisely, to

✩

Research is partly supported by a US NSF grant DMS#1301595.

* Corresponding author at: Department of Mathematics, Wayne State University, Detroit, MI 48202, USA.

E-mail addresses:xiaoyue.cui@wayne.edu (X Cui), nguyenlam@wayne.edu (N Lam), gzlu@wayne.edu

(G Lu).

http://dx.doi.org/10.1016/j.jfa.2014.08.004

0022-1236/© 2014 Elsevier Inc All rights reserved.

deal with the general caseσ∈ SN−1 , it is often assumed that

σ = e N = (0, , 0,1) and, hence, one just needs to work on the one-dimensional case This can be done by using the ro- tation in the Euclidean spaces Due to the non-commutative nature of the Heisenberg group, the absence of this unifor- mity on the Heisenberg group creates extra difficulties for us

to handle Hence, we need to find a different approach to tablish this characterization.

es-© 2014 Elsevier Inc All rights reserved.

Theclassicaldefinitionof Sobolevspaceisas follows:TheSobolevspaceW k,p (Ω) is

definedto betheset of allfunctionsu ∈ L p (Ω) suchthatforeverymulti-index αwith

|α| ≤ k,theweakpartialderivativeD α ubelongstoL p (Ω),i.e

W k,p (Ω) =

u ∈ L p (Ω) : D α u ∈ L p (Ω), ∀|α| ≤ k

.

Here,ΩisanopensetinRn and1≤ p≤ +∞.Thenaturalnumberkiscalledtheorder

ofthe Sobolevspace W k,p (Ω). This definitioncanbe extendedeasily to other settingssuchasRiemannianmanifolds,sincethegradientiswell-definedthere[18].Moreover,wecanalsodefinethefractionalSobolevspace,wheretheorderkisnotanaturalnumber,viaBesselpotentials [33]

Sobolevspaces onRiemannianmanifoldsor with metricmeasure spaces as targetedspaceshavebeenstudiedby,e.g.,KorevaarandSchoen[19],Hebey[18],etc.TherehavebeencharacterizationsofSobolevspacesindoublingmetricmeasurespaces.Forinstance,variouscharacterizationsoffirstorderSobolevspacesinmetricmeasurespaceshavebeengiven using a Lipschitztype (pointwise) estimate by Hajlasz [17], then using PoincarétypeinequalitiesbyFranchi,LuandWheeden[16]forthefirstorderSobolevspaces(seealsoFranchi,HajlaszandKoskela[15]),andsubsequentlybyLiu,LuandWheeden [21]forhighorderSobolevspaces,etc.TheHeisenberggroup(andmoregenerally, stratifiedgroups) is aspecial case of metric measure spaces with doubling measures The char-acterizations given in [17,16] and [21] also give alternative definitions of non-isotropicSobolevspacesontheHeisenberggroup.Indeed,itwasshownin[21]thatthedefinition

ofnon-isotropicFolland–Steinspaces[14]isequivalenttotheSobolevspacesonstratifiedgroupsusing thehigherorder Poincaréinequalities(seealso[23,24,26,27,10])

Nevertheless,themainpurposeofourpaperfocusesonthosetypesofcharacterizations

of Sobolev spaces on the Heisenberg group in the spirit of characterizations given by

To thisend,wewill firstrecallthoseresultsof[5]and[29]

Theorem A.(See Bourgain, Brezis and Mironescu [5] ) Let g ∈ L p(RN ), 1 < p < ∞ Then g ∈ W 1,p(RN ) iff

This resultis studiedfurtherandextendedin[2–4,7,20,28,31]

Recently, Hoai-Minh Nguyen [29] established some new characterizations of theSobolev spaceW 1,p(RN) that areclosely relatedto Theorem A Moreprecisely, itwasconjecturedbyBrezisandconfirmedin[29]that

Theorem B.(See H.M Nguyen [29] ) Let1< p < ∞ Then

(a) There exists a positive constant C N,p depending only on N and p such that

Inthispaper,wewillestablishresultsofthetypesimilartoTheorem Binthesetting

of Heisenberg groups Let H = Cn× R be the n-dimensionalHeisenberg groupwhosegroupstructure isgivenby

However,forsimplicitywe willwrite ruto denoteδ r u TheJacobiandeterminantof δ r

isr Q,where Q = 2n+ 2 isthehomogeneousdimensionof H

Weuseξ = (z, t) todenoteanypoint(z, t)∈ H andρ (ξ) = (|z|4+ t2)1 todenotethehomogeneousnormofξ∈ H.Withthis norm,we candefine aHeisenberg ballcentered

at ξ = (z, t) withradius R: B (ξ, R)= {v ∈ H : ρ(ξ−1· v) < R}.Thevolumeof suchaball isσ Q = C Q R Q forsomeconstant C Q depending onlyon Q Wealso define Σ theunitsphereintheHeisenberggroupH:

Σ=

ξ ∈ H : ρ(ξ) = 1

.

Weuse∇Hf toexpressthehorizontalsubgradientofthefunctionf : H→ R:

Theorem 1.1.Let1< p < ∞ and f ∈ W 1,p (H) Then

(a) There exists a positive constant C Q,p depending only on Q, p such that

Σ

|

(e, 0), σ

|p dσ

for any (e,0)∈ Σ.

(c) There exists a positive constant C Q,p such that

OurTheorem 1.1 extendsTheorem 3 of H.M Nguyen[29] in theEuclidean spaces.UsingTheorem 1.1,wesetupnewcharacterizationsoftheSobolevspaceW 1,p(H) which

isoneof themain purposesofthispaper Moreprecisely,weprovethat

Theorem1.2 Let1< p < ∞ and f ∈ L p (H) Then the following are equivalent:

for some 1≤ q < p, where r (B) is the radius of the ball B.

Theequivalenceof(1),(2)and(3)intheaboveTheorem 1.2intheEuclideanspaceswasgiveninTheorem 3of[29]

The following remarks are in order First, the proofs of the main theorems (e.g.,Theorem B) in [29] rely on the underlying geometry of the Euclidean spaces, such asthat any two points can be connected by a line-segment Second, it is worth to notethatoneof themain techniques inthe proof ofTheorem B is to usethe uniformityineverydirectionsoftheunitsphereintheEuclidean spaces.Moreprecisely, todealwiththe generalcase σ∈ SN −1, it is oftento be assumed thatσ = e N = (0, , 0,1) and,hence, one just needs to work on 1-dimensional case This can be done by using therotationintheEuclideanspaces.InthecaseofHeisenberggroups,thistypeofproperty

is not available because of the structure of the Heisenberg groups, in particular, the

dilation Hence, we need to find a different approach to this characterization In fact,

wewill usetherepresentationformulaontheHeisenberggroupprovedin[22]to obtainestimate (2.1) This estimate will allow us to establish auseful lemma (Lemma 2.2 inSection2) Third,aswehaveshownin[21],(1),(4)and(5)areallequivalent.Therefore,the new ingredienthere is that (1), (2) and (3) are equivalent Fourth, results of thispapertogetherwithcharacterizationsofsecondorderSobolevspacesinEuclideanspacesestablishedin[12]havebeenpresentedin[11].Wehavealsoextendedresultsinthispaper

to generalstratifiedgroupsin[13]

The plan of this paper is as follows: In Section2, we will study some helpful mas andusethem toproveTheorem 1.1whichgivespropertiesofSobolevfunctionsin

lem-W 1,p(H) for 1< p <∞ Theorem 1.2 givesthe characterizationsof Sobolev functions

in W 1,p(H) for 1 < p < ∞ and will be considered in Section 3 The borderline case

p = 1 (i.e., for BV functions) will be investigated in Section 4 We also study somegeneralizationsandvariantsinSection5whichextendsTheorems 1.1 and1.2

2 ProofofTheorem 1.1

2.1 Some preliminary lemmas

Wefirstrecallanelementarylemmafrom [29]andincludeaproof

Lemma 2.1 Let Ω be a measurable set in Rm , Φ and Ψ be two measurable nonnegative functions on Ω, and α > −1 Then

thateverytwopointscanbeconnectedbyaline-segmentandthenusedthemean-valuetheoremtocontrolthedifferenceof|f (x) − f (x + he)| (where h∈ RN ande∈ SN −1)bythe Hardy–Littlewood maximal function of the partial derivative of f inthe direction

ofe Such anargument doesnotworkon theHeisenberg group.Therefore, weneed toadaptanewargumentbyusing therepresentation formulaonthe HeisenberggroupHestablishedin[22]

Lemma2.2 Let f ∈ W 1,p (H),1< p < ∞ Then we have

where C Q,p is a positive constant depending only on Q and p.

Proof First,werecallthefollowingpointwiseestimateonstratifiedgroupsprovedin[22](seeLemma 3.1onpage 382there),foranymetricball B in H andeveryu ∈ B,wehave

andA Q,p istheuniversalconstantdependingonlyonQandp

Nownotingthatby(2.1):

weget

Theproofnowiscompleted ✷

Beforewestateandprovethenextlemma,welike tomakethefollowingremark.Let

ByLemma 2.2,lim infδ→0 I (δ) doesexist.InthesettingofEuclideanspaces[29],thislimitis rather easyto evaluate Moreprecisely, bypolar coordinatesand therotations

inthe Euclidean spaces, it is often assumed in[29] thatσ = e N = (0, , 0,1) Thus,

to deal with the general case σ ∈ SN −1, the author in [29] just needs to consider theone-dimensionalcase.Then,usingrealanalysistechniquessuchastheMaximalfunction,Lebesgue’s dominated convergence theorem, Hoai-Minh Nguyen finds successfully theexactvalueoflim infδ→0 I (δ).InoursettingofHeisenberggroupH,thisapproachisnotavailablebecause oftheunderlyinggeometry ontheHeisenberg group.Hence,weneed

toproposeanewmethodinordertocalculatelim infδ→0 I (δ). Indeed,ourmain ideaisthatwewill firststudytherelationsofI (δ) and J (ε).

Infact,wecanprovethefollowingresultsontheHeisenberggroupwhichareimportantfor us to establish the characterizationsof Sobolev spaces W 1,p(H) on the Heisenberggroup.

Lemma 2.3.Let f ∈ W 1,p (H), 1< p < ∞ There hold

Proof First,wenoticebythechangeof variablethat

Lemma 2.5.Let f ∈ C1(H) Then

Inthefollowing,Cwill beaconstantindependentofu, h, σ, ε

Since f ∈ C1(H), bytriangleinequalityand Taylorexpansion[25],wehave

(b)From Lemmas 2.4,2.3,2.5andthedensityargument,wehave (b).

(c) ThisisLemma 2.2

(d)This isaconsequenceofLemmas 2.4,2.3,2.5andthedensityargument

3 ProofofTheorem 1.2

Theproof isdividedinto six steps

Step1:(1)⇒ (2).Thisisaconsequenceofpart(a)ofTheorem 1.1andthefactthat

Thus,withanextraassumptionf ∈ L∞(H), wehavethat(2)⇒ (1).

Forthegeneralcase,wemakeuseofthetruncated function.ForR >0,define

 ≤f(u) − f(v) for all u, v ∈ H.

Asaconsequence,onehas

Since R >0 isarbitrary,we candeducethatf ∈ W 1,p(H).

Step3:(1)⇒ (3).Thisisaconsequenceofpart(c) ofTheorem 1.1andthefactthat

Multiplying (3.1)byεδ ε−1, 0< ε <1,and integrating with respect toδ over (0,1),

byLemma 2.1withα = p + ε− 1,Φ (u, v)= |f (u) − f (v)|, Ψ (u, v)=ρ(u−11·v) Q +p,onehas

As aconsequenceofStep 2,wehavef ∈ W 1,p(H)

Theproof isnowcompleted

4 Thecasep= 1 andBVfunctionsonthe Heisenberggroup

Inthissection,wewillinvestigatethespecialcasep= 1.First,werecallthedefinition

of thespace BV (Ω) offunctionswithbounded variationinΩ⊂ H

Definition4.1(Horizontal vector fields).Thespaceof smoothsectionsofHΩ,thezontalsubbundleonΩ,isdenotedbyΓ (HΩ).ThespaceΓ c (HΩ) denotesalltheelements

hori-ofΓ (HΩ) withsupportcontainedinΩ.ElementsofΓ (HΩ) arecalledhorizontalvectorfields

Definition 4.2 (H-BV functions). We say that a function u ∈ L1(Ω) is afunction of

where the symbol div denotes the Riemannian divergence We denote by BV (Ω) the

spaceofallfunctionsofH-bounded variation

See[1,8,9] fordefinitionsofBV spacesonmoregeneralsettings

Inthissection,wewillprovethefollowingproperty:

Theorem4.1 Let f be a function in L1(H) satisfying

forsomepositiveconstantC >0

Proceeding similarly as in Step 4of the proof of Theorem 1.2, multiplying (4.1) by

εδ ε−1 ,0< ε <1,integrating with respect to δ over (0,1),and then using Lemma 2.1,

Now, wealsosplittheproofinto two steps:

Step 1:Wesuppose furtherthatf ∈ L∞(H) Now, we define f k as inStep 2 of theproof of Theorem 1.2 Notingthatthefunctiont 1+ε isstill convex onR+, wecanalsohave

Then onehasf R ∈ L∞(H), f R (u) R→∞ −→ f (u) pointwisefora.e.u∈ H,and



f R (u) − f R (v)

 ≤f(u) − f(v) for all u, v ∈ H.

As aconsequence,onegets

SinceR >0 isarbitrary,wecandeducethatf ∈ BV (H) ✷

UsingTheorem 4.1,wecanalsohavethefollowingLipschitztypecharacterizationof

BV space:

Theorem4.2 Let f ∈ L1(H) be such that there exists a nonnegative function F ∈ L1(H)

Beforewebeginourproofofthistheorem,weliketomakesomeremarks.Wenotethat

in ourTheorem 1.2, part(4), theSobolev spaces W 1,p(H) for p >1 was characterized

if the above estimate (4.2) holds for F ∈ L p(H) But this characterization does nothold for p= 1 (see also the paper [17]) Therefore, our theorem can be viewed as theborderline case of the Sobolev space when p = 1 on the Heisenberg group H Morerecently,ithasbeenshownin[32]thatiftheaboveestimate(4.2)holdsforF ∈ L1(H),then f ∈ W 1,1(H)

ProofofTheorem4.2 First,wenoteherethatforallδ ∈ (0,1):

h2dhdudσ

= 2ˆ

5 Somegeneralizationsandvariantsofcharacterizations

In this section, we will study some generalizations of the above results The nextTheoremisageneralizedresultofTheorem 1.2:

Theorem 5.1 Let f ∈ L p (H), 1< p < ∞ and F : [0,∞)→ [0, ∞) be continuous such that

∞

ˆ

0

F (t)t −p−1 dt = 1.

andusingTheorem 1.2,wegetf∈ W 1,p(H) Hence, f ∈ W 1,p(H) ✷

Thesecondresultinthissectionistoweakenthestatement(3)inTheorem 1.2.Moreprecisely,wewill provethat

Theorem5.2 Let1< p < ∞ and f ∈ L p (H) be such that

sincebyTheorem 1.2,wegettheassertion.

Now, foreveryε ∈ (0,1),since

ρ (u−1· v) Q+p χ {|f (u)−f (v)|>δ n}(u, v)dudv. (5.7)

Now,fix(u, v) suchthat

0 <

f(u) − f(v) ≤ 1anddenoten (u,v) thesmallestintegernumbersuchthat

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