prescribed centroaffine curvature problem A priori estimates and existence of solutions to the Journal of Functional Analysis

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Journal of Functional Analysis

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A priori estimates and existence of solutions to the prescribed centroaffine curvature problem

Huaiyu Jian^a,b, Jian Lu^b,c,∗, Xu-Jia Wang^b

a DepartmentofMathematics,TsinghuaUniversity,Beijing100084,China

b CentreforMathematicsandItsApplications,AustralianNationalUniversity, Canberra, ACT0200,Australia

cDepartmentofMathematics,ZhejiangUniversityofTechnology,Hangzhou 310023,China

a r t i c l e i n f o a bs t r a c t

Article history:

Received1April2017 Accepted30August2017 Availableonline5September2017 CommunicatedbyCédricVillani

Keywords:

Monge–Ampèreequation Centroaffinecurvature Minkowskiproblem

Inthispaper westudytheprescribedcentroaffinecurvature problem in the Euclidean space Rⁿ⁺¹. This problem is equivalentto solvingaMonge–Ampèreequationontheunit sphere. It correspondsto the critical case of the Blaschke–

Santaló inequality. By approximation from the subcritical case, and using an obstruction condition and a blow-up analysis, we obtain sufficient conditions for the a priori estimates, andthe existence of solutions up to a Lagrange multiplier.

©2017ElsevierInc.Allrightsreserved.

1. Introduction

Given ahypersurfaceM intheEuclideanspaceRⁿ⁺¹,thecentroaffinecurvatureκof M atpointpisbydefinitionequaltoK/dⁿ⁺²,whereK istheGausscurvatureanddis

✩ ThefirstandthesecondauthorsweresupportedbytheNaturalScienceFoundationofChina(11771237, 11401527),thesecondandthirdauthorsweresupportedbytheAustralianResearchCouncil.

* Correspondingauthor.

E-mailaddresses:hjian@math.tsinghua.edu.cn(H. Jian),lj-tshu04@163.com(J. Lu), Xu-Jia.Wang@anu.edu.au(X.-J. Wang).

http://dx.doi.org/10.1016/j.jfa.2017.08.024 0022-1236/©2017ElsevierInc.Allrightsreserved.

thedistancefrom theorigintothetangentplaneofM atp.Thecentroaffinecurvature κwasfirstdiscoveredbyTzitzéica[31] in1908. Itisinvariantundertransformationsin SL(n+ 1) and is anelementary quantity inthe affine differentialgeometry and inthe theoryofconvex bodies[8,13,14,21–23,25,27].It appearsnaturallyingeometric objects suchas affine normaland affine spheres, and plays afundamental role inthe study of manygeometricproblems[1–3,9,15,16,19].

In this paper, we consider the following prescribed centroaffine curvature problem.

Givenapositivefunctionf inRⁿ⁺¹,findproperconditionsonf suchthatthereexistsa closedconvexhypersurfaceM inRⁿ⁺¹ surroundingtheorigin,ofwhichthecentroaffine curvatureκatapoint p∈M isequalto f(p).

ThecorrespondingprescribedmeancurvatureandGausscurvatureproblems,namely the problems with the centroaffine curvature replaced by the mean curvature or the Gauss curvature, wereraised byYau [34]. Inthecase ofGauss curvature, the problem wasstudiedin[10,26,30,33].

LetH be the supportfunction of the polarbody of M. We show in Section2that theproblemisequivalentto solvingthefollowingMonge–Ampèreequation,

det(∇²H+HI)(x) =f(x/H)

Hⁿ⁺² x∈Sⁿ, (1.1)

where∇²H = (∇ijH) is thecovariantderivativesof H with respectto anorthonormal frameontheunitsphereSⁿ,and Iistheunitmatrix.

Problem(1.1)iscloselyrelatedtotheLp-Minkowskiproblem[22],

det(∇²H+HI)(x) =f(x)H^p−1 x∈Sⁿ, (1.2) wheref isapositivefunctiononSⁿ.TheL_p-Minkowskiproblemis anextension ofthe famous Minkowski problem (the case when p = 1), and has been extensively studied recently[5,9,11,18,24,32]. In particular asolutionto the Lp-Minkowskiproblem is also aself-similarsolutiontoananisotrophicGausscurvatureflow, ofwhichtheasymptotic behaviourof solutionshasattractedmuchattentions[4,12].

Equation (1.1), or equation (1.2) inthe case p=−n−1,is referred to as the cen- troaffineMinkowskiproblem[9]. ThecentroaffineMinkowskiproblemis alsoofinterest intheimage processing. Inimageprocessing, onehopes thatthedeformationofimage is invariant when one looks at the picture from different angles. In other words, the imageprocessingshouldbeinvariantunderprojectivetransformations.Forthispurpose an evolution equation was introduced in [2], which becomes a centroaffine Minkowski problemifonedealswithself-similarsolutions.

Inthispaperweestablishtheaprioriestimatesandexistenceofsolutionstoequation (1.1).AsthecentroaffinecurvatureisinvariantunderprojectivetransformationsonSⁿ, allellipsoidsofvolume|B₁(0)|haveconstantcentroaffinecurvature1,namelytheyareall solutionsto(1.1)withf ≡1.Thereforeconditionsareneededfortheuniform estimate

ofH.Infactanobstructionwasfoundin[9]fortheexistenceofsolutions,whichmeans thatforsomef there isnosolutiontoequation(1.1).

On the other hand, equation (1.1) corresponds to the critical exponent case of the Blaschke–Santaló inequality[9].Therefore onemayemploythevariationalapproachto study theproblem.As iswell known,thecritical exponentcaseisusually verycompli- cated.Sowewillfirstproveanexistenceresultinthesubcriticalcase,andthenestablish theaprioriestimatesandprovetheexistenceofsolutionsbyapproximation.

ForasupportfunctionH,denotethevolumeoftheassociatedconvexbodybyvol(H), namely

vol(H) = 1 n+ 1

Sⁿ

Hdet(∇²H+HI). (1.3)

Then wehave

Theorem 1.1. Let f be abounded and positive functionin Rⁿ⁺¹. For any positive con- stants v >0and q∈[n+ 1,n+ 2), there exista number λ>0 anda positive support function H ∈ C^1,γ(Sⁿ) for some γ ∈ (0,1) with volume vol(H) = v, which solve the equation

det(∇²H+HI)(x) =λf(x/H)

H^q x∈Sⁿ. (1.4)

The above theorem dealswith the subcritical case q < n+ 2. Byapproximation to the critical exponent caseq = n+ 2, and using a blow-up analysis and the necessary conditionin[9],wefindsufficientconditionsfortheaprioriestimatesofsolutions.

Theorem 1.2.Letf ∈C¹(Rⁿ⁺¹)be apositivefunction.Assume that f(x) =f(∞) +β+o(1)

|x|^α as|x| → ∞, (1.5) forconstantsα >0,f(∞)>0,andβ= 0,and

either f(x)> f(∞) or f(x)< f(∞) ∀x∈Rⁿ⁺¹. (1.6) LetH be asolution to(1.1).Then wehavetheapriori estimates

C⁻¹≤H ≤C, (1.7)

where C isapositiveconstant depending onlyon nandf.

ByTheorems 1.1 and1.2,andusingapproximation,weobtainthefollowingexistence resultforequation (1.1).

Theorem1.3. Letf ∈C¹(Rⁿ⁺¹)beapositivefunction.Assumethatf satisfies(1.5)with β >0and

f(x)> f(∞)∀x∈Rⁿ⁺¹. (1.8) Then for any positive constant v > 0, there exists a number λ and a support function H ∈C^2,γ (∀ γ∈(0,1))with vol(H)=v whichsolve theequation

det(∇²H+HI)(x) = λf(x/H)

Hⁿ⁺² x∈Sⁿ. (1.9)

Remark.InTheorem 1.1,itsufficestoassumethatf isapositiveandboundedfunction.

Inthiscase,thesolutionisC^1,γ forsomeγ∈(0,1)[7].InTheorems 1.2 and1.3,weuse condition(2.7)and so f ∈C¹(Rⁿ⁺¹) isneeded. Theassumption f ∈C¹(Rⁿ⁺¹) implies thatH ∈C^2,γ(Sⁿ) forallγ∈(0,1).

Theprescribedcentroaffinecurvatureproblem isrelatedto theBlaschke–Santalóin- equality

sup

H

ξinf∈Kvol(H)

Sⁿ

1

(H−ξ·x)ⁿ⁺¹dσ_Sⁿ ≤ ω_n²

n+ 1, (1.10)

justas theprescribed scalarcurvature problem related to theSobolevinequality.Here dσ_Sⁿ denotes the volume element of Sⁿ, and ω_n =

Sⁿdσ_Sⁿ. The prescribed scalar curvatureequationonthesphere

−Δ_Sⁿu+1

2n(n−2)u=R(x)u^p onSⁿ (1.11) hasbeen studiedby numerousauthors (see, e.g.[28]), where p= ⁿ⁺²_n−2.To prove thea priori estimates and the existence of solutions to (1.1), we use the blow-up approach.

Thefirststepis toprovetheexistenceofasolutionHqto (1.4)inthesub-criticalcase q ∈ [n+ 1,n+ 2). Even in the sub-criticalcase, the Monge–Ampère equation (1.2) is morecomplicatedthan(1.11).Itiswellknownthatthesolutionsto(1.11)areuniformly bounded inthe sub-critical case1< p < ⁿ⁺²_n−2. Butthis isnot true for equation (1.2).

There mayexistinfinitely manysolutionsto (1.2)whichare notuniformly boundedin thesubcriticalcase[17].

The second step is to find conditions on f such that Hq is uniformly bounded for all q∈ [n+ 1,n+ 2). Suppose thatsup_x∈SnHq(x)→ ∞ as q → n+ 2. Wenormalize the associated convex body to get anew support functionH˜q, of which the limit H˜_∞ satisfiesequation(1.1)foradifferentfunctionf_∞.Inorderthatthis blow-upargument works,itiscrucialtohaveaclassificationofthelimitshapeH˜_∞.

However, unlike the prescribed scalarcurvature equation (1.11), where the limit of theblow-upsequenceisunique(byaLiouvilletypetheorem),theproblem(1.1)ismore

difficult,becausethelimitf_∞ isafunction,notaconstantingeneral.Toovercomethis difficulty, weassumethat

|p|→+∞lim f(p) =const, (1.12) so thatf_∞is aconstantandhenceH˜_∞ mustbeasphere.

Condition(1.12)aloneisnotsufficient,asthereisnouniform estimateeveniff ≡1.

To establish the a priori estimates (1.7), we use the necessary condition (2.7) and a blow-up analysis,bywhichwearriveat thecondition(1.5),whichisastrengtheningof (1.12).Theblow-upargument wasalsoused in[20] fortherotationallysymmetric case oftheLp-Minkowskiproblem.Inthispaperweconsiderthenon-symmetriccaseandthe analysis ismoredelicate,astherearemanydifferentcasestodealwith.SeeLemma 4.1.

This paper is organized as follows. In Section 2, we show that the prescribed cen- troaffine curvature problem is equivalent to equation (1.1). In Section 3, we prove the existence ofsolutions inthesubcriticalcase, namelyTheorem 1.1. InSection4,we es- tablish theaprioriestimatesinTheorem 1.2byadelicateblow-upanalysis.Finallywe proveTheorem 1.3inSection5.

2. TheMonge–Ampère equation

Given abounded, positivefunctionf inRⁿ⁺¹,the prescribedcentroaffinecurvature problem istofindaconvexhypersurfaceM suchthat

κ(p) =f(p) ∀p∈M, (2.1)

where κ(p) is the centroaffine curvature of M at p. When M is smooth and strictly convex, κ(p) canbe expressedasin[9]

κ(p) = 1

Hⁿ⁺²(x) det(∇²H+HI)(x),

where x∈Sⁿ isthe unitouternormal ofM atpand H isthesupportfunction ofM, given by

H(x) = sup{x, p |p∈M} x∈Sⁿ.

Extend H to Rⁿ⁺¹ such thatit ishomogeneous of degree1.Denote the gradientof H inRⁿ⁺¹ by∇.Itiswellknownthat

p=∇H(x) =∇H(x) +H(x)x.

Thus (2.1)canbewrittenas

Hⁿ⁺²(x) det(∇²H+HI)(x) = f

∇H(x)−1

x∈Sⁿ. (2.2)

LetM^∗betheboundaryofthepolarsetoftheconvexbodyenclosedbyM [27].Let ρ^∗be theradialfunctionofM^∗,suchthat

M^∗={xρ^∗(x)|x∈Sⁿ}. DenotebyH^∗ thesupportfunctionofM^∗.Thenbydefinition,

ρ^∗(x) = 1/H(x), whichimpliesthat

∇H =−∇ρ^∗ (ρ^∗)². Henceintermsofρ^∗, equation(2.2)canbe rewrittenas

det

−ρ^∗∇²ρ^∗+ 2(∇ρ^∗)^T∇ρ^∗+ (ρ^∗)²I

(ρ^∗)⁴ⁿ⁺² =

f −∇ρ^∗ (ρ^∗)²

−1

. (2.3)

Ontheotherhand,letx^∗ betheunitouternormalofM^∗ atpointρ^∗(x)x.Wehave x^∗=− ∇ρ^∗

∇ρ^∗(x), whichimpliesthat

x^∗

H^∗(x^∗) =−∇ρ^∗

(ρ^∗)²(x). (2.4)

ItisknownthattheGausscurvature ofM^∗ atthispoint isgivenby

K^∗= 1

det (∇²H^∗+H^∗I) (x^∗) =det

−ρ^∗∇²ρ^∗+ 2(∇ρ^∗)^T∇ρ^∗+ (ρ^∗)²I (ρ^∗)²ⁿ⁻²|∇ρ^∗|ⁿ⁺² (x).

Henceequation (2.3)canalsobe writtenas det

∇²H^∗+H^∗I

(x^∗) =f

−∇ρ^∗/(ρ^∗)² |∇ρ^∗| (ρ^∗)²

n+2

(x)

= f(x^∗/H^∗) (H^∗)ⁿ⁺² (x^∗),

wherethesecondequalityisdueto(2.4).ThusifH isasolutiontoequation(2.2),then thesupportfunctionofitspolarbody,H^∗,satisfies thefollowing equation

det

∇²H^∗+H^∗I

(x^∗) =f(x^∗/H^∗)

(H^∗)ⁿ⁺² x^∗ ∈Sⁿ, whichisexactlytheequation (1.1).

An important property of equation (1.1) is its invarianceunder projective transfor- mationsonSⁿ [9].ThatisifH isasolutionto(1.1),thenHA withA∈SL(n+ 1) given by

H_A(x) =|Ax| ·H Ax

|Ax|

x∈Sⁿ, (2.5)

satisfies thefollowingequation

det(∇²HA+HAI) =f(Ax/HA)

H_Aⁿ⁺² x∈Sⁿ. (2.6)

Inthepaper[9]theauthorsalsofoundanecessaryconditionfortheexistenceofsolutions to (1.1).Thatis, ifH isasolutionto (1.1),then

Sⁿ

∇ξ[f(x/H)](x)

Hⁿ⁺¹ dσ_Sⁿ = 0 (2.7)

foranyprojectivevectorfieldξgeneratedbyanysquarematrixBofordern+ 1,namely ξ(x) =Bx−(x^TBx)x, x∈Sⁿ. (2.8) In the following we may drop dσSⁿ, when the integral over Sⁿ is under the standard metric.

3. Existenceofsolutionsinthesubcriticalcase

InthissectionwesketchtheproofofTheorem 1.1.Theproof issimilar tothatin[9, Section5],wheretheexistenceofsolutionsforthecasef =f(x) wasobtained.Werefer thereaderto [9]formoredetails.

Forq∈[n+ 1,n+ 2),we denote

F(x, t) =

+∞

t

f(x/s)

s^q ds, (3.1)

where x∈Sⁿ andt>0,and

J[H] =

Sⁿ

F(x, H(x)). (3.2)

Considerthemaximizingproblem M_v=: sup

H∈Sv

y∈KinfH

J[H(x)−y·x], (3.3)

wherevisapositiveconstant,andSvisthesetofsupportfunctionssuchthatthevolume oftheassociatedconvexbodyisequaltov.

Observethat

finf

q−1· 1

t^q−1 ≤F(x, t)≤ fsup

q−1· 1

t^q−1 ∀x∈Sⁿ. (3.4) Given H ∈ Sv, since q ∈ [n+ 1,n+ 2), one easily sees that J[H(x)−y ·x] → ∞ whenever y ∈ KH and y converges to aboundarypoint of∂KH. Hence there exists a pointy=y(H)∈K_H suchthattheinfimuminf_y∈KHJ[H(x)−y·x] isattainedat y.

Weclaimthatthereexisttwo positiveconstantsC₁,C₂,dependingonlyonnandf, suchthat

C₁

q−1v^−q−1n+1 ≤Mv ≤ C₂

q−1v^−n+1q−1. (3.5) Infact,bytheHölderinequality,wehave

J[H(x)−y·x]≤ fsup

q−1

Sⁿ

1

(H(x)−y·x)^q−1

≤ Cnfsup

q−1

⎛

⎝

Sⁿ

1

(H(x)−y·x)ⁿ⁺¹

⎞

⎠

q−1 n+1

.

BytheBlaschke–Santalóinequality(1.10),we obtain

y∈KinfH

J[H(x)−y·x]≤Cnfsup

q−1 ·vol(KH)^−q−1n+1 = C2

q−1v^−q−1n+1,

whichimpliesthesecond inequalityof(3.5).LettingH ≡[(n+ 1)v/ω_n]^1/(n+1),wehave Mv ≥ inf

y∈KH

J[H(x)−y·x]

≥ f_inf q−1

Sⁿ

1 H^q−1

≥Cnfinf

q−1v^−n+1q−1, whichisthefirstinequalityof(3.5).

Lemma3.1. Foragivenpositiveconstant v >0,thereexistsasupportfunctionH ∈ Sv

suchthat Mv= infy∈KHJ[H(x)−y·x].

Proof. Let {Hj} ⊂ Sv be a maximizing sequence. We show that {Hj} is uniformly bounded. Supposetothecontrarythatmaxx∈SⁿHj(x)→+∞as j→+∞.Denote the minimum ellipsoidofKHj byEj.Letzj bethecentreof Ej.Wehave

y∈infK_HjJ[H_j(x)−y·x]≤J[H_j−z_j·x]

≤ fsup

q−1

Sⁿ

1

(Hj(x)−zj·x)^q−1.

By a translation of coordinates, we assume that zj is the origin. Then the support function of Ej canbe expressed as |Bjx|for somepositive definite matrixBj of order n+ 1.As E_j istheminimumellipsoidof K_Hj, wehave

1

n+ 1|B_jx| ≤H_j(x)≤ |B_jx| ∀x∈Sⁿ. (3.6) It followsthat

y∈infK_HjJ[H_j(x)−y·x]≤C_nf_sup

Sⁿ

1

|Bjx|^q−1. (3.7) From (3.6), we seethatC_n⁻¹v ≤detB_j ≤C_nv. Byassumption,max_x∈Sⁿ|B_jx| →+∞ as j→+∞.Hencewhenq < n+ 2,oneinfersthat[9]

j→lim+∞

Sⁿ

1

|B_jx|^q−1 = 0,

whichtogetherwith(3.7)impliesthatMv= 0,contradictingwith(3.5).Therefore{Hj} is uniformlybounded.

By Blaschke’s selection theorem, there is a subsequence of {H_j} which converges uniformly toamaximizerH oftheproblem (3.3)and H isuniformly bounded. 2 ProofofTheorem 1.1. Foranygivenpositiveconstantv >0,byLemma 3.1,thereexists asupportfunctionH ∈ Sv suchthatMv= infy∈KHJ[H(x)−y·x]. Fromtheproofsof Corollary5.4andLemmas5.5–5.6in[9],oneseesthatthemaximizerH isageneralized solutionto

det(∇²H+HI)(x) =λf(x/H)

H^q ∀x∈Sⁿ, where λistheLagrangemultiplier.

Whenq∈[n+ 1,n+ 2),oneeasily verifiesthatH ispositive.ForifinfH = 0,then J[H]=∞, whichisincontradiction with (3.5).Henceby [7], H is strictly convexand C^1,γ smooth,forsomeγ∈(0,1).ThiscompletestheproofofTheorem 1.1. 2

Weremarkthatiffurthermoref ∈C¹(Rⁿ⁺¹),thenby[6,29], wehaveH ∈C^2,γ(Sⁿ) foranyγ∈(0,1).

4. Aprioriestimates inthe criticalcase

Inthis sectionwe usethe necessary condition(2.7)and ablow-up analysis toprove theaprioriestimates(1.7)inTheorem 1.2.

LetA_k∈SL(n+ 1) beasequenceofdiagonalmatrices,

Ak = diag (s1,k,· · ·, sn+1,k), (4.1) where s1,k ≥ · · · ≥ sn+1,k > 0 and s1,k → +∞ and sn+1,k → 0 as k → +∞. We successivelydefine integersl1,l2,· · · asfollows:

l1:= max

j

j : lim

k sj,k= +∞

, l_i:= max

j

j: lim

k

s_j,k sl_i−1,k

= +∞

fori= 2,3,· · ·.

(4.2)

By choosing a subsequence we may assume all the limits exist or equal infinity. The procedurein(4.2)mustendinfinitesteps,say,atstepm.Thenwe have

1≤l_m<· · ·<· · ·< l₁≤n < n+ 1.

Lemma 4.1. Assume that ϕ_k, ψ_k are twosequences of uniformly bounded functions on Sⁿ,converging uniformly tofunctions ϕ,ψ ask→+∞.Assume that ϕ∈C¹(Sⁿ) and ψ isapositiveconstant.Consider thefollowingintegral

Λk :=

Sⁿ

ϕk(x)ζ(ψk(x)Akx) 1

|A_kx|^α, (4.3)

whereα >0isaconstant,andζ∈C(Rⁿ⁺¹)isa boundedfunctionsatisfying (a) ζ(y)=O(|y|^α) asy →0,

(b) ζ(y)=ζ_∞+o(1) asy→ ∞.

Thenask→+∞,we havethefollowingestimates.

(i) Whenα > l₁,

Λ_k= 1

s_1,k· · ·s_l1,k

⎛

⎝ψ^α−l1

S^n−l1

ϕ(0, v)dσ_Sn−l1

u∈R^l1

ζ(u, ψN v)

|(u, ψN v)|^αdu+o(1)

⎞

⎠,

(4.4) whereN = limk→∞diag (sl1+1,k,· · ·, sn,k,0) (weallow allthelimits tobe zero).

(ii) When α=l1,

Λk= Clogsl1,k

s1,k· · ·sl1,k

⎛

⎝ζ_∞

S^n−l1

ϕ(0, v)dσ_Sn−l1 +o(1)

⎞

⎠. (4.5)

(iii) When li< α < li−1 fori= 2,· · ·,m,

Λ_k= C

s_1,k· · ·s_li,k·s^α−l_l ⁱ

i−1,k

⎛

⎝ζ_∞

S^n−li

ϕ(0, v)

|N v|^α−lidσ_Sⁿ−li+o(1)

⎞

⎠, (4.6)

where N = lim_k→∞diag _s^sli+1,k

li−1,k,^s_s^li+2,k

li−1,k,· · ·,^{sli−1−1,k}_s

li−1,k ,^s_s^li−1,k

li−1,k,0,· · ·,0

isa matrix of ordern+ 1−li.

(iv) When α=l_i fori= 2,· · ·,m,

Λ_k =Clog(sli,k/sl_i−1,k) s1,k· · ·sli,k

⎛

⎝ζ_∞

S^n−li

ϕ(0, v)dσ_Sn−li +o(1)

⎞

⎠. (4.7)

(v) When α < lm,

Λk= 1 s^α_l

m,k

⎛

⎝ζ_∞

Sⁿ

ϕ(x)

|Ax˜ |^αdσSⁿ+o(1)

⎞

⎠, (4.8)

where A˜ = limk→∞diag _s^s1,k

lm,k,_s^s2,k

lm,k,· · ·,^slm−1,k_s

lm,k ,^s_s^lm,k

lm,k,0,· · ·,0

is a matrix of order n+ 1.

In theabove,C isa positiveconstant.

Proof. Weprovethislemma casebycase.

Case(i):α > l1.Forconvenience, wedenotel:=l1,x= (u,v) and A_k =

Mk 0 0 Nk

, (4.9)

where

u= (x₁,· · ·, x_l), v= (x_l+1,· · ·, x_n+1), (4.10) and Mk andNk arediagonalmatricesoforder land n+ 1−l,respectively.Denote

Sⁿ_∗ ={x= (u, v)∈Sⁿ |1/2≤ |u| ≤1}. (4.11) Bythecoareaformula,

Sⁿ_∗

1

|M_ku|^α =

1/2≤|u|≤1

du ρ(u)

|v|=ρ(u)

1

|M_ku|^αdσ

=ωn−l

1/2≤|u|≤1

1

|M_ku|^αρ(u)^n−l−1du,

(4.12)

whereρ(u)=

1− |u|².Wehave

1/2≤|u|≤1

1

|Mku|^αρ(u)^n−l−1du= 1

1 2

ρ(r)^n−l−1dr

|u|=r

1

|Mku|^αdσ

= 1

1 2

r^l−α−1ρ(r)^n−l−1dr

|u|=1

1

|M_ku|^αdσ

=C_l,α

S^l−1

1

|Mku|^αdσ,

whereCl,α isapositiveconstantdepending onlyonn, landα.By[18,Lemma3.1],

S^l−1

1

|Mku|^αdσ = 1 detMk

S^l−1

M_k⁻¹u^α−ldσ.

Hencefrom (4.12)weobtain

S_∗ⁿ

1

|M_ku|^α =ωn−lCl,α

detM_k

S^l−1

M_k⁻¹u^α−ldσ. (4.13)

Whenα−l >0,thereexist positiveconstantC˜ dependingonlyonlandα,suchthat

C˜⁻¹M_k⁻¹u^α−l≤ u1

s1,k

^α−l+· · ·+ ul

sl,k

^α−l≤C˜M_k⁻¹u^α−l. Henceby(4.13),

S_∗ⁿ

1

|Mku|^α = C_k detMk

1

s^α_1,k^−l +· · ·+ 1 s^α_l,k^−l

= Ck

detM_k

trM_k⁻¹α−l

,

(4.14)

where C_k isapositiveconstantindependentofM_k,andC˜⁻¹≤ _ωn−^C_l^k_Cl,α ≤C.˜ Now wecomputeΛk.Bythecoareaformula,wehave

I_k:=

Sⁿ\Sⁿ_∗

ϕ_k(x)ζ(ψ_k(x)A_kx) 1

|Akx|^α

=

|u|<1/2

du ρ(u)

|v|=ρ(u)

ϕ_k(x)ζ(ψ_k(x)A_kx) 1

|Akx|^αdσ.

Letv=ρ(u)˜v.Wehave

Ik =

|u|<1/2

ρ^n−l−1(u)du

|v˜|=1

ϕk(x)ζ(ψk(x)Akx) 1

|A_kx|^αdσ

=

|˜v|=1

dσ

|u|<1/2

ϕk(x)ζ(ψk(x)Akx) 1

|Akx|^αρ^n−l−1(u)du

=:

|˜v|=1

Φ_k(˜v)dσ.

(4.15)

Letu=M_k⁻¹u.˜ Then

Φk(˜v) = 1 detMk

|M_k⁻¹u˜|<1/2

ϕk(x)ζ(ψk(x)(˜u, Nkv)) 1

|u, N˜ kv|^αρ^n−l−1(u)d˜u,

(4.16)

where |u, N˜ _kv|isanabbreviationof|(˜u, N_kv)|.Therefore

|detMk·Φk(˜v)| ≤C

|M_k⁻¹u˜|<1/2

|ζ(ψk(x)(˜u, Nkv))| 1

|u, N˜ _kv|^αd˜u

≤C

u∈R˜ ^l

|ζ(ψk(x)(˜u, Nkv))| 1

|u, N˜ _kv|^αd˜u,

which is integrable by our assumptions (a), (b) and α > l. Applying the dominated convergence theoremto(4.16), weobtain,ask→+∞,

Φ_k(˜v) = 1 detM_k

⎛

⎝

u∈R˜ ^l

ϕ(0,v)ζ˜ (ψ·(˜u, Nv))˜ 1

|u, N˜ v˜|^αd˜u+o(1)

⎞

⎠,

whereN := lim_kN_k.Inserting theaboveformulainto(4.15),weobtain

I_k = 1 detMk

⎛

⎜⎝

|˜v|=1

ϕ(0,v)dσ(˜˜ v)

˜ u∈R^l

ζ(ψ·(˜u, Nv))˜ 1

|u, N˜ v˜|^αd˜u+o(1)

⎞

⎟⎠. (4.17)

Notethat

|Λ_k−I_k| ≤C

S_∗ⁿ

1

|Mku|^α

≤ C

detMk

trM_k⁻¹α−l

= o(1) detMk

. Hencefrom (4.17),

Λk = 1 detMk

⎛

⎜⎝

|˜v|=1

ϕ(0,v)dσ(˜˜ v)

u∈R˜ ^l

ζ(ψ·(˜u, Nv))˜

|u, N˜ v˜|^α d˜u+o(1)

⎞

⎟⎠

= 1

detMk

⎛

⎜⎝ψ^α−l

|v˜|=1

ϕ(0,˜v)dσ(˜v)

˜ u∈R^l

ζ(˜u, ψNv)˜

|u, ψN˜ ˜v|^αd˜u+o(1)

⎞

⎟⎠.

Weobtain(4.4).

Case(ii):α=l1.Forconvenience,wedenoteagainl:=l1,andusethesamenotations in(4.10) and(4.9).

BeforecomputingΛk,weestimateanintegralfirst. Denote

T_k :={x= (u, v)∈Sⁿ | |u|<1/2,|M_ku| ≥s_l,k/2}. (4.18) Bythecoareaformula,similarly to(4.12),wehave

Tk

1

|M_ku|^α =ωn−l

|u|<1/2,|Mku|≥sl,k/2

1

|M_ku|^αρ(u)^n−l−1du

=Cω_n−l

|u|<1/2,|Mku|≥sl,k/2

1

|Mku|^αdu

(4.19)

foraconstantC∈(1/2ⁿ,2).Let

u:=sl,kM_k⁻¹u.˜ (4.20)

Then

Tk

1

|M_ku|^α = Cωn−l

detMk·s^α−l_l,k

|sl,kM_k⁻¹˜u|<1/2,|u˜|≥1/2

1

|u˜|^αd˜u. (4.21)

Observing that sl,kM_k⁻¹ is a diagonal matrix whose diagonal entries are in ascending order andthelastoneisequalto 1,wehave

|sl,kM_k⁻¹u|<1/2,˜ |˜u|≥1/2

1

|u˜|^αd˜u≤

u∈R˜ ^l−1×(−1/2,1/2),|˜u|≥1/2

1

|u˜|^αd˜u

<+∞,

where thelast inequalityholdswhenα≥l.(Althoughinthiscaseα=l,wededucethe following (4.22)and (4.27)forα≥l,whichwillbe usedincase(iii).) Thus thereexists apositiveconstantC dependingonlyonnandl,suchthat

Tk

1

|Mku|^α ≤ C

detM_k·s^α−l_l,k . (4.22) To estimateΛ_k, withoutloss ofgenerality,weassumethat

s_l,k>2, s_l+1,k ≤1, ∀k≥1.

Denote

Fk :={x= (u, v)∈Sⁿ | |Mku| ≥1}, (4.23) G_k:={x= (u, v)∈Sⁿ | |M_ku|< s_l,k/2}. (4.24) Then Fk∪Gk =Sⁿ,and

|u|<1/2 ∀x= (u, v)∈Gk, (4.25)

Sⁿ\Gk=S_∗ⁿ∪Tk. (4.26)

By(4.14)and (4.22)andsinceα≥l,there existsapositiveconstantCdependingonly onn,l andα,suchthat

Sⁿ\Gk

1

|Mku|^α ≤ C

detM_k·s^α−l_l,k . (4.27) Observethat

|Akx| ≤√

2|Mku|, ∀x= (u, v)∈Fk. (4.28)

Similarlyto(4.19),we have

|Sⁿ\Fk|=ωn−l

|Mku|<1

ρ^n−l−1(u)du

=Cω_n−l

|Mku|<1

du (4.29)

= C

detM_k. SinceF_k∪G_k =Sⁿ,wecanwriteΛ_k as

Λk=

⎛

⎜⎝

Sⁿ\Fk

+

Fk∩Gk

+

Sⁿ\Gk

⎞

⎟⎠ϕk(x)ζ(ψk(x)Akx) 1

|Akx|^α

=:I_k+II_k+III_k.

(4.30)

Notingthattheintegrandisboundedbyourassumptions(a)and(b),weseefrom(4.29) that

|I_k| ≤C

Sⁿ\Fk

dσ_Sⁿ≤ C detMk

.

By(4.27) wealsohave

|III_k| ≤C

Sⁿ\Gk

1

|Mku|^α ≤ C detMk

.

Therefore(4.30)canbewrittenas

Λk =IIk+ O(1)

detM_k ask→+∞. (4.31)

ToestimateIIk,firstcomputingas in(4.19),(4.20)and(4.21),wehave

Fk∩Gk

1

|M_ku|^α ≤

Fk∩Gk

1

|M_ku|^l

= Cωn−l

detM_k

1/sl,k≤|u˜|<1/2

1

|u˜|^ld˜u

= Cω_n−lω_l−1 detMk

1/2 1/sl,k

1

rdr (4.32)

= Cωn−lωl−1

detMk

log sl,k

2

= (C+o(1))logsl,k

detM_k, as k→+∞,where C isapositiveconstantindependentofk.

Weclaimthatforanyboundedfunctionη∈C(Rⁿ⁺¹) satisfying

ylim→∞η(y) = 0, (4.33)

and anypositiveconstantλ_k≥1,

Fk∩Gk

η(ψk(x)λkAkx) 1

|A_kx|^α =o(1)logsl,k

detM_k, as k→+∞. (4.34) Infact,denote

q(r) := sup

|y|≥r|η(y)| r∈[0,+∞).

Thenqisboundedandmonotonicallydecreasing.By(4.33)itsatisfieslimr→+∞q(r)= 0.

Observing that

Fk∩Gk

|η(ψk(x)λkAkx)| 1

|A_kx|^α ≤

Fk∩Gk

q(|ψk(x)λkAkx|) 1

|A_kx|^α

≤

Fk∩Gk

q ψ

2|A_kx| 1

|Akx|^α

≤

Fk∩Gk

q(|M_ku|) 1

|Mku|^α,

(4.35)

where withoutloss of generality, we haveassumed thatψ ≥2.Again computing as in (4.19),(4.20)and (4.21),wehave

Fk∩Gk

q(|M_ku|) 1

|Mku|^α ≤

Fk∩Gk

q(|M_ku|) 1

|M_ku|^l

= Cωn−l

detM_k

1/sl,k≤|u˜|<1/2

q(|sl,ku˜|)

|u˜|^l d˜u (4.36)

= Cω_n−lω_l−1 detMk

1/2 1/sl,k

q(s_l,kt) t dt

= Cω_n−lω_l−1 detMk

sl,k/2 1

q(r) r dr.

Sinceq(t)→0 ast→ ∞,itis easytoseethat

sl,k/2 1

q(r)

r dr=o(1) logs_l,k ask→ ∞.Hence(4.36)becomes

Fk∩Gk

q(|Mku|) 1

|Mku|^α =o(1)logs_l,k detMk

,

whichtogetherwith(4.35)implies(4.34).

WecannowcomputeII_k.Write

ζ(y) =ζ_∞+η(y).

Thenη satisfies(4.33).Byourassumptions,

IIk =

Fk∩Gk

(ϕ(x) +o(1)) (ζ_∞+η(ψk(x)Akx)) 1

|A_kx|^α

=ζ_∞

Fk∩Gk

ϕ(x)

|A_kx|^α +

Fk∩Gk

ϕ(x)η(ψk(x)Akx)

|A_kx|^α +o(1)

Fk∩Gk

1

|A_kx|^α

=ζ_∞

Fk∩Gk

ϕ(x)

|A_kx|^α +o(1)logsl,k

detM_k,

(4.37)

where(4.32)and(4.34)areusedinthelastequality.Furthermore,bythecoareaformula wehave

Fk∩Gk

ϕ(x)

|A_kx|^α =

1≤|Mku|<sl,k/2

du ρ(u)

|v|=ρ(u)

ϕ(x)

|A_kx|^αdσ

=

1≤|Mku|<sl,k/2

ρ(u)^n−l−1du

|˜v|=1

ϕ(u, ρ(u)˜v)

|A_kx|^α dσ.

(4.38)