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Advances
in
Mathematics
www.elsevier.com/locate/aim
Sobolev
inequalities
in
arbitrary
domains
✩Andrea Cianchia,∗, Vladimir Maz’yab,c
a
DipartimentodiMatematicaeInformatica“U.Dini”,UniversitàdiFirenze, VialeMorgagni67/A,50134Firenze,Italy
bDepartmentofMathematics,LinköpingUniversity,SE-58183Linköping,Sweden cDepartmentofMathematicalSciences,M&OBuilding,UniversityofLiverpool,
LiverpoolL693BX,UK a r t i c l e i n f o a bs t r a c t Article history: Received15September2015 Accepted1February2016 Availableonlinexxxx
CommunicatedbyErwinLutwak MSC: 46E35 46E30 Keywords: Sobolevinequalities Irregulardomains Boundarytraces Optimalnorms Representationformulas
AtheoryofSobolevinequalitiesinarbitraryopensetsinRnis established.Boundaryregularityofdomainsisreplacedwith informationonboundarytracesoftrialfunctionsandoftheir derivatives upto someexplicitminimal order.Therelevant Sobolev inequalities exhibit the same critical exponents as intheclassicalframework.Moreover,they involveconstants independentofthegeometryofthedomain,andhenceyield genuinely new results even in the case when just smooth domains are considered. Our approach relies upon new representationformulasforSobolevfunctions,andonensuing pointwiseestimateswhichholdinanyopenset.
© 2016ElsevierInc.All rights reserved.
✩ This research was partly supported by the Research Project of Italian Ministry of University and
Research (MIUR) Prin 2012 (grant number 2012TC7588) “Elliptic and parabolic partial differential equations:geometricaspects,relatedinequalities,andapplications”,andbyGNAMPAoftheItalianINdAM (NationalInstituteofHighMathematics).
* Correspondingauthor.
E-mailaddress:cianchi@unifi.it(A. Cianchi). http://dx.doi.org/10.1016/j.aim.2016.02.012 0001-8708/© 2016ElsevierInc.All rights reserved.
1. Introduction
The aim of this paper is to develop a theory of Sobolev embeddings, of any order
m∈ N,inarbitrary opensets Ω inRn,withn≥ 2.Asusual, byanm-th orderSobolev embeddingwemeananestimateforanormoftheh-thorderweakderivatives(0≤ h≤
m− 1) ofanym timesweaklydifferentiablefunctioninΩ in termsofnormsofsomeof itsderivativesuptotheorder m.
The classical theory of Sobolev embeddings involves ground domains Ω satisfying suitableregularity assumptions. For instance,a formulationof theoriginal theorem by Sobolevreadsasfollows. AssumethatΩ isaboundeddomainsatisfyingthecone prop-erty,m∈ N,1< p< mn, andF(·) is any continuousseminorm inWm,p(Ω) whichdoes notvanishonany(non-zero)polynomialofdegreenotexceedingm−1.Thenthereexists aconstantC = C(Ω) suchthat
u L np n−mp(Ω)≤ C ∇mu Lp(Ω)+F(u) (1.1) foreveryu∈ Wm,p(Ω).Here,Wm,p(Ω) denotes theusual Sobolevspace ofthose func-tionsinΩ whoseweakderivativesupto theorder m belongtoLp(Ω),and∇mu stands forthevectorofall(weak)derivativesof u oforder m.
ItiswellknownthatstandardSobolevembeddingsarespoiledinpresenceofdomains with“bad”boundaries.Inparticular,inequalitiesoftheform(1.1)donothold,atleast withthesamecriticalexponent n−mpnp ,inirregulardomains. Theinterplaybetweenthe geometryofthedomainandSobolevinequalities,eveninframeworksmoregeneralthan theEuclideanone,hasovertheyearsbeenthesubjectofextensiveinvestigations,along diversedirections,byanumberofauthors.Theirresultsaretheobjectofarichliterature, which includes the papers [3,5,6,16,7,8,12,13,18,19,25–27,29,28,31,32,37,39,43,49,47,48, 50–52,54–56,60–62,64,65,68,69] and the monographs [21–24,41,62,66]. Various classical problemsinthetheoryof spacesofdifferentiable functions,involvingpossiblyirregular sets, have attracted newinterest inthe last two decades, and have been considered in suchcontributionsas[10,17,20,33,34,45,40,46,63,67,70].
In order to avoid any assumption on Ω, we deal with Sobolev inequalities from an unconventionalperspective.Theunderlingideaofourresultsisthatsuitableinformation onboundarytraces oftrialfunctionscanreplaceboundaryregularityof Ω.
Theinequalitiesthatwillbe establishedhavetheform
∇hu Y (Ω,μ)≤ C ∇mu X(Ω)+N∂Ω(u) , (1.2)
wherem∈ N,h∈ N0,·X(Ω)isaBanachfunctionnormonΩ withrespecttoLebesgue measureLn,·
Y (Ω,μ)isaBanachfunctionnormwithrespecttoapossiblymoregeneral measure μ,andN∂Ω(·) isa(non-standard)seminormon∂Ω,dependingonthetraceof u, andofitsderivativesuptotheorderm−12 .Here,N0=N∪ {0},and[·] denotesinteger part.Moreover,∇0u standsjustforu,andweshalldenote∇1u alsoby∇u.
Somedistinctivetraitsoftheinequalitiestobe presentedcanbe itemizedasfollows:
• No regularity on Ω is a priori assumed. In particular, the constants in (1.2) are independentofthegeometryofΩ.
• ThecriticalSobolevexponents,or,moregenerally,theoptimaltargetnorms,arethe sameasinthecaseofregulardomains.
• The order m2−1 of the derivatives, on which the seminorm N∂Ω(·) depends, is minimalforaninequalityoftheform(1.2)toholdwithoutanyadditionalassumption onΩ.
Let us emphasize that, in view of these features, our inequalities provide original conclusionsevenwhenjustappliedtosmoothtrialfunctionsu onregulardomains Ω.
A first-order Sobolev inequality on arbitrary domains Ω ⊂ Rn, of the form (1.2), where X(Ω)= Lp(Ω),Y (Ω,μ)= Lq(Ω),andN∂Ω(·)=· Lr(∂Ω),with1≤ p< n,r≥ 1
and q = min{ rn
n−1,n−pnp } was established in [60] via isoperimetricinequalities. Sobolev inequalities of this kind, but still involving only first-order derivatives and Lebesgue measure, haverecentlyreceived renewedattention. Inparticular, the paper[57] makes useofmasstransportationtechniquestoaddresstheproblemoftheoptimalconstantsfor
p∈ (1,n),theproblem whenp= 1 having alreadybeen solvedin[60].Sharp constants ininequalitiesintheborderlinecasewhenp= n areexhibitedin[58].
In the present paper, we develop a completely different approach, which not only enables us to establish arbitrary-order inequalities, which cannot just be derived via iterationoffirst-orderones,butalsoaugmentsthefirst-ordertheory,inthatmoregeneral measures andnormsareallowed.
Ourpointofdepartureisanewpointwiseestimateforfunctions,andtheirderivatives, onarbitrary –possiblyunboundedandwithinfinitemeasure –domains Ω.Suchestimate involves a novel class of double-integral operators, where integration is extended over Ω× Sn−1. Therelevant operators act on akind of higher-order difference quotientsof thetraces offunctionsandoftheirderivativeson∂Ω.
In view of applications to norm inequalities, the next step calls for an analysis of boundednesspropertiesoftheseoperatorsinfunctionspaces.Tothispurpose,weprove their boundednessbetweenoptimalendpointspaces.Incombination withinterpolation argumentsbasedontheuseofPeetreK-functional,theseendpointresultsleadto point-wise bounds, forSobolevfunctions, inrearrangementform. Asaconsequence, Sobolev inequalities onan arbitraryn-dimensional domain arereduced to considerably simpler one-dimensional inequalitiesforHardytypeoperators.
With this apparatus at disposal, we are able to establish inequalities involving Lebesguenorms,withrespecttoquitegeneralmeasures,aswellasYudovich–Pohozaev– Trudinger type inequalities in exponential Orlicz spaces for limiting situations. The compactness of corresponding Rellich–Kondrashov type embeddings, with subcritical exponents,is alsoshown.Inequalities forotherrearrangement-invariantnorms, suchas
Lorentz and Orlicz norms, could be derived. However, in order to avoid unnecessary additionaltechnicalcomplications,thisissueisnotaddressedhere.
Onemajormotivationforourresearchistoprovideaneffectivefunctionalframework forboundaryvalueproblemsforpartialdifferentialequations,andvariationalproblems, indomainslackinganyregularity,towhichthetheoryofSobolevspacesavailableinthe existingliteraturedoesnotapply.Inthisconnection,letusmentionthefollowingsimple instance,whichyetgivestheflavorofthegeneralityallowedbytheuseofaninequality like(1.2).Considertheproblem
Δ2u = div F in Ω,
u = 0, B u = 0 on ∂Ω, (1.3)
where Ω is any open set in Rn, n ≥ 3, Δ2 is the bi-Laplace operator, F : Ω → Rn is a given function, and B is the (second-order) boundary operator generated by the minimizationofthequadraticfunctional
Ω
|∇2u|2+ 2F· ∇udx
among functionsu vanishingon ∂Ω.Here,thedot“·” standsfor scalarproduct inRn. Assumethat F ∈ Ln+22n (Ω),where 2n
n+2 is the Hölderconjugate of the critical Sobolev exponent 2n
n−2. A special case of Theorem 6.3, Section 6, tells us that there exists a constantC = C(n) suchthattheSobolevinequality
∇u L 2n n−2(Ω)≤ C∇ 2u L2(Ω)
holdsforeveryfunctionu vanishingon∂Ω.(Incidentally,noticethat,instead,an inequal-ity of this kind fails if ∇2u is replaced just with Δu, unless Ω is sufficiently regular.) AstandardargumentrelyinguponRiesz’representationtheoreminHilbertspacesthen yieldstheexistenceofauniquesolutionu to(1.3), whateverΩ is.
Ananalysis ofmoregeneralproblems inarbitrarydomainsgoesbeyond thescope of thepresentcontribution,andisthesubjectof aworkinprogress.
The paper is organized as follows. In the next section we offer a brief overview of some Sobolev type inequalities, in basic cases, which follow from our results, and we discusstheirnoveltyandoptimality.Section3containssomepreliminarydefinitionsand results. The statement of our main results starts with Section 4, which is devoted to our key pointwise inequalities for Sobolev functions on arbitrary open sets. Estimates in rearrangement form are derivedin the subsequent Section5. In Section 6, Sobolev typeinequalitiesinarbitraryopensetsareshowntofollowviasuchestimates.Examples whichdemonstratethesharpnessofourresultsareexhibited inSection7.Inparticular,
Example 7.4shows thatinequalitiesof theform (1.2) maypossiblyfail ifN∂Ω(u) only depends on derivatives of u on ∂Ω up to an order smaller than [m−12 ]. Finally, in the
Appendix, somenewnotions,whichare introducedinthedefinitionsof theseminorms
N∂Ω(·),arelinkedto classicalpropertiesofSobolevfunctions. 2. Atasteofresults
Inorder togiveanoverallideaofthecontent ofthispaper, weenucleatehereaftera few basicinstancesof theinequalitiesthatcanbe derivedviaourapproach.
We begin with two examples which demonstrate that our conclusions lead to new resultsalsointhecaseoffirst-orderinequalities,namelyinthecasewhenm= 1 in(1.2).
LetΩ beanyopensetinRn,andletμ beaBorelmeasureonΩ suchthatμ(B
r∩Ω)≤
Crα forsomeC > 0,andα∈ (n− 1,n],andfor everyball Br withradius r.Clearly, if
μ=Ln,thenthis conditionholds withα = n.
Assumethat1< p< n andr > 1,andlets= min{n−1rα ,n−pαp }.Then
uLs(Ω,μ)≤ C
∇uLp(Ω)+uLr(∂Ω)
(2.1) for some constant C and every function u with bounded support, provided that
Ln(Ω)<∞,μ(Ω)<∞ andHn−1(∂Ω)<∞.Here,Hn−1denotesthe(n−1)-dimensional Hausdorffmeasure.Inparticular,ifr = p(n−1)n−p ,andhences=nαp−p,then(2.1)holdseven iftheassumptiononthefinitenessofthesemeasuresisdropped;inthiscase,theconstant
C depends onlyonn, p,α.Inequality(2.1)follows viaageneralprinciplecontainedin
Theorem 6.1, Section6.It extendsaversionof theSobolevinequalityformeasures,on regulardomains[62,Theorem 1.4.5].Italsoaugments,asfarasmeasuresareconcerned, theresultof[57],whoseapproach,thoughyieldingsharpconstants,isconfinedtonorms evaluated with respect to the Lebesgue measure. Let us point out that, by contrast, ourmethod,beingbasedonrepresentationformulas,neednotleadtooptimalnormsin inequalitieswithp= 1.
Considernowtheborderlinecasecorresponding top= n.Asaconsequenceof Theo-rem 6.1 again,onecanshowthat
uexp L n n−1(Ω,μ)≤ C ∇uLn(Ω)+u exp Ln−1n (∂Ω) , (2.2)
for some constant C and every function u with bounded support, provided that
Ln(Ω)<∞,μ(Ω)<∞ andHn−1(∂Ω)<∞.Here,· exp L
n
n−1(Ω,μ)and·exp Ln−1n (∂Ω)
denote norms in Orlicz spaces of exponential type on Ω and ∂Ω, respectively. In-equality (2.2) on the one hand extends the Yudovich–Pohozaev–Trudinger inequality to possiblyirregulardomains;ontheotherhand,itenhances,undersomerespect,a re-sult of [58], where optimal constants are exhibited in estimates involving the weaker norm inexp L(Ω),and justfortheLebesguemeasure.
Letusnowturntohigher-orderinequalities.Focusing,forthetimebeing,on second-orderinequalitiesmayhelptograspthequality andsharpnessofourconclusionsinthis
framework.In theremaining partof this section, wethus assumethat m= 2 in (1.2); wealsoassume,forsimplicity,thatμ=Ln.
First, assume thath = 0. Then we canprove (among other possible choices of the exponents)that,if1< p<n2,then
uL pn n−2p(Ω)≤ C ∇2u Lp(Ω)+u V1,0L p(n−1) n−p (∂Ω)+uL p(n−1) n−2p (∂Ω) , (2.3) for some constant C = C(p,n), in particular independent of Ω, and every function u
withbounded support.Note that n−2ppn is thesamecritical Sobolevexponentas inthe caseofregulardomains.Here,· V1,0Lr(∂Ω)denotes,forr∈ [1,∞],theseminorm given
by
uV1,0Lr(∂Ω)= inf
g gLr(∂Ω), (2.4)
wheretheinfimumistakenamongallBorelfunctionsg on ∂Ω suchthat
|u(x) − u(y)| ≤ |x − y|(g(x) + g(y)) for Hn−1-a.e. x, y∈ ∂Ω, (2.5) and Lr(∂Ω) denotes aLebesgue space on ∂Ω with respect to the measure Hn−1. The functiong appearingin(2.5)isanuppergradient,inthesenseof[38],fortherestriction of u to ∂Ω, endowed with the metric inherited from the Euclidean metric in Rn, and with themeasure Hn−1. In[38],a definition of this kind, and anassociated seminorm givenas in(2.4),wereintroduced todefine first-orderSobolevtypespacesonarbitrary metricmeasure spaces.In thelast twodecades, various notionsofupper gradientsand of Sobolevspaces of functionsdefined onmetric measure spaces, havebeen the object ofinvestigationsandapplications.Theyconstitutethetopicofanumberofpapersand monographs,including[4,11,35,39,42,53].
Letus stress that,although thenew termu
V1,0Lp(n−1)n−p (∂Ω) on theright-hand side of(2.3) canbe droppedwhen Ω is aregular,say Lipschitz,domain,it isindispensable ifΩ isarbitrary. This canbe shownby taking into account adomain as in Fig. 1 (see
Example 7.1,Section7).
Asinthecaseofregulardomains,ifp> n2,thentheLebesguenormontheleft-hand sideof(2.3)canbe replacedbythenorminL∞.Indeed,ifr > n− 1,then
uL∞(Ω)≤ C
∇2u
Lp(Ω)+uV1,0Lr(∂Ω)+uL∞(∂Ω) (2.6)
foranyopensetΩ suchthatLn(Ω)<∞ andHn−1(∂Ω)<∞,forsomeconstantC,and for any functionu with bounded support. Inparticular, the constantC depends on Ω onlythroughLn(Ω) andHn−1(∂Ω).
Inthe limitingsituationwhen n ≥ 3,p= n2 and r > n− 1, a Yudovich–Pohozaev– Trudingertypeinequalityoftheform
uexp L n
n−2(Ω)≤ C
∇2u
Ln2(Ω)+uV1,0Lr(∂Ω)+uexp Ln−2n (∂Ω)
Fig. 1.Example 7.1, Section7.
holds for some constant C independent of the geometry of Ω, and every function u
with bounded support, provided that Ln(Ω) < ∞ and Hn−1(∂Ω) < ∞. The norms
· exp L n
n−2(Ω) and · exp Ln−2n (∂Ω) are thesame exponential norms appearingin the Yudovich–Pohozaev–Trudingerinequalityonregulardomains,andinitsboundarytrace counterpart.
Considernextthecasewhenstill m= 2 in (1.2), buth= 1.From ourestimatesone can infer that, if 1< p < n and r ≥ 1, and Ω is any open set with Ln(Ω) < ∞ and
Hn−1(∂Ω)<∞,then ∇uLq(Ω)≤ C ∇2u Lp(Ω)+uV1,0Lr(∂Ω) (2.8) for some constant C independent of the geometry of Ω, and every function u with
bounded support,where
q = min rn n−1,n−pnp
. (2.9)
In particular, ifr = p(n−1)n−p , and henceq = nnp−p, then theconstantC in (2.8)depends onlyonn andp.
Inequality(2.8)isoptimalundervariousrespects.Forinstance,ifΩ isregular,then,as aconsequenceof(1.1), theseminorm uV1,0Lr(∂Ω) canbereplacedjustwithuLr(∂Ω)
on the right-hand side. Bycontrast, this is impossible ina domain Ω as inFig. 2, for any q∈ [1,n−pnp ], whateverr is –seeExample 7.2, Section7.
Fig. 2.Example 7.2, Section7.
Fig. 3.Example 7.3, Section7.
The questionof the optimalityof the exponent q given by (2.9)can also be raised. Theanswerisaffirmative.Actually,domainslikethatofFig. 3showthatsuchexponent
q isthelargestpossiblein(2.8)ifnoregularityisimposedonΩ (Example 7.3,Section7). Whenp> n,inequality(2.8)canbe replacedwith
∇uL∞(Ω)≤ C
∇2u
Lp(Ω)+uV1,0L∞(∂Ω), (2.10)
for some constant C independent of the geometry of Ω, and every function u with
Finally,intheborderlinecasecorrespondingtop= n,anexponentialnormisinvolved again. UndertheassumptionthatLn(Ω)<∞ andHn−1(∂Ω)<∞,onehasthat
∇uexp L n n−1(Ω)≤ C ∇2u Ln(Ω)+u V1,0exp Ln−1n (∂Ω) (2.11) forsomeconstantC,dependingonΩ onlythroughLn(Ω) andHn−1(∂Ω),andforevery functionu withbounded support.Here,theseminorm ·
V1,0exp Ln−1n (∂Ω) isdefinedas
in(2.4), withthenorm · Lr(∂Ω) replacedwith thenorm ·
exp Ln−1n (∂Ω).Again, the exponential normsin(2.11) are thesameoptimal Orlicz target normsfor Sobolevand traceinequalities,respectively,onregulardomains.
3. Preliminaries
LetΩ be anyopenset inRn,n≥ 2.Givenx∈ Ω,define
Ωx={y ∈ Ω : (1 − t)x + ty ⊂ Ω for every t ∈ (0, 1)}, (3.1) and
(∂Ω)x={y ∈ ∂Ω : (1 − t)x + ty ⊂ Ω for every t ∈ (0, 1)}. (3.2) TheyarethelargestsubsetofΩ and∂Ω,respectively,whichcanbe“seen”from x.Itis easily verifiedthatΩx isanopenset.Thefollowingpropositiontellsusthat(∂Ω)x isa Borelset.
Proposition 3.1. Assume that Ω is an open set in Rn, n≥ 2. Let x∈ Ω. Then theset (∂Ω)x,defined by(3.2),isBorelmeasurable.
Proof. Givenany r∈ Q∩ (0,1),define
(∂Ω)x(r) ={y ∈ ∂Ω : (1 − t)x + ty ⊂ Ω for every t ∈ (0, r)}.
If y ∈ (∂Ω)x(r), then there exists δ > 0 such that Bδ(y)∩ ∂Ω ⊂ (∂Ω)x(r). Thus, for eachr∈ Q∩ (0,1),theset(∂Ω)x(r) isopenin∂Ω,inthetopologyinducedbyRn. The conclusionthenfollows fromthefactthat(∂Ω)x=∩r∈Q∩(0,1)(∂Ω)x(r). 2
Next,wedefine thesets
(Ω× Sn−1)0={(x, ϑ) ∈ Ω × Sn−1 : x + tϑ∈ ∂Ω for some t > 0}, (3.3) and
Clearly,
(Ω× Sn−1)0= Ω× Sn−1 if Ω is bounded. (3.5) Let
ζ : (Ω× Sn−1)0→ Rn (3.6) bethefunctiondefinedas
ζ(x, ϑ) = x + tϑ, where t is such that x + tϑ∈ (∂Ω)x.
Inother words,ζ(x,ϑ) is thefirstpoint ofintersectionof thehalf-line {x+ tϑ: t> 0}
with∂Ω.
Given afunctiong : ∂Ω → R,with compact support,we adoptthe conventionthat
g(ζ(x,ϑ)) isdefinedforevery(x,ϑ)∈ Ω× Sn−1, onextendingitby0 on(Ω× Sn−1)∞; namely,weset
g(ζ(x, ϑ)) = 0 if (x, ϑ)∈ (Ω × Sn−1)∞. (3.7) Letusnextintroducethefunctions
a : Ω× Sn−1→ [−∞, 0) and b : Ω × Sn−1→ (0, ∞] (3.8) givenby b(x, ϑ) = |ζ(x,ϑ) − x| if (x,ϑ) ∈ (Ω × S n−1) 0, ∞ otherwise, (3.9) and a(x, ϑ) =−b(x, −ϑ) if (x, ϑ) ∈ Ω × Sn−1. (3.10) Proposition 3.2. The functionζ is Borelmeasurable. Hence, the functions a and b are
Borelmeasurableaswell.
Proof. Assume first that Ω is bounded, so that (Ω× Sn−1)0 = Ω× Sn−1. Consider a sequenceofnestedpolyhedra{Qk} invadingΩ,andthecorresponding sequenceof func-tions{ζk},definedasζ,withΩ replacedwithQk.SuchfunctionsareBorelmeasurable, byelementaryconsiderations, andhenceζ is also Borelmeasurable,sinceζk converges toζ pointwise.
Next,assumethatΩ isunbounded.Foreachh∈ N,considerthesetΩh= Ω∩ Bh(0), where Bh(0) is the ball, centered at 0, with radius h.Let ζh and bh be the functions, definedasζ andb,withΩ replacedwithΩh.SinceΩhisbounded,thenwealreadyknow
that bh is Borel measurable. Moreover, bh converges to b pointwise. Hence, b is Borel measurable as well,andinparticular theset(Ω× Sn−1)0, whichagreeswith {b<∞}, is Borel measurable.Finally, the functionζh is Borel measurable,inasmuch as Ωh is a bounded set.Moreover,ζh convergestoζ pointwiseontheBorelset(Ω× Sn−1)0.Thus,
ζ isBorelmeasurable. 2
Given m ∈ N and p∈ [1,∞], we denote byVm,p(Ω) the Sobolevtypespace defined as
Vm,p(Ω) =u : u is m-times weakly differentiable in Ω, and|∇mu| ∈ Lp(Ω) .
(3.11) Let usnotice that,inthedefinitionofVm,p(Ω),it isonlyrequired thatthederivatives of u of the highest order m belong to Lp(Ω). Replacing Lp(Ω) in (3.11) with a more general Banach function space X(Ω) leads to the notion of m-th order Sobolev type space VmX(Ω) builtuponX(Ω).
For k ∈ N0, we denote as usual byCk(Ω) the space of real-valued functionswhose
k-th orderderivativesinΩ arecontinuousuptotheboundary.Wealsoset
Cbk(Ω) ={u ∈ Ck(Ω) : u has bounded support}. (3.12) Clearly,
Cbk(Ω) = Ck(Ω) if Ω is bounded.
Let α = (α1,. . . ,αn) bea multi-indexwith αi ∈ N0 for i = 1,. . . ,n. Weadopt the notations|α|= α1+· · ·+αn,α!= α1!· · · αn!,andϑα= ϑα11· · · ϑαnnforϑ∈ Rn.Moreover, we setDαu= ∂|α|u
∂xα11 ...∂xαnn foru: Ω→ R.
We need to extend the notion of upper gradient g for the restriction of u to ∂Ω
appearingin(2.5)tothecaseofhigher-orderderivatives.Tothispurpose,letusdenote bygk,j,wherek∈ N0andj = 0,1,(k,j) = (0,0),any Borelfunctionon∂Ω such that:
(i) Ifk∈ N, j = 0,andu∈ Cbk−1(Ω), |α|≤k−1 (2k− 2 − |α|)! (k− 1 − |α|)!α! (y− x)α |y − x|2k−1 (−1)|α|Dαu(y)− Dαu(x) ≤ gk,0(x) + gk,0(y) (3.13) forHn−1-a.e. x,y∈ ∂Ω.
(ii) Ifk∈ N,j = 1,andu∈ Cbk(Ω), n i=1 |α|≤k−1 (2k− 2 − |α|)! (k− 1 − |α|)!α! (y− x)α |y − x|2k−1 (−1)|α|Dα ∂u∂x i(y)− D α ∂u ∂xi(x) ≤ gk,1(x) + gk,1(y) (3.14) forHn−1-a.e.x,y∈ ∂Ω. (iii) Ifk = 0,j = 1,andu∈ C0 b(Ω), |u(x)| ≤ g0,1(x) (3.15) forHn−1-a.e.x∈ ∂Ω.
Notethatinequality(3.13), withk = 1,agrees with(2.5),and henceg1,0 hasthesame roleasg in(2.5).Letusalsopoint outthat,as(2.5)extendsaclassicalpropertyofthe gradientofweaklydifferentiablefunctionsinRn,likewiseitshigher-orderversions(3.13) and(3.14)extendaparallelpropertyoffunctionsinRnendowedwithhigher-orderweak derivatives.ThisisshowninProposition A.1oftheAppendix.
Inanalogywith(2.4),weintroducetheseminormgiven, forr∈ [1,∞], by
uVk,jLr(∂Ω)= inf
gk,jg
k,j
Lr(∂Ω) (3.16)
where k, j and u are as above, and the infimum is extended over all functions gk,j fulfillingtheappropriatedefinitionamong(3.13),(3.14)and(3.15).Moregenerally,given aBanachfunctionspaceZ(∂Ω) on∂Ω withrespecttotheHausdorffmeasure Hn−1,we define
uVk,jZ(∂Ω)= inf
gk,jg
k,j
Z(∂Ω). (3.17)
Observethat,inparticular,
uV0,1Z(∂Ω)=uZ(∂Ω).
4. Pointwiseestimates
In the present section we establish our first main result: a pointwise estimate for Sobolevfunctions,andtheirderivatives,inarbitraryopensets.Inwhatfollowswedefine, fork∈ N,
(k) =
0 if k is odd,
Theorem 4.1 (Pointwise estimate). Let Ω be any open set in Rn, n ≥ 2. Assume that m∈ N andh∈ N0aresuchthat0< m−h< n.ThenthereexistsaconstantC = C(n,m)
such that |∇hu(x)| ≤ C Ω |∇mu(y)| |x − y|n−m+hdy + m−h−1 k=1 Ω Sn−1 g[k+h+12 ],(k+h)(ζ(y, ϑ)) |x − y|n−k dHn−1(ϑ) dy + Sn−1 g[h+12 ],(h)(ζ(x, ϑ)) dHn−1(ϑ) for a.e. x∈ Ω, (4.2) for every u ∈ Vm,1(Ω) ∩ C[m−12 ] b (Ω). Here, g[ k+h+1 2 ],(k+h) is any function as in (3.13)–(3.15),andconvention(3.7)is adopted.
Remark 4.2. In the case when m− h = n, and Ω is bounded, an estimate analogous to(4.2)canbeproved,withthekernel |x−y|n−m+h1 inthefirstintegralontheright-hand
side replaced with log|x−y|C . The constant C depends on n andthe diameter of Ω. If
m− h> n,and Ω isbounded, then thekernel isbounded byaconstantdepending on
n, m andthediameterof Ω.
Remark4.3.Undertheassumptionthat
u =∇u = · · · = ∇[m−12 ]u = 0 on ∂Ω, (4.3)
onecanchooseg[h+1
2 ],(h)= 0 fork = 0,. . . ,m− h− 1 in(4.2).Hence, |∇hu(x)| ≤ C Ω |∇mu(y)| |x − y|n−m+hdy for a.e. x∈ Ω. (4.4) Aspecialcaseof(4.4),correspondingtoh= m− 1,istheobjectof[62,Theorem 1.6.2]. Remark 4.4. As already mentioned in Section 1, the order m2−1 of the derivatives prescribed on∂Ω,whichappearsontheright-handsideof(4.2),isminimal forSobolev type inequalities to hold in arbitrarydomains. This issue isdiscussed inExample 7.4, Section7below.
A keystep inthe proofof Theorem 4.1iscontained inthe nextlemma, whichdeals with thecasewhenh= m− 1 inTheorem 4.1.
Lemma 4.5.LetΩ be any opensetin Rn,n≥ 2. (i) Ifu∈ V2−1,1(Ω)∩ Cb−1(Ω) for some∈ N,then
|∇2−2u(x)| ≤ C Ω |∇2−1u(y)| |x − y|n−1 dy + Sn−1 g−1,1(ζ(x, ϑ)) dHn−1(ϑ) for a.e. x∈ Ω, (4.5)
forsomeconstant C = C(n,).
(ii) If u∈ V2,1(Ω)∩ C−1
b (Ω) for some∈ N,then
|∇2−1u(x)| ≤ C Ω |∇2u(y)| |x − y|n−1 dy + Sn−1 g,0(ζ(x, ϑ)) dHn−1(ϑ) (4.6)
for some constant C = C(n,). Here, g−1,1 and g,0 are functions as in
(3.13)–(3.15),andconvention (3.7)isadopted.
Ourproof ofLemma 4.5 inturnrequires thefollowingrepresentationformulaforthe (2− 1)-th order derivative of a one-dimensional function in an interval, in terms of its2-th derivativeintherelevant interval, andof itsderivatives upto theorder − 1
evaluatedattheendpoints.
Lemma4.6. Let−∞< a< b<∞.Assumethat ψ∈ W2,1(a,b) for some∈ N.Then ψ(2−1)(t) = t a Q2−12τ− a − b b− a ψ(2)(τ ) dτ− b t Q2−1a + b− 2τ b− a ψ(2)(τ ) dτ + (2− 1)!(−1) −1 k=0 (2− k − 2)! k!(− k − 1)! 1 (b− a)2−k−1 (−1)k+1ψ(k)(b) + ψ(k)(a) (4.7)
fort ∈ (a,b). Here, ψ(k) denotes thek-th order derivative of ψ,and Q
2−1 isthe
poly-nomial ofdegree 2− 1, obeying
Q2−1(t) + Q2−1(−t) = 1 for t ∈ R, (4.8)
and
Q2−1(−1) = Q(1)2−1(−1) = · · · = Q(2−1−1)(−1) = 0. (4.9) Proof. Letusrepresentψ as
where andς arethesolutionstotheproblems (2)(t) = ψ(2)(t) in (a, b), (k)(a) = (k)(b) = 0 for k = 0, 1, . . . , − 1, (4.11) and ς(2)(t) = 0 in (a, b),
ς(k)(a) = ψ(k)(a), ς(k)(b) = ψ(k)(b) for k = 0, 1, . . . , − 1, (4.12) respectively.Letusfirstfocus onproblem(4.11).Weclaimthat
(2−1)(t) = t a Q2−1 2τ− a − b b− a ψ(2)(τ ) dτ − b t Q2−1a + b− 2τ b− a ψ(2)(τ ) dτ for t∈ (a, b), (4.13)
whereQ2−1isasinthestatement.Inordertoverify(4.13),letusconsidertheauxiliary problem
ω2(s) = φ(s) in (−1, 1),
ω(k)(±1) = 0, k = 0, 1, . . . , − 1, (4.14) where φ ∈ L1(a,b) is any given function. Let κ: [−1,1]2 → R be the Green function associated withproblem(4.14),sothat
ω(s) =
1 −1
κ(s, r)φ(r) dr for s∈ [−1, 1]. (4.15)
Thefunctionκ takesanexplicitform([14];seealso[36,Section 2.6]),givenby
κ(s, r) = C|s − r|2−1
1−sr |s−r|
1
(t2− 1)−1dt for s = r, (4.16)
whereC = C() isasuitableconstant.Onecaneasilyseefromformula(4.16)thatκ(s,r)
is apolynomialofdegree2− 1 in s forfixedr, andapolynomialofdegree2− 1 inr
forfixeds,bothin{(s,r)∈ [−1,1]2: s> r},andin{(s,r)∈ [−1,1]2: s< r}.Moreover,
κ(s, r) = C −1 j=0 − 1 j (−1)−1−j 2j + 1 (1− sr) 2j+1(s− r)2−2k−2 − (s − r)2−1−1 j=0 − 1 j (−1)−1−j 2j + 1 . (4.17) Thus,ifs> r, ∂2−1κ ∂s2−1(s, r) = C(2− 1)! −1 j=0 − 1 j (−1)+j 2j + 1 r 2j+1− −1 j=0 − 1 j (−1)−1−j 2j + 1 , (4.18) apolynomialof degree 2− 1 in r, dependingonly on oddpowers of r. Letus denote thispolynomialbyQ2−1(r).It followsfrom(4.18) thatQ2−1(−1)= 0.Moreover,
∂Q2−1 ∂r (r) = ∂ ∂r ∂2−1κ ∂s2−1(s, r) = C(2− 1)! −1 j=0 − 1 j (−1)+jr2j =−C(2 − 1)!(r2− 1)−1. (4.19) Thus, Q2−1 vanishes, together with all its derivatives up to the order − 1, at −1, namelyQ2−1 fulfills(4.9).Equation(4.18)alsotellsus thatQ2−1(s)− Q2−1(0) is an oddfunction,and hence
Q2−1(s) + Q2−1(−s) = 2Q2−1(0) for s∈ R. (4.20) Sinceκ isaneven function,
∂2−1κ
∂s2−1(s, r) = ∂ 2−1κ
∂s2−1(−s, −r) = −Q2−1(−r) if −1 ≤ s < r ≤ 1.
Thus,(2− 1)-timesdifferentiationofequation(4.15) yields
ω(2−1)(s) = s −1 Q2−1(r)φ(r) dr− 1 s Q2−1(−r)φ(r) dr for s ∈ [−1, 1]. (4.21)
Since ω(2) = φ, an integration by parts in (4.21), equation (4.20), and the fact that
Q2−1(−1)= 0,tellusthat ω(2−1)(s) = 2Q2−1(0)ω(2−1)(s)− s −1 Q2−1(r)ω(2−1)(r) dr − 1 s Q2−1(−r)ω(2−1)(r) dr for s∈ [−1, 1]. (4.22)
Owing to the arbitrariness of ω, equation (4.22) ensures that 2Q2−1(0) = 1. Equa-tion(4.8)thusfollowsfrom (4.20).
Thefunction definedas
(t) = ω 2t−a−b b−a for t∈ [a, b]
isthusthesolutiontoproblem(4.11),andtherepresentationformula(4.13) followsvia achangeofvariablesin(4.21).
Consider nextproblem (4.12). Thefunction ς is apolynomialof degree 2− 1, and
ς(2−1) is a constant which, owing to the two-point Taylor interpolation formula (see e.g.[30,Chapter 2,Section 2.5,Ex. 3]),isgivenby
ς(2−1)= (2− 1)! d−1 dt−1 ς(t) (t− b) |t=a + d −1 dt−1 ς(t) (t− a) |t=b = (2− 1)! d−1 dt−1 ψ(t) (t− b) |t=a + d −1 dt−1 ψ(t) (t− a) |t=b . (4.23)
Leibnitz’differentiationruleforproductsyields d−1 dt−1 ψ(t) (t− b) |t=a + d −1 dt−1 ψ(t) (t− a) |t=b = (−1) −1 k=0 (2− k − 2)! k!(− k − 1)! 1 (b− a)2−k−1 (−1)k+1ψ(k)(b) + ψ(k)(a). (4.24)
Equation(4.7)followsfrom (4.13),(4.23) and(4.24). 2
Proof of Lemma 4.5. Given x∈ Ω and ϑ∈ Sn−1, leta(x,ϑ) andb(x,ϑ) be definedas in(3.10)and (3.9),respectively.
Webeginwiththeproofof(4.5)for= 1.Ifu∈ V1,1(Ω)∩C0
b(Ω),then,byastandard property ofSobolevfunctions,fora.e.x∈ Ω thefunction
[0, b(x, ϑ)] t → u(x + tϑ) belongsto V1,1(0,b(x,ϑ)) forHn−1-a.e.ϑ∈ Sn−1,and
d
dtu(x + tϑ) =∇u(x + tϑ) · ϑ for a.e. t ∈ [0, b(x, ϑ)].
Hence,forany suchx andϑ,
u(ζ(x, ϑ))− u(x) =
b(x,ϑ)
0
where convention (3.7)is adopted.Integrating both sidesof equation (4.25) over Sn−1 yields nωnu(x) = Sn−1 u(ζ(x, ϑ)) dHn−1(ϑ)− Sn−1 b(x,ϑ) 0 ∇u(x + tϑ) · ϑ dt dHn−1(ϑ), (4.26) whereωn= π n 2/Γ(1+n
2),theLebesguemeasureof theunitballinR
n.Onehasthat Sn−1 b(x,ϑ) 0 ∇u(x + tϑ) · ϑ dt dHn−1(ϑ) = Sn−1 b(x,ϑ) 0 1 tn−1∇u(x + ϑ) · ϑ t n−1d dHn−1(ϑ) = Ωx ∇u(y) · (y − x) |x − y|n dy. (4.27) Inequality(4.5),with= 1,followsfrom(4.26) and(4.27).
Letusnextprove(4.6).Ifu∈ V2,1(Ω)∩ Cb−1(Ω),thenfora.e.x∈ Ω,thefunction [a(x, ϑ), b(x, ϑ)] t → u(x + tϑ)
belongstoV2,1(a(x,ϑ),b(x,ϑ)) forHn−1-a.e. ϑ∈ Sn−1.Consideranysuchx and ϑ.If (x,ϑ)∈ (Ω× Sn−1) 0,then, byLemma 4.6, d2−1 dt2−1u(x + tϑ) = t a(x,ϑ) Q2−1 2τ− a(x, ϑ) − b(x, ϑ) b(x, ϑ)− a(x, ϑ) d2 dτ2u(x + τ ϑ) dτ − b(x,ϑ) t Q2−1a(x, ϑ) + b(x, ϑ)− 2τ b(x, ϑ)− a(x, ϑ) d2 dτ2u(x + τ ϑ) dτ + (2− 1)!(−1) × −1 k=0 (2− k − 2)! k!(− k − 1)! (−1)k+1dku(x+tϑ) dtk |t=b(x,ϑ)+ dku(x+tϑ) dtk |t=a(x,ϑ) (b(x, ϑ)− a(x, ϑ))2−k−1 (4.28) fort∈ (a(x,ϑ),b(x,ϑ)).If,instead,(x,ϑ)∈ (Ω× Sn−1)∞, then
d2−1 dt2−1u(x + tϑ) = −t∞ d2 dτ2u(x + τ ϑ) dτ, if b(x, ϑ) =∞, t −∞ d 2 dτ2u(x + τ ϑ) dτ, if a(x, ϑ) =−∞, (4.29)
fort∈ (a(x,ϑ),b(x,ϑ)) (ifboth b(x,ϑ)=∞ anda(x,ϑ)=−∞,then eitherexpression ontheright-handsideof(4.29) canbe exploited).
Wehavethat dk dtku(x + tϑ) = |α|=k k! α!ϑ
αDαu(x + tϑ) for a.e. t∈ (a(x, ϑ), b(x, ϑ)), (4.30)
fork = 1,. . . ,2.From (4.28)–(4.30)weinferthat |α|=2−1 (2− 1)! α! ϑ αDαu(x) =−χ(Ω×Sn−1)∞(x, ϑ) |α|=2−1 (2− 1)! α! ϑ α ±∞ 0 |γ|=1 ϑγDα+γu(x + τ ϑ) dτ + χ(Ω×Sn−1)0(x, ϑ) 0 a(x,ϑ) Q2−1 2τ− a(x, ϑ) − b(x, ϑ) b(x, ϑ)− a(x, ϑ) d2 dτ2u(x + τ ϑ)dτ − b(x,ϑ) 0 Q2−1 a(x, ϑ) + b(x, ϑ)− 2τ b(x, ϑ)− a(x, ϑ) d2 dτ2u(x + τ ϑ) dτ + (2− 1)!(−1) −1 k=0 (2− k − 2)! k!(− k − 1)! × |α|=k k! α!ϑ α
(−1)k+1Dαu(x + b(x, ϑ)ϑ) + Dαu(x + a(x, ϑ)ϑ)
(b(x, ϑ)− a(x, ϑ))2−k−1
, (4.31)
where theupperlimitofintegrationinthefirstintegralontheright-handsideiseither +∞,or −∞ accordingtowhetherb(x,ϑ)=∞ or a(x,ϑ)=−∞.
Denote by{Pβ} the system of allhomogeneous polynomials of degree2− 1 in the variables ϑ1,. . . ,ϑn suchthat
Sn−1
Pβ(ϑ)ϑαdHn−1(ϑ) = δαβ,
whereδαβstandsfortheKroneckerdelta.Onmultiplyingequation(4.31)by (2Pβ−1)!(ϑ),and integrating overSn−1 oneobtainsthat
Dβu(x) β! =− Sn−1 χ(Ω×Sn−1)∞(x, ϑ)Pβ(ϑ) |α|=2−1 |γ|=1 ϑα+γ α! ±∞ 0 Dα+γu(x + τ ϑ) dτ dHn−1(ϑ)
+ Sn−1 χ(Ω×Sn−1)0(x, ϑ) Pβ(ϑ) (2− 1)! × 0 a(x,ϑ) Q2−1 2τ− a(x, ϑ) − b(x, ϑ) b(x, ϑ)− a(x, ϑ) d2 dτ2u(x + τ ϑ)dτ − b(x,ϑ) 0 Q2−1a(x, ϑ) + b(x, ϑ)− 2τ b(x, ϑ)− a(x, ϑ) d2 dτ2u(x + τ ϑ) dτ dHn−1(ϑ) + (−1) Sn−1 Pβ(ϑ) × |α|≤−1 (2− |α| − 2)! ϑα (− |α| − 1)!α!
(−1)|α|+1Dαu(x + b(x, ϑ)ϑ) + Dαu(x + a(x, ϑ)ϑ)
(b(x, ϑ)− a(x, ϑ))2−|α|−1 dH
n−1(ϑ).
(4.32) Thereexist constants C = C(n,) andC = C(n,) suchthat
Sn−1 χ(Ω×Sn−1)∞(x, ϑ)Pβ(ϑ) |α|=2−1 |γ|=1 ϑα+γ α! ±∞ 0 Dα+γu(x + τ ϑ) dτ dHn−1(ϑ) ≤ C Sn−1 ∞ 0 |∇2u(x + τ ϑ)| dτ dHn−1(ϑ) = C Sn−1 ∞ 0 |∇2u(x + τ ϑ)| τn−1 τn−1dτ dHn−1(ϑ) ≤ C Ω |∇2u(y)| |x − y|n−1 dy. (4.33)
Next,weclaimthatthere existsaconstantC = C(,n) suchthat Sn−1 χ(Ω×Sn−1)0(x, ϑ) Pβ(ϑ) 0 a(x,ϑ) Q2−1 2τ− a(x, ϑ) − b(x, ϑ) b(x, ϑ)− a(x, ϑ) d2 dτ2u(x + τ ϑ)dτ − Pβ(ϑ) b(x,ϑ) 0 Q2−1 a(x, ϑ) + b(x, ϑ)− 2τ b(x, ϑ)− a(x, ϑ) d2 dτ2u(x + τ ϑ)dτ dHn−1(ϑ) ≤ C Ω |∇2u(y)| |x − y|n−1dy. (4.34)
Sn−1 χ(Ω×Sn−1)0(x, ϑ)Pβ(ϑ) × 0 a(x,ϑ) Q2−12τ− a(x, ϑ) − b(x, ϑ) b(x, ϑ)− a(x, ϑ) d2 dr2u(x + τ ϑ)dτ dH n−1(ϑ) = Sn−1 χ(Ω×Sn−1)0(x, ϑ)Pβ(ϑ) × −a(x,ϑ) 0 Q2−1 −2r + a(x, ϑ) + b(x, ϑ) b(x, ϑ)− a(x, ϑ) d2 dτ2u(x− rϑ)dr dH n−1(ϑ) = Sn−1 χ(Ω×Sn−1)0(x,−θ)Pβ(−θ) × −a(x,−θ) 0 Q2−1 −2r + a(x,−θ) + b(x, −θ) b(x,−θ) − a(x, −θ) d2 dr2u(x + rθ)dr dH n−1(θ) =− Sn−1 χ(Ω×Sn−1)0(x, θ)Pβ(θ) × b(x,θ) 0 Q2−1a(x, θ) + b(x, θ)− 2r b(x, θ)− a(x, θ) d2 dr2u(x + rθ)dr dH n−1(θ),
wherewehavemadeuseofthefactthatPβ(−θ)=−Pβ(θ) if|β|= 2− 1,andof(3.10). Thus, Sn−1 χ(Ω×Sn−1)0(x, ϑ) Pβ(ϑ) 0 a(x,ϑ) Q2−1 2τ− a(x, ϑ) − b(x, ϑ) b(x, ϑ)− a(x, ϑ) d2 dτ2u(x + τ ϑ)dτ − Pβ(ϑ) b(x,ϑ) 0 Q2−1a(x, ϑ) + b(x, ϑ)− 2τ b(x, ϑ)− a(x, ϑ) d2 dτ2u(x + τ ϑ)dτ dHn−1(ϑ) = 2 Sn−1 Pβ(ϑ) b(x,ϑ) 0 Q2−1 a(x, ϑ) + b(x, ϑ)− 2r b(x, ϑ)− a(x, ϑ) d2 dr2u(x + rϑ) drdH n−1(ϑ) = 2 Sn−1 Pβ(ϑ)
× b(x,ϑ) 0 Q2−1a(x, ϑ) + b(x, ϑ)− 2r b(x, ϑ)− a(x, ϑ) |α|=2 (2)! α! ϑ αDαu(x + rϑ) drdHn−1(ϑ) ≤ C Sn−1 b(x,ϑ) 0 |∇2u(x + rϑ)| drdHn−1(ϑ)≤ C Ωx |∇2u(y)| |x − y|n−1 dy
forsomeconstants C = C(n,) andC= C(n,).Hence,inequality(4.34) follows. Finally,bydefinition(3.13),thereexists aconstantC = C(n,) suchthat
Sn−1 χ(ω×Sn−1)0(x, ϑ)(−1)Pβ(ϑ) × |α|≤−1 (2− |α| − 2)! (− |α| − 1)!α!ϑ α ×
(−1)|α|+1Dαu(x + b(x, ϑ)ϑ) + Dαu(x + a(x, ϑ)ϑ)
(b(x, ϑ)− a(x, ϑ))2−|α|−1 dHn−1(ϑ) ≤ C Sn−1 χ(ω×Sn−1)0(x, ϑ)g,0(x + ϑa(x, ϑ)) + g,0(x + ϑb(x, ϑ))dHn−1(ϑ) = 2C Sn−1 g,0(ζ(x, ϑ)) dHn−1(ϑ). (4.35) Combining(4.32)–(4.35)yields (4.6).
Inequality(4.5), with≥ 2, follows onapplying (4.6)with u replaced withits first-orderderivatives. 2
ProofofTheorem 4.1. Forsimplicityofnotation,weconsider thecasewhenh= 0,the proof in the generalcase being analogous. Let u ∈ Vm,1(Ω)∩ C[m−12 ]
b (Ω). By inequal-ity(4.5)with= 1, |u(x)| ≤ C Ω |∇u(y)| |x − y|n−1dy + Sn−1 g0,1(ζ(x, ϑ)) dHn−1(ϑ) for a.e. x∈ Ω. (4.36)
From(4.36) andanapplicationofinequality(4.6)with= 1 oneobtainsthat
|u(x)| ≤ C Ω Ω |∇2u(z)| |y − z|n−1 dz dy |x − y|n−1 + Ω Sn−1 g1,0(ζ(y, ϑ))dH n−1(ϑ) dy |x − y|n−1 + Sn−1 g0,1(ζ(x, ϑ)) dHn−1(ϑ)
≤ C Ω |∇2u(z)| |x − z|n−2dz + Ω Sn−1 g1,0(ζ(y, ϑ))dH n−1(ϑ) dy |x − y|n−1 + Sn−1 g0,1(ζ(x, ϑ)) dHn−1(ϑ) for a.e. x∈ Ω, (4.37)
for someconstantsC = C(n) and C = C(n).Notethatinthelast inequalitywehave made useof aspecialcaseofthewellknownidentity
Rn 1 |x − y|n−σ Rn f (z) |y − z|n−γdz dy = C Rn f (z) |x − z|n−σ−γ dz for a.e. x∈ Rn, (4.38) which holds forsomeconstantC = C(n,σ,γ) and foreverycompactly supported inte-grable functionf ,providedthatσ > 0,γ > 0 and σ + γ < n.
Inequality(4.37) inturnyields, viaanapplicationofinequality(4.5)with= 2,
|u(x)| ≤ C Ω |∇3u(z)| |x − z|n−3 dz + Ω Sn−1 g1,1(ζ(y, ϑ))dH n−1(ϑ) dy |x − y|n−2 + Ω Sn−1 g1,0(ζ(y, ϑ))dH n−1(ϑ) dy |x − y|n−1 + Sn−1 g0,1(ζ(x, ϑ)) dHn−1(ϑ) for a.e. x∈ Ω, (4.39)
for some constant C = C(n). A finite induction argument, relying upon an alternate iterateduseofinequalities(4.6)and(4.5)asabove,eventuallyleadsto (4.2). 2 5. Estimatesinrearrangementform
Thepointwiseboundsestablishedintheprevioussectionenableustoderive rearrange-mentestimatesforfunctions,andtheirderivatives,withrespecttoanyBorelmeasureμ
onΩ suchthat
μ(Br(x)∩ Ω) ≤ Cμrα for x∈ Ω and r > 0, (5.1) forsomeα∈ (n−1,n] andsomeconstantCμ> 0.Here,Br(x) denotestheball,centered at x, withradius r.
Recall that, given a measure space R, endowed with apositive measure ν, the de-creasing rearrangement φ∗ν : [0,∞)→ [0,∞] of a ν-measurable functionφ : R→ R is definedas
φ∗ν(s) = inf{t ≥ 0 : ν(|φ| > t}) ≤ s} for s ∈ [0, ∞).
(φ + ψ)∗ν(s)≤ φ∗ν(s/2) + ψν∗(s/2) for s≥ 0, (5.2) foreverymeasurable functionsφ andψ onR.
Anyfunctionφ sharesitsintegrabilitypropertieswithitsdecreasingrearrangement φ∗ν, since
ν({|φ| > t}) = L1({φ∗ν> t}) for every t ≥ 0.
As a consequence, any norm inequality, involving rearrangement-invariant norms, be-tweentherearrangementsofthederivativesofSobolevfunctionsandtherearrangements ofitslower-orderderivatives,immediatelyyieldsacorrespondinginequalityforthe orig-inalSobolevfunctions.Thus,therearrangementinequalitiesto beestablishedhereafter reducetheproblem ofn-dimensionalSobolevtypeinequalitiesinarbitraryopensetsto considerably simpler one-dimensional Hardy type inequalities – see Theorem 6.1, Sec-tion6below.
Theorem 5.1 (Rearrangement estimates). Let Ω be any open bounded open set in Rn,
n≥ 2. Let m∈ N and h∈ N0 be such that 0< m− h< n. Assumethat μ isa Borel
measurein Ω fulfilling (5.1)for some α∈ (n− 1,n] andfor someCμ > 0. Then there
exist constantsc= c(n,m) and C = C(n,m,α,Cμ) suchthat
|∇hu|∗ μ(cs)≤ C s−n−m+hα snα 0 |∇mu|∗ Ln(r)dr + ∞ snα r−n−m+hn |∇mu|∗ Ln(r)dr + m−h−1 k=1 s−n−1−kα sn−1α 0 g[k+h+12 ],(k+h)∗ Hn−1(r)dr + ∞ sn−1α r−n−1−kn−1 g[k+h+12 ],(k+h)∗ Hn−1(r)dr + s−n−1α sn−1α 0 g[h+12 ],(h)∗ Hn−1(r)dr for s > 0, (5.3) foreveryu∈ Vm,1(Ω)∩ C[m2−1]
b (Ω).Here, (·) is definedasin (4.1),andg[
k+h+1 2 ],(k+h)
denotesanyBorelfunctionon∂Ω fulfillingtheappropriateconditionfrom (3.13)–(3.15).
Remark5.2.Ininequality(5.3), andinwhatfollows, whenconsideringrearrangements and norms with respect to ameasure μ, Sobolev functions and their derivatives have tobe interpretedas theirtraces withrespect toμ. Suchtraces arewell defined,thanks to standard (local) Sobolev inequalities with measures, owing to the assumption that
α∈ (n− 1,n] in(5.1). Ananalogousconvention appliesto theintegraloperatorsto be consideredbelow.
In preparation for theproof of Theorem 5.1, weintroduce afew integral operators, and provepointwise estimatesfortheirrearrangements.
LetΩ be anyopenset inRn.WedefinetheoperatorT as Tg(x) =
Sn−1
|g(ζ(x, ϑ))| dHn−1(ϑ) for x∈ Ω, (5.4) atanyBorelfunction g : ∂Ω→ R.Here,andinwhatfollows,weadoptconvention(3.7). Note that,owing toFubini’s theorem,Tg isameasurable functionwith respectto any Borelmeasure in Ω.
Forγ∈ (0,n),wedenote byIγ theclassicalRiesz potentialoperatorgivenby
Iγf (x) = Ω
f (y)
|y − x|n−γ dy for x∈ Ω, (5.5) at anyf ∈ L1(Ω),andwecall N
γ theoperatordefinedas
Nγg(x) = ∂Ω g(y) |x − y|n−γdHn−1(y) for x∈ Ω, (5.6) at anyfunctiong∈ L1(∂Ω).
Finally,wedefine theoperatorQγ asthecomposition
Qγ = Iγ◦ T. (5.7) Namely, Qγg(x) = Ω Sn−1 |g(ζ(y, ϑ))|dHn−1(ϑ) dy |x − y|n−γ for x∈ Ω, (5.8) forany Borelfunctiong : ∂Ω→ R.
Ouranalysisofthese operatorsrequiresafewnotationsandpropertiesfrom interpo-lationtheory.
AssumethatR isameasurespace,endowedwithapositivemeasureν.Givenapair
X1(R) andX2(R) ofnormedfunctionspaces,a functionφ∈ X1(R)+X2(R),ands∈ R, we denotebyK(φ,s;X1(R),X2(R)) theassociated Peetre’sK-functional, definedas
K(s, φ; X1(R), X2(R)) = inf φ=φ1+φ2 φ1X1(R)+ sφ2X2(R) for s > 0.
WeneedanexpressionfortheK-functional(uptoequivalence)inthecasewhenX1(R) andX2(R) arecertainLebesgueorLorentzspaces, andR isoneofthemeasure spaces mentioned above. Recall that, given σ > 1, the Lorentz space Lσ,1(R) is the Banach functionspaceofthosemeasurable functionsφ onR forwhichthenorm
φLσ,1(R)=
∞ 0
φ∗ν(s)s−1+1σds
isfinite.TheLorentzspaceLσ,∞(R),alsocalledMarcinkiewiczspaceorweak-Lσspace, istheBanachfunctionspaceofthosemeasurablefunctionsφ onR forwhichthequantity
φLσ,∞(R)= sup s>0
sσ1φ∗
ν(s)
isfinite.Notethat,inspiteofthenotation,thisisnotanorm. However,itisequivalent to anorm,upto multiplicativeconstants depending onσ, obtainedonreplacing φ∗ν(s) with 1s0sφ∗ν(r)dr.
Itiswellknownthat
K(φ, s; L1(R), L∞(R)) = s 0
φ∗ν(r) dr for s > 0, (5.9)
foreveryφ∈ L1(R)+ L∞(R)[9,Chapter 5, Theorem 1.6].Ifσ > 1,then
K(φ, s; Lσ,∞(R), L∞(R)) ≈ rσ1φ∗
ν(r)L∞(0,sσ) for s > 0, (5.10)
foreveryφ∈ Lσ,∞(R)+ L∞(R)[44,Equation (4.8)].Moreover,
K(φ, s; L1(R), Lσ,1(R)) ≈ sσ 0 φ∗ν(r) dr + s ∞ sσ r−σ1φ∗ ν(r) dr for s > 0, (5.11)
for every φ ∈ L1(R)+ Lσ,1(R) [44, Theorem 4.2]. In (5.10) and (5.11), the notation “≈”meansthatthetwo sidesareboundedbyeachotheruptomultiplicativeconstants dependingonσ.
Let R and S be positive measure spaces. An operator L defined on a linear space ofmeasurablefunctionsonR,andtakingvaluesinto thespaceofmeasurablefunctions onS,iscalledsub-linearif,foreveryφ1andφ2inthedomainofL andeveryλ1,λ2∈ R,
|L(λ1φ1+ λ2φ2)| ≤ |λ1||Lφ1| + |λ2||Lφ2|.
A basicresult inthe theory of real interpolation tellsus what follows. Assumethat L
is asub-linear operatoras above,and Xi(R) andYi(S), i= 1,2,are normed function spacesonR suchthat
L : Xi(R) → Yi(S) (5.12) withnormsnotexceedingNi,i= 1,2.Here,thearrow“→”denotesaboundedoperator. Then,
K(Lφ, s; Y1(S), Y2(S)) ≤ max{N1, N2}K(φ, s; X1(R), X2(R)) for s > 0, (5.13) foreveryφ∈ X1(R)+ X2(R).
Lemma 5.3.LetΩ beanopensetinRn,n≥ 2,andletγ∈ (0,n). Assumethatμ isany
Borel measurein Ω fulfilling (5.1)for someα∈ (n− γ,n] and forsome Cμ> 0. Then
there existsaconstant C = C(n,α,γ,Cμ) such that
Nγg L
α
n−γ,∞(Ω,μ)≤ CgL1(∂Ω) (5.14)
forevery g∈ L1(∂Ω).
Proof. Wemakeuseofanargumentrelatedto[1,2].Giveng∈ L1(∂Ω) andt> 0,define
Et={x ∈ Ω:|Nγg(x)|> t}, anddenote byμt therestriction ofthe measure μ toEt. ByFubini’stheorem, tμ(Lt) = t Ω dμt(x)≤ Ω |Nγg(x)|dμt(x)≤ Ω ∂Ω |g(y)| |x − y|n−γdHn−1(y) dμt(x) ≤ ∂Ω |g(y)| Ω dμt(x)dHn−1(y) |x − y|n−γ . (5.15) Next, Ω dμt(x) |x − y|n−γ = (n− γ) ∞ 0 −n+γ−1 {x∈Ω:|x−y|γ−n>γ−n} dμt(x) d ≤ (n − γ) ∞ 0 −n+γ−1μt(B(y)) d. (5.16)
From (5.15)and(5.16) wededucethat,foreachfixed r > 0,
tμ(Lt)≤ (n − γ) ∂Ω |g(y)| ∞ 0 −n+γ−1μt(B(y)) d dHn−1(y) = (n− γ) r 0 −n+γ−1 ∂Ω
+ (n− γ) ∞ r −n+γ−1 ∂Ω
|g(y)|μt(B(y)) dHn−1(y) d. (5.17) Wehavethat r 0 −n+γ−1 ∂Ω
|g(y)|μt(B(y)) dHn−1(y) d
≤ CμgL1(∂Ω) r 0 −n+γ−1+αd = Cμr α−n+γ α−n+γgL1(∂Ω). (5.18)
Ontheotherhand, ∞ r −n+γ−1 ∂Ω
|g(y)|μt(B(y)) dHn−1(y) d≤ μ(Lt) ∞ r −n+γ−1 ∂Ω |g(y)| dHn−1(y) d = μ(Lt)r −n+γ n−γ gL1(∂Ω). (5.19)
Combining(5.17)–(5.19), andchoosingr =μ(Lt)
Cμ 1 α ,yield tμ(Lt)≤ (n − γ)gL1(∂Ω) Cμ rα−n+γ α− n + γ + μ(Lt) r−n+γ n− γ = α−n+γα gL1(∂Ω)C n−γ α μ μ(Lt)1− n−γ α . (5.20) Thus, tμ(Lt) n−γ α ≤ α α−n+γC n−γ α μ gL1(∂Ω) for t > 0. (5.21)
Hence,inequality(5.14) follows. 2
Proposition5.4. LetΩ bean openset inRn.Then Sn−1 |g(ζ(x, ϑ))| dHn−1(ϑ)≤ 2n ∂Ω |g(y)| |x − y|n−1dHn−1(y) for x∈ Ω, (5.22)
foreveryBorelfunctiong : ∂Ω→ R.Here, convention(3.7) isadopted.
Proof. Wesplittheproofinsteps.Fixx∈ Ω.
Step1.DenotebyΠ:Rn\ {x}→ Sn−1 theprojectionfunctioninto Sn−1 givenby Π(y) = y− x
Then
Hn−1(Π(E))≤ 1
dist(x, E)n−1H
n−1(E) (5.23)
foreveryset E⊂ ∂Ω.
ThefunctionΠ isdifferentiable,and|∇Π(y)|≤ |y − x|−1 fory∈ Rn\ {x}. Thus,the restrictionofΠ toE isLipschitzcontinuous,and
|∇Π(y)| ≤ 1
dist(x, E) for y∈ E. (5.24) Inequality (5.24) implies (5.23), by a standard property of Hausdorff measure – see e.g.[59,Theorem 7.5].
Step 2.Wehavethat
Hn−1(Π(E))≤ 2n E
dHn−1(y)
|x − y|n−1 for every Borel set E⊂ ∂Ω. (5.25) Thefollowing chainholds:
E dHn−1(y) |x − y|n−1 = ∞ 0 Hn−1({y ∈ E : |x − y|−n+1> t}) dt = ∞ 0 Hn−1({y ∈ E : |x − y| < τ}) d(−τ1−n) = k∈Z 2k+1 2k Hn−1({y ∈ E : |x − y| < τ}) d(−τ1−n) ≥ k∈Z Hn−1({y ∈ E : |x − y| < 2k}) 2k+1 2k d(−τ1−n) = (1− 2−n+1) k∈Z 2−k(n−1)Hn−1({y ∈ E : |x − y| < 2k}) ≥ (1 − 2−n+1) k∈Z 2−k(n−1)Hn−1({y ∈ E : 2k−1≤ |x − y| < 2k}). (5.26) Since dist(x,{y ∈ E : 2k−1 ≤ |x− y|< 2k})≥ 2k−1,by(5.23) Hn−1({y ∈ E : 2k−1≤ |x − y| < 2k}) ≥ C2(k−1)(n−1)Hn−1(Π({y ∈ E : 2k−1≤ |x − y| < 2k})) (5.27)
fork∈ Z. From(5.26)and(5.27) wededucethat E dHn−1(y) |x − y|n−1 ≥ 12 k∈Z 2−k(n−1)2(k−1)(n−1)Hn−1(Π({y ∈ E : 2k−1≤ |x − y| < 2k})) = 2−n k∈Z Hn−1(Π({y ∈ E : 2k−1 ≤ |x − y| < 2k})) ≥ 2−nHn−1∪ k∈ZΠ({y ∈ E : 2k−1≤ |x − y| < 2k}) = 2−nHn−1Π(∪k∈Z{y ∈ E : 2k−1≤ |x − y| < 2k}) = 2−nHn−1(Π(E)), (5.28)
whenceinequality(5.25)follows.
Step3.Conclusion. Wehavethat Π({y ∈ (∂Ω)x:|g(y)| > t}) = {ϑ ∈ Sn−1:|g(ζ(x, ϑ))| > t} for t > 0. Thus,by(5.25), {y∈(∂Ω)x:|g(y)|>t} dHn−1(y) |x − y|n−1 ≥ 2−nHn−1({ϑ ∈ Sn−1:|g(ζ(x, ϑ))| > t}) for t > 0. Hence, (∂Ω)x |g(y)| |x − y|n−1dHn−1(y) = ∞ 0 {y∈(∂Ω)x:|g(y)|>t} dHn−1(y) |x − y|n−1 dt ≥ 2−n ∞ 0 Hn−1({ϑ ∈ Sn−1:|g(ζ(x, ϑ))| > t}) dt = 2−n Sn−1 |g(ζ(x, ϑ))| dHn−1(ϑ). (5.29) Inequality(5.22)isthusestablished. 2
Lemma 5.5. Let Ω be an open set in Rn, n ≥ 2. Assume that μ is any Borel measure
in Ω fulfilling(5.1) for some α∈ (n− 1,n] and forsome Cμ > 0. Then there exists a
constant C = C(n,α,Cμ) suchthat
(Tg)∗μ(s)≤ Cs−n−1α
sn−1α
0
g∗Hn−1(r) dr for s > 0, (5.30)
Proof. ByProposition 5.4,there existsaconstantC = C(n) suchthat
Tg(x)≤ CN1|g|(x) for x ∈ Ω, (5.31) for everyBorelfunction g : ∂Ω→ R. Hence,byLemma 5.3, with γ = 1,there exists a constantC = C(n,α,Cμ) suchthat
T gL α
n−1,∞(Ω,μ)≤ CgL1(∂Ω) (5.32) forevery Borel functiong : ∂Ω→ R.
Ontheother hand,
0≤ T g(x) ≤ gL∞(∂Ω) Sn−1 dHn−1(ϑ) = nωngL∞(∂Ω) for x∈ Ω, (5.33) and hence T gL∞(Ω,μ)≤ nωngL∞(∂Ω) (5.34) for everyBorel functiong : ∂Ω→ R. We thusdeduce from (5.9), (5.10), (5.13), (5.32)
and (5.34)that s(Tg)∗μ(snα−1)≤ rn−1α (Tg)∗ μ(r)L∞(0,sn−1α )≈ K(Tg, s; L α n−1,∞(Ω, μ), L∞(Ω, μ)) ≤ CK(g, s; L1(∂Ω), L∞(∂Ω)) = C s 0 gH∗n−1(r) dr for s > 0, (5.35)
for some constant C = C(n,α,Cμ), and for every Borel function g : ∂Ω → R. Hence, inequality(5.30)follows. 2
Lemma 5.6.LetΩ beanopensetinRn,n≥ 2,andletγ∈ (0,n). Assumethatμ isany
Borel measureinΩ fulfilling (5.1)forsome α∈ (n− γ,n] and forsomeCμ> 0. Then,
there existsaconstant C = C(n,γ,α,Cμ) such that
(Iγf )∗μ(s)≤ C s−n−γα snα 0 fL∗n(r) dr + ∞ sαn r−n−γn f∗ Ln(r) dr for s > 0, (5.36) forevery f ∈ L1(Ω).
Proof. Astandardweak-type inequalityforRiesz potentialstellsus thatthereexists a constantC1= C1(n,γ,α,Cμ) suchthat
