CR manifolds of dimension 3 Solving the Kohn Laplacian on asymptotically flat Advances in Mathematics

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Advances in Mathematics

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Solving the Kohn Laplacian on asymptotically flat CR manifolds of dimension 3

Chin-Yu Hsiao^a,∗,1, Po-Lam Yung^b,2

a Instituteof Mathematics,AcademiaSinica,6F,Astronomy-Mathematics Building,No. 1,Sec. 4,RooseveltRoad,Taipei10617,Taiwan

b DepartmentofMathematics,theChineseUniversityofHongKong,Shatin, Hong Kong

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

Article history:

Received8December2014 Accepted28April2015 Availableonline12June2015 CommunicatedbyCharlesFefferman

Keywords:

KohnLaplacian

Stronglypseudoconvex3manifolds CRpositivemasstheorems Severalcomplexvariables

Let ( ˆX,T^1,0Xˆ) be a compact orientable CR embeddable threedimensionalstronglypseudoconvexCRmanifold,where T^1,0Xˆ is a CR structure on X.ˆ Fix a point p ∈ Xˆ and take aglobalcontactformθˆsothat θˆisasymptotically flat nearp.Then( ˆX,T^1,0X,ˆ θ) isˆ apseudohermitian3-manifold.

Let Gp ∈ C^∞( ˆX \ {p}), Gp > 0, with Gp(x) ∼ ϑ(x,p)⁻² near p, where ϑ(x,y) denotes the natural pseudohermitian distanceonX.ˆ Considerthenewpseudohermitian3-manifold withablow-upofcontactform( ˆX\ {p},T^1,0X,ˆ G²_pθ) andˆ let 2bdenotethecorrespondingKohnLaplacianonXˆ\ {p}. In this paper, weprove that theweighted Kohn Laplacian G²_p2b has closed range in L² with respect to theweighted volumeformG²_pθˆ∧dθ,ˆ andthattheassociatedpartialinverse and the Szegö projection enjoy some regularity properties near p. As an application, we prove the existence of some special functionsinthekernelof 2b thatgrowat aspecific rate at p. The existence of such functions provides an importantingredientfortheproofofapositivemasstheorem

* Correspondingauthor.

E-mailaddresses:chsiao@math.sinica.edu.tw(C.-Y. Hsiao),plyung@math.cuhk.edu.hk(P.-L. Yung).

1 Supported inpartbytheDFGfundedprojectMA2469/2-1,TaiwanMinistryofScienceandTechnology project103-2115-M-001-001andtheGolden-JadefellowshipofKendaFoundation.

2 Supported inpartbyNSFgrantDMS-1201474,aTitchmarshFellowshipattheUniversityofOxford, a JuniorResearchFellowshipfromSt. Hilda’sCollege,andadirectgrant forresearchfromthe Chinese UniversityofHongKong(4053120).

http://dx.doi.org/10.1016/j.aim.2015.04.028 0001-8708/© 2015ElsevierInc.All rights reserved.

in 3-dimensional CR geometry by Cheng, Malchiodi and Yang[5].

© 2015ElsevierInc.All rights reserved.

1. Introduction

1.1. Motivationfrom CRgeometry

Thestudydescribed inthispaperwasmotivatedbythatonapositivemasstheorem in3-dimensional CR geometry by Cheng,Malchiodi andYang [5], where one needsto findsomespecial functionsinthekernel ofthe KohnLaplacianthatgrow at aspecific rate at agiven point onan asymptotically flat pseudohermitian3-manifold. We begin bygivingabriefdescriptionoftherelevanceofourresultwith theirworkbelow.

Consider a compact orientable 3-dimensional strongly pseudoconvex CR manifold ( ˆX,T^1,0Xˆ) withCRstructureT^1,0Xˆ.Weassumethroughout thatitisCRembeddable insomeC^N.Bychoosingacontactformθˆ₀onXˆthatiscompatiblewithitsCRstructure, onecanmake( ˆX,T^1,0X,ˆ θˆ₀) apseudohermitian3-manifold;inparticular,onecandefine aHermitian innerproductonT^1,0Xˆ,by

Z1|Z2θˆ0 = 1

2dθˆ0(Z1, iZ2).

Now fix p ∈Xˆ. By conformallychanging the contact form, we mayfind another con- tact form θˆ(which is amultiple of θˆ0 by apositive smooth function), so that near p, there exists CR normalcoordinates (z,t). In other words, the contact form θˆand the coordinates(z,t) arechosen,so that

(i) thepoint pcorrespondsto (z,t)= (0,0);

(ii) one canfind alocal section Zˆ1 of T^1,0Xˆ near p, withZˆ1|Zˆ1θˆ= 1, such thatZˆ1

admitsanexpansionnear pas describedin(1.31) below;and

(iii) theReebvectorfieldTˆwithrespecttoθˆadmitsanexpansionasdescribedin(1.32).

Then ( ˆX,T^1,0X,ˆ θ) isˆ another pseudohermitian 3-manifold. Assume that this pseudo- hermitian 3-manifold is of positive Tanaka–Webster class: this means that the lowest eigenvalueoftheconformalsublaplacian

L_b:=−4Δ_b+R

is strictly positive. Here R = Rθˆ is the Tanaka–Webster curvature of Xˆ, and Δb is thesublaplacianon X.ˆ (Theaboveassumption onLb willhold when e.g.Rθˆisstrictly positiveonX.)ˆ ThenLb isinvertible,so onecanwritedowntheGreen’s functionGp of Lb withpoleat p.Wenormalize Gp so that

LbGp= 16δp.

Then nearp,Gpadmits thefollowingexpansion Gp= 1

2πρˆ⁻²+A+f, f ∈ E( ˆρ¹), (1.1) where A is some real constant, ρˆ⁴(z,t) = |z|⁴+t², and for m ∈ R, E( ˆρ^m) denotes, roughlyspeaking,theset ofallsmoothfunctionsg∈C^∞( ˆX\ {p}) suchthat|g(z, t)| ≤

ˆ

ρ(z,t)^{m−p−q−2r} near p, along with some suitable control of the growth of derivatives near p(see(1.34) fortheprecisemeaningoftheFréchetspaceE( ˆρ^m)).

Letnow

X = ˆX\ {p} and θ=G²_pθˆ

andletT^1,0X betherestrictionofT^1,0XˆtoX.Then(X,T^1,0X,θ) isanewnon-compact pseudohermitianmanifold,whichwethinkofastheblow-upofouroriginal( ˆX,T^1,0X,ˆ θ)ˆ at p. Wesay thatthis pseudohermitianmanifold isasymptotically flat,since underan inversion of coordinates, X has asymptotically the geometry of the Heisenberg group at infinity. We note that the Tanaka–Webster scalar curvature Rθ of X is identically zero, since the conformal factor Gp we used is theGreen’s function for the conformal sublaplacian on Xˆ. Let ∂b and 2b denote the associated tangential Cauchy–Riemann operatorandKohnLaplacianrespectively.In[5],Cheng,MalchiodiandYangintroduced thepseudohermitianp-massfor(X,T^1,0X,θ),given by

m(θ) := lim

Λ→0i

{ρ=Λˆ }

ω¹₁∧θ,

where ω¹₁ standsfor theconnectionform of thegiven pseudohermitianstructure.Then theyprovedthatthereisaspecificβ∈ E( ˆρ⁻¹),with2bβ=E( ˆρ⁴),suchthat

m(θ) =−3 2

X

2bβ²θ∧dθ+ 3

X

|β_,¯1¯1|²θ∧dθ+3 4

X

β·Pβθ ∧dθ, (1.2)

where β_,¯1¯1 is some derivative of the function β, and P is the CR Paneitz operatorof (X,T^1,0X,θ). (Note Rθ = 0 in our current set-up, so the term involving Rθ in the corresponding identityofmassin[5]isnotpresentabove.)Moreover,itwasshownthat (1.2) holds for any β in place of β, as long as β−β ∈ E( ˆρ^1+δ), and 2bβ = E( ˆρ^3+δ) for some δ > 0. Thus, if we could find such a β in the kernel of 2b, then under the assumption thattheCRPaneitzoperatorP isnon-negative,onecanconcludethatthe mass m(θ) is non-negative.Theconstruction ofsuchβ isthemotivation ofthecurrent paper. (SeeCorollary 1.2 inthenextsubsection.)

Classically,ifonewantsto solve2b onsayacompactCRmanifold, oneproceeds by showing firstthat2b extendstoaclosedlinearoperatoronL²,and thatthis extended

2b has closed range on L². Then one solves 2b in a weak sense, and shows that the solutionisclassicaliftheright handside oftheequation issmooth.This strategydoes notdirectly apply in oursituation, since our CRmanifold X is non-compact. In fact, thenatural volumeform on X is given by θ∧dθ, and even ifwe extend2b so that it becomesaclosedlinearoperatoronL²(θ∧dθ),thisoperatormaynothaveclosedrange inL²(θ∧dθ),asisseeninsomesimpleexamples(e.g.whenXˆ istheunitsphereinC²).

Toovercomethis difficulty,weintroduceinthis paperaweightedvolumeform m1:=G⁻²_p θ∧dθ,

aswell asaweightedKohn Laplacian,namely 2b,1:=G²_p2b.

Wewillshowthat2b,1extendsas adenselydefinedclosedlinearoperator 2b,1: Dom2b,1⊂L²(m₁)→L²(m₁),

andthisextendedoperatorhasclosedrangeinL²(m1).(SeeTheorem 1.3below.)Here L²(m1) isthespaceofL² functionswithrespecttothevolumeformm1.Asaresult,we havethefollowing L² decomposition:

2b,1N+ Π =I on L²(m₁)

Here N : L²(m1) → Dom2b,1 is the partial inverse of 2b,1 and Π : L²(m1) → (Ran2b,1)^⊥ isthe orthogonalprojection ontothe orthogonalcomplement of therange of 2b,1 inL²(m1). Wewill show thatfor every 0< δ < 2,N and Π can be extended continuouslyto

N :E( ˆρ^−2+δ)→ E( ˆρ^δ),

Π :E( ˆρ^−2+δ)→ E( ˆρ^−2+δ) (1.3) (seeTheorem 1.4below). Hence

2b,1N+ Π =I on E( ˆρ^−2+δ), (1.4) forevery0< δ <2.Now,letβbeas in(1.2).Put

f :=2b,1β=G²_p2bβ.

Then by theexpansion of G_p and theassumption on2bβ, we have f ∈ E( ˆρ^−1+δ), for every0 < δ <1. From (1.3), we know thatΠf is well-defined and N f ∈ E( ˆρ^1+δ), for every0< δ <1.Moreover,wewillshow, inTheorem 1.5below,that

Πf = 0.

Thus from(1.4),wehave

2b,1(N f) =f =2b,1β.

Ifwe put

β:=β−N f, then

2bβ = 0 and β−β∈ E( ˆρ^1+δ), forevery0< δ <1.Withthiswe haveachieved ourgoal.

It turnsoutthatalargepartofouranalysis doesnotdependonthefactthatG_pis theGreen’sfunctionofaconformalsublaplacian.AllthatweneedisthatG_p admitsan expansionas in(1.1),thatit issmoothonX,and thatitis positiveeverywhere onX.

Wewill formulateourresultinthisframeworkinthenextsubsection.

We expect that it is possible to approach the same problem by proceeding via L^p spaces rather than weighted L² spaces. In [13], we solved the 2b equation in someL^p spacesinaspecialcase.

Theoperator2b,1canbeseenastheKohnLaplacianonthenon-compactCRmanifold X = ˆX\ {p}definedwithrespecttothenaturalCRstructureT^1,0Xˆ andthe“singular”

volumeform m1. Thecoefficientsof2b,1 aresmoothonX butsingularatp. Thiswork canbe seenas afirststudy of thiskind of“singularKohn Laplacians”.It will be quite interesting todevelop somekindof “singular”functional calculusforpseudodifferential operators and Fourierintegraloperators andestablishacompletely microlocalanalysis for 2b,1 along the lines of Beals and Greiner [1], Boutet de Monvel and Sjöstrand [3]

and [12].Wehopethatthe“singularKohn Laplacians”will beinterestingforanalysts.

1.2. Ourmainresult

Let us now formulate our main results intheir full generality. Consider acompact orientable3-dimensional stronglypseudoconvexpseudohermitianmanifoldXˆ,withCR structure T^1,0Xˆ andcontactformθˆ0.WeassumethroughoutthatitisCRembeddable insomeC^N.Byconformallychangingthecontactformθˆ0,wemayfindanothercontact form θ,ˆ so thatnear p, onecan find CR normal coordinates (z,t) as described in the previoussubsection.Wewillwrite

ˆ

ρ(z, t) = (|z|⁴+t²)^1/4,

and foreverym∈R,wecandefine aFréchetfunctionspaceE( ˆρ^m) as in(1.34).

Now fixapoint p∈Xˆ,andlet

X = ˆX\ {p}.

WefixfromnowonaneverywherepositivefunctionGp∈C^∞(X),suchthatGpadmits anexpansion

G_p= 1

2πρˆ⁻²+A+f, f ∈ E( ˆρ¹), (1.5) where A is some real constant. (Again Gp need not be the Green’s function of the conformalsublaplacianany more.)Define now

θ=G²_pθ.ˆ

Then(X,T^1,0X,ˆ θ) isanon-compact pseudohermitian3-manifold,whichwethinkofas theblow-upoftheoriginalX.ˆ TheθdefinesforusapointwiseHermitianinner product onT^1,0X, givenby

Z₁|Z₂θ=1

2dθ(Z₁, iZ₂);

wedenotethedualpointwiseinner productonthespace(0,1) formsonX bythesame notation·|·θ.Let∂b bethetangentialCauchy–RiemannoperatoronX.Thisisdefined depending onlyonthe CRstructure on X. Now there isanaturalvolume form onX, givenby

m:=θ∧dθ.

Thisinduces aninner productonfunctionsonX,givenby (f|g)_m=

X

f g m,

andaninnerproduct on(0,1) formsonX,givenby (α|β)_m,θ=

X

α|βθm.

Wewrite∂^∗,f_b fortheformaladjointof∂bunderthesetwoinnerproducts.Inotherwords,

∂^∗,f_b satisfies

(∂bu|v)m,θ= (u|∂^∗,f_b v)m

forallfunctionsuand (0,1) formsv onX thatare smoothwith compactsupport.We cannow definetheKohnLaplacianonX,namely

2b :=∂^∗,f_b ∂b,

at leastonsmoothfunctionswith compactsupportonX;then (2bu|f)m= (u|2bf)m

forallfunctionsu,f onX thataresmoothwithcompactsupport,so wecanextend2b

to distributionsonX byduality.

Our goal is then to solve a specific equation involving 2b. First, let χ(z,t) be a smoothfunctionwithcompactsupportonX,ˆ sothatitssupportiscontainedinthelocal coordinate chartgiven bytheCRnormal coordinates(z,t), andthatit isidentically1 inaneighborhood ofp.Let

β0=χ(z, t) iz

|z|²−it∈ E( ˆρ⁻¹).

Then2bβ₀∈ E( ˆρ³).Furthermore,aswasshownin[5],thereexistsβ₁∈ E( ˆρ¹),suchthat if

β˜:=β0+β1, then

F :=2bβ˜∈ E( ˆρ⁴).

Actually,inwhatfollows,allwewilluseisthatF∈ E( ˆρ^3+δ) forall0< δ <1.Ourmain theorem cannow bestatedas follows:

Theorem 1.1. LetF be as defined above. Then there exists asmooth function uon X, such thatu∈ E( ˆρ^1+δ)forany 0< δ <1,and

2bu=F.

Bytakingβ :=β−u,wethenhave:

Corollary 1.2. There exists β ∈ E( ˆρ⁻¹) with β−β∈ E( ˆρ^1+δ) for any 0< δ < 1, such that

2bβ = 0.

Thisprovidesakeytoolintheproofofapositivemasstheoremin3-dimensionalCR geometry in the work of Cheng, Malchiodi and Yang [5], as was explained in the last subsection.

Some remarks are inorder. Thefirst is about numerology.Considerations of homo- geneity showsthat2btakesafunctioninE( ˆρ^k) toE( ˆρ^k+2).Thusthehomogeneityabove works outright;the onlysmall surpriseis thatwhileβ0 isinE( ˆρ⁻¹), 2bβ0 is inE( ˆρ³),

whichis 2orders betterthan expected.Butthatis areflection of thefactthatour β0

hasbeenchosensuchthat∂bβ0isalmost annihilatedby∂^∗,f_b .

Next,β1 aboveis anexplicitcorrection inE( ˆρ¹) suchthat2bβ1∈ E( ˆρ³) cancelsout themain contributionof2bβ₀∈ E( ˆρ³).ThisensuresthatF ∈ E( ˆρ^3+δ) forall0< δ <1, whichinturnguaranteesthattheβweconstructinCorollary 1.2isdeterminedexplicitly uptoE( ˆρ^1+δ) forall(in particular,forsome) 0< δ <1.Thelatterisimportantinthe proof of the positive mass theorem in [5], since any term in the expansion of β that is in E( ˆρ¹) would enter into the calculation of mass in (1.2). But for the purpose of solving2b inthe current paper, the correction term β1 is not essential; inparticular, if F0 :=2bβ0 ∈ E( ˆρ³), then our proof belowcarries over, and shows thatthere exists u₀∈ E( ˆρ¹) suchthat2bu₀=F₀.

Finally, in Corollary 1.2, note that we do not claim ∂bβ = 0. It is only 2bβ that vanishes,ascanbeshownbysaytheexamplewhenXˆ isthestandardCRsphereinC². 1.3. Ourstrategy

Aswe mentionedearlier, thedifficultyinestablishingthe abovetheorem isthatthe CRmanifold weare workingon, namelyX, isnon-compact; also, thenaturalmeasure on X, namely m = θ∧dθ, has infinite volume on X. Let L²(m) be the space of L² functionsonX with respectto m. Evenifweextend 2b to be aclosedlinearoperator on L²(m) →L²(m), ingeneral the extended2b may nothave closed range inL²(m).

Thustheclassicalmethodsofsolving2b failinoursituation.

WethusproceedbyintroducingaweightedL²space,andaweightedKohnLaplacian.

Let

m1:=G⁻²_p θ∧dθ.

WedefineL²(m₁) tobethespaceofL²functionsonX withrespecttotheinnerproduct (f|g)m1 :=

X

f g m1,

anddefineL²_(0,1)(m₁,θ) toˆ bethespaceofL²(0,1) formsonX withrespecttotheinner product

(α|β)_m

1,θˆ:=

X

α|βθˆm₁.

Weextendthetangential Cauchy–Riemannoperatorso that

Dom∂b,1:={u∈L²(m1): the distributional∂b ofuonX is inL²_(0,1)(m1,θ)ˆ},

and define

∂_b,1u:= the distributional∂_b ofuonX ifu∈Dom∂b,1.Then

∂b,1: Dom∂b,1⊂L²(m1)→L²_(0,1)(m1,θ),ˆ is adenselydefinedclosedlinearoperator.Let

∂^∗_b,1: Dom∂^∗_b,1⊂L²_(0,1)(m1,θ)ˆ →L²(m1)

be its adjoint. Let 2b,1 denote the Gaffney extension of the singular Kohn Laplacian given by

Dom2b,1=

s∈L²(m1):s∈Dom∂b,1, ∂b,1s∈Dom∂^∗_b,1 ,

and 2b,1s =∂^∗_b,1∂b,1s fors ∈Dom2b,1. By aresult of Gaffney,2b,1 is anon-negative self-adjoint operator(see[19, Prop. 3.1.2]).Weextend2b,1 todistributionsonX by

2b,1:D(X)→D(X),

(2b,1u|f)m1 = (u|2b,1f)m1, u∈D(X), f ∈C₀^∞(X).

This iswell-defined,sinceiff isatestfunctiononX,thensois2b,1f.Onecanshow 2b,1u=G²_p2bu, ∀u∈D(X).

Infact,byduality,itsufficestocheckthisforatestfunctionu∈C₀^∞(X).Ifuisassuch, then foranytestfunctionv∈C₀^∞(X),

(2b,1u|v)m1= (∂b,1u|∂b,1v)_m

1,θˆ

= (∂_bu|∂_bv)_m

1,θˆ

= (∂bu|G⁻²_p ∂bv)_m,θˆ

= (∂bu|∂bv)m,θ

= (2bu|v)m

= (G²_p2bu|v)m1, and thedesiredequality follows.

Thus solving 2b is essentiallythe sameas solvingfor 2b,1, and it isthe latter that forms theheartofourpaper.

The key here isthen three-fold, as is representedby the nextthree theorems. First we willshowthat

Theorem1.3. 2b,1:Dom2b,1⊂L²(m1)→L²(m1) has closedrange in L²(m1).

Oncethisisshown,wehavethefollowingL² decomposition:

2b,1N+ Π =I on L²(m1).

Here N : L²(m₁) → Dom2b,1 is the partial inverse of 2b,1 and Π : L²(m₁) → (Ran2b,1)^⊥ istheorthogonalprojectiononto(Ran2b,1)^⊥.Wenow need:

Theorem1.4. Forevery0< δ <2,ΠandN can beextended continuouslyto Π :E( ˆρ^−2+δ)→ E( ˆρ^−2+δ),

N :E( ˆρ^−2+δ)→ E( ˆρ^δ).

Itthenfollowsthat

2b,1N+ Π =I on E( ˆρ^−2+δ), forevery0< δ <2.Now,letF beasinTheorem 1.1. Put

f :=G²_pF.

Then f ∈ E( ˆρ^−1+δ), for every 0 < δ < 1. From Theorem 1.4, we know that Πf is well-defined;wewillshow that

Theorem1.5.

Πf = 0.

Itthenfollowsthatu:=N f satisfies

u∈ E( ˆρ^1+δ), 2b,1u=f =G²_pF.

Fromtherelationbetween2b,1and2b,weobtainthedesiredconclusioninTheorem 1.1.

1.4. Outline ofproofs

To prove the theorems in the previous subsection, we need to introduce two other KohnLaplacians,whichwedenoteby2ˆb and2b,asfollows.

First,2ˆb isthenaturalKohn Laplacianon( ˆX,T^1,0X,ˆ θ).ˆ There wehavethenatural measure

ˆ

m:= ˆθ∧dθ.ˆ

One can then define L²( ˆm) to be the space of L² functions on Xˆ with respect to the inner product

(f|g)_mˆ :=

Xˆ

f gm,ˆ

anddefine L²_(0,1)( ˆm,θ) toˆ bethespaceofL²(0,1) formsonXˆ withrespecttotheinner product

(α|β)_{m,ˆ θˆ}:=

Xˆ

α|βθˆm.ˆ

WeextendthetangentialCauchy–Riemannoperatorsothat

Dom∂ˆ_b:={u∈L²( ˆm): the distributional∂_b ofuon ˆX is inL²_(0,1)( ˆm,θ)ˆ}, and define

∂ˆ_bu:= the distributional∂_b ofuon ˆX ifu∈Domˆ∂b. Then

∂ˆb: Domˆ∂b ⊂L²( ˆm)→L²_(0,1)( ˆm,θ)ˆ is adenselydefinedclosedlinearoperator.Let

∂ˆ^∗_b: Domˆ∂^∗_b ⊂L²_(0,1)( ˆm,θ)ˆ →L²( ˆm)

be itsadjoint.Let2ˆb denotetheGaffneyextensionof theKohn Laplaciangiven by Dom ˆ2b={s∈L²( ˆm):s∈Dom∂ˆb, ∂ˆbs∈Domˆ∂^∗_b},

2ˆbs=∂ˆ^∗_b∂ˆbs fors∈Dom ˆ2b.

It is then a non-negative self-adjoint operator on L²( ˆm) (see e.g. [19, Prop. 3.1.2]).

The analysis of this 2ˆb is very well-understood; see work of Kohn [16,17], Boas–Shaw [2],Christ[6,7]andFefferman–Kohn[9]intheCRembeddablecase,andworkofKohn–

Rossi[18],Folland–Stein[10],Rothschild–Stein[26],Greiner–Stein[11],Nagel–Stein[22], Fefferman[8],BoutetdeMonvel–Sjöstrand[3],Nagel–Stein–Wainger[23],Nagel–Rosay–

Stein–Wainger [24,25] and Machedon [20,21] for some earlier work or related results.

On the other hand,it isnot verystraightforward to reducethe analysis of2b,1 to the analysis of this 2ˆb; we go through an intermediate Kohn Laplacian, which we denote by ˜2b.

Tointroduce2˜b,firstweneed toconstructaspecialCRfunctionψ onXˆ,suchthat

∂bψ= 0 on X,ˆ ψ= 0 onX,andnearp,wehave

ψ(z, t) = 2π(|z|²+it) + error,

wheretheerrorvanisheslike ρˆ⁴ nearp.(Theprecise constructionisgiven inSection4.) Onecanthendefine thefollowing volumeform onX:ˆ

˜

m:=G²_p|ψ|²θˆ∧dθ.ˆ

NotethatXˆ hasfinitevolumewithrespecttothisvolumeform,sinceG²_p|ψ|²isbounded near p. However, this volume form does not havea smooth density against m;ˆ this is oneof thebiggestsources of difficultiesinwhatwe dobelow.The keyturns outto be thefollowing:theasymptotics(1.5)weassumed ofG_p allowsusto obtainsomecrucial asymptoticsofthedensityofm˜ againstmˆ nearp.Thisinturnimpliesacrucialrelation betweenthe2b wewillintroduce,andthe2ˆb wedefinedabove(see(1.9)below).

NowletL²(m) be thespaceofL² functionsonXˆ withrespectto theinnerproduct (f|g)m˜ :=

Xˆ

f gm,˜

anddefineL²_(0,1)( ˜m,θ) toˆ be thespaceofL² (0,1) formsonXˆ withrespecttotheinner product

(α|β)_m,˜ θˆ:=

Xˆ

α|βθˆm.˜

Weextendthetangential Cauchy–Riemannoperatorso that

Dom∂b:={u∈L²(m): the distributional ∂b ofuonX is inL²_(0,1)(m, θ)ˆ}, anddefine

∂bu:= the distributional ∂b ofuonX ifu∈Dom∂_b.Then

∂_b: Dom∂_b⊂L²(m) →L²_(0,1)(m, θ)ˆ isadenselydefinedclosedlinearoperator.Let

˜∂^∗_b: Dom∂˜^∗_b ⊂L²_(0,1)( ˜m,θ)ˆ →L²( ˜m)

be itsadjoint.Let2˜b denotetheGaffneyextensionof theKohn Laplaciangiven by Dom ˜2b={s∈L²( ˜m):s∈Dom∂˜_b, ∂˜_bs∈Dom˜∂^∗_b},

2˜bs=∂˜^∗_b∂˜_bs fors∈Dom ˜2b.

It is then a non-negative self-adjoint operatoron L²( ˜m). The analysis of 2˜b is not as well-understood, since this Kohn Laplacian (in particular, the operator∂˜^∗_b) is defined with respect to a non-smooth measure m.˜ Nonetheless, it is this Kohn Laplacian that can be related to our operator of interest, namely 2b,1, in a simple manner. We will provethatsincem =|ψ|²m₁,

u∈Dom2b,1 if and only if _ψ^u ∈Dom2b, (1.6) 2b,1u=ψ2b(ψ⁻¹u), ∀u∈Dom2b,1. (1.7) Thus wecanunderstandthesolutionsof2b,1, oncewe understandthe solutionsof2˜b. Inorder tocarryoutthelatter,werelate2˜b to 2ˆb:wewillshow that

Dom ˜2b= Dom ˆ2b, (1.8)

and thereexistssomeg∈ E( ˆρ¹,T^0,1Xˆ) (possiblynon-smooth atp)suchthat

2bu= ˆ2bu+gˆ∂bu, ∀u∈Dom ˜2b. (1.9) (Here g∂ˆ_bu is thepointwise pairing of the (0,1) vector g with the (0,1) form ˆ∂_bu; see thediscussion inSection1.5fortheprecisemeaningofE( ˆρ¹,T^0,1Xˆ).)

WecannowoutlinetheproofsofTheorems 1.3,1.4 and 1.5.

First, from (1.6) and (1.7), it is clear that 2b,1 has closed range in L²(m1), if and only if 2b has closed range in L²( ˜m). On the other hand, one can check that

˜∂_b:Dom∂_b ⊂ L²( ˜m) → L²_(0,1)( ˜m,θ) isˆ the identical as an operator to ˆ∂_b:Dom∂ˆ_b ⊂ L²( ˆm) → L²_(0,1)( ˆm,θ).ˆ The latter is known to haveclosed range in L²( ˆm) by the CR embeddabilityofX;ˆ see[17](also[4]).Hencethesameholdsfortheformer,anditfollows that2˜b hasclosedrangeinL²( ˜m).ThisprovesTheorem 1.3.

Now from theaboveargument, we see thatnot only 2˜b has closed rangein L²( ˜m), butalso2ˆb hasclosedrangeinL²( ˆm).Thusthereexistpartial inverses

N˜:L²( ˜m)→Dom( ˜2b)⊂L²(m), and

Nˆ:L²( ˆm)→Dom( ˆ2b)⊂L²( ˆm)

to2˜b and 2ˆb respectively,so thatif

Π:˜ L²( ˜m)→L²( ˜m)

istheorthogonalprojectionontothekernel of2˜b inL²( ˜m),and Π:ˆ L²( ˆm)→L²( ˆm)

istheorthogonalprojectionontothekernel of2ˆb inL²( ˆm),then 2bN+Π = I onL²(m),

and

2ˆbNˆ+ ˆΠ =I onL²( ˆm).

Fromtherelation

m1=|ψ|⁻²m˜ and(1.7),itiseasytosee that

Π =ψΠψ ⁻¹ and N =ψN ψ ⁻¹, (1.10) at leastwhen appliedto functionsinL²(m1);thus to proveTheorem 1.4,it sufficesto proveinsteadthatΠ and N extendas continuousoperators

Π: E( ˆρ^−4+δ)→ E( ˆρ^−4+δ) (1.11) N:E( ˆρ^−4+δ)→ E( ˆρ^−2+δ) (1.12) forevery0< δ <2.Inordertodoso,werelateΠ to Π,ˆ andN toNˆ,sinceΠ andˆ Nˆ are muchbetterunderstood.Wewillshow, onL²(m), that

Π(I + ˆR) = ˆΠ (1.13)

N(I + ˆR) = (I−Π) ˆ N , (1.14) where

Rˆ:=g∂ˆbNˆ:L²(m) →L²(m)

isacontinuouslinearoperator;infact,theseidentitiesarealmost immediatefrom(1.8) and(1.9).Furthermore,onecanshowthatNˆ andΠ extendˆ ascontinuousoperators

Π:ˆ E( ˆρ^−4+δ)→ E( ˆρ^−4+δ) (1.15) N:ˆ E( ˆρ^−4+δ)→ E( ˆρ^−2+δ) (1.16) for every0< δ <2;thus ifonecanshowthat(I+ ˆR) isinvertibleonL²(m), and that the inverse extends to acontinuous operatorE( ˆρ^−4+δ) → E( ˆρ^−4+δ), then from (1.13), at least onecanconcludetheassertionabout Π in Theorem 1.4.It turns outthatitis unclear whetheror not the lattercan be done; so we chooseto proceed differently, by somebootstrap argument.Itisthisargument thatweexplainbelow.

First, so far Π and N are definedonly onL²(m). Since E( ˆρ^−4+δ) isnota subsetof L²(m) when 0< δ <2,we needto firstextendΠ and N to E( ˆρ^−4+δ),0< δ <2.This is donebyrewriting (1.13)and(1.14)as

Π = ˆ Π−Π ˆR, N = (I−Π) ˆ N−NR.ˆ (1.17) Note thatRˆ extendsto acontinuousoperator

R:ˆ E( ˆρ^−4+δ)→ E( ˆρ^−2+δ)⊂L²(m)

for every0< δ <2,sinceRˆ=gˆ∂bNˆ, andNˆ satisfiestheanalogous property.Thus the secondtermontherighthandsidesoftheequationsin(1.17)mapE( ˆρ^−4+δ) continuously to L²(m). It follows that the domains of definition of Π and N can be extended to E( ˆρ^−4+δ),0< δ <2.

To proceedfurther,let’swrite Πˆ^∗,m,Nˆ^∗,mand Rˆ^∗,m fortheadjointsofΠ,ˆ Nˆ and Rˆ withrespecttotheinnerproductofL²(m). SinceΠ and N areself-adjointoperatorson L²(m), wehave,by(1.13)and (1.14),that

(I+ ˆR^∗,m)Π = ˆΠ^∗,m (1.18) (I+ ˆR^∗,m)N = ˆN^∗,m(I−Π) (1.19) on L²(m). Now we need to understand some mapping properties of Πˆ^∗,m, Nˆ^∗,m and Rˆ^∗,m;to doso,wenotethat

Πˆ^∗,m=mˆ m

Πˆm ˆ m, Nˆ^∗,m=mˆ

mNˆm

ˆ m, Rˆ^∗,m= mˆ

mNˆ∂ˆ^∗_b(g^∗m ˆ m),

where m/ mˆ := G²_p|ψ|² is the density of m with respect to m,ˆ and similarly m/ˆ m :=

G⁻_p²|ψ|⁻². Here g^∗ is the (0,1) formdual to g. Note thatm/ m,ˆ m/ˆ m ∈ E( ˆρ⁰). Hence onecanshowthat

Πˆ^∗,m:E( ˆρ^−4+δ)→ E( ˆρ^−4+δ) (1.20) Nˆ^∗,m:E( ˆρ^−4+δ)→ E( ˆρ^−2+δ) (1.21) Rˆ^∗,m:E( ˆρ^−4+δ)→ E( ˆρ^−2+δ) (1.22) for every 0 < δ < 2; these are easy consequences of the analogous properties of Π,ˆ Nˆ and ˆR. It then follows that (1.18) and (1.19) continue to hold on E( ˆρ^−4+δ) for all 0< δ <2.

Now we returnto (1.18) and (1.19). The problem facingus there isthat we donot knowwhetherI+ ˆR^∗,misinvertibleonE( ˆρ^−4+δ);ifitis,thenwecanconclude,sayfrom (1.18),thatatleastΠ satisfies theconclusionofTheorem 1.4.Inordertogetaroundthis problem,we haveto proceeddifferently;thetrickhereistointroduceasuitablecut-off function,asfollows.

LetχbeasmoothfunctiononX,ˆ suchthatχisidentically1inaneighborhoodofp, andvanishesoutsideasmallneighborhood ofp.Then by(1.18)and(1.19),we have

(I+ ˆR^∗,mχ)Π = ˆΠ^∗,m−Rˆ^∗,m(1−χ)Π, (1.23) (I+ ˆR^∗,mχ)N = ˆN^∗,m(I−Π) −Rˆ^∗,m(1−χ)N (1.24) onE( ˆρ^−4+δ) for all0< δ <2. Theupshot hereis thefollowing: ifthe supportof χ is sufficientlysmall,then(I+ ˆR^∗,mχ) isinvertibleonL²(m), andextendstoalinearmap (I+ ˆR^∗,mχ)⁻¹:E( ˆρ^−4+δ)→ E( ˆρ^−4+δ) (1.25) forevery0< δ <4.Roughlyspeaking,thisispossible,because

I+ ˆR^∗,mχ=I+mˆ

mNˆˆ∂^∗_b(χg^∗m ˆ m), andbecause

χg^∗ ∈ E( ˆρ¹,Λ^0,1T^∗Xˆ)

has compact support in a sufficiently small neighborhood of p. (In particular, χg^∗ is small.)Furthermore,forany 0< δ <2,onecanshowthat

(1−χ)Π:E( ˆρ^−4+δ)→C₀^∞(X) (1.26) (1−χ)N:E( ˆρ^−4+δ)→C₀^∞(X) (1.27) where C₀^∞(X) is the space of all smooth functions on X that has compact support inX. These canbe used to controlthelast termon the right handside of (1.23) and (1.24).By(1.20) and(1.22),onethenconcludesthattherighthandsideof(1.23)maps E( ˆρ^−4+δ) intoitself for 0< δ <2;thus (1.25)shows that(1.11) holds as desired.This

inturncontrolsthefirsttermof(1.24);by(1.21),(1.22) and(1.27),oneconcludesthat the right hand side of (1.24) maps E( ˆρ^−4+δ) into E( ˆρ^−2+δ) for 0 < δ < 2. Finally, anotherapplicationof(1.25)showsthatN satisfies(1.12)asdesired.ThusTheorem 1.4 is established,modulo(1.25),(1.26)and(1.27).

It may help to reiterate here the reason for the introduction of the cut-off χ: that was introduced so that one can invert I + ˆR^∗,mχ. In fact, since the coefficient of g^∗ ∈ E( ˆρ¹,Λ^0,1T^∗Xˆ),χg^∗ hassufficientlysmall L^∞ norm, ifthesupportofχ ischosen sufficiently small. As aresult,onecould make Rˆ^∗,mχL²(m)→L²(m) ≤1/2, bycontrol- ling the support of χ. This allows one to invert I+ ˆR^∗,mχ on L²(m) via aNeumann series.WenoteinpassingthatitisonlybecausewehavetakentheadjointofRˆthatthe productχg^∗appearsintheexpressionforI+ ˆR^∗,mχ;thatisessentiallywhywewantto take theadjoints of (1.13)and (1.14). Onecanthen proceed to extend (I+ ˆR^∗,mχ)⁻¹ so that it satisfies (1.25); the precise detail is rather involved, and we leave this until Section3.

It then remains to prove(1.26) and (1.27). To do so, we need yet to introduceyet anotherKohn Laplacian,namely2ˆb,ε.Letη∈C₀^∞(R³) beanon-negativefunctionsuch that η(z,t) = 1 if ρ(z,ˆ t) ≤ 1/2, and η(z,t) = 0 if ρ(z,ˆ t) ≥ 1. For 0 < ε < 1, let ηε(z,t)=η(ε⁻¹z,ε⁻²t),and let

ˆ

m_ε:=η_εmˆ+ (1−η_ε)m.

Thisvolumeformhasasmoothdensityagainstm,ˆ andthevolumeofXˆ withrespectto this volumeform isfinite.SoifweextendtheCauchy–Riemannoperatorsuchthat

Domˆ∂b,ε :={u∈L²( ˆmε): the distributional∂b ofuis inL²_(0,1)( ˆmε,θ)ˆ}, and define

∂ˆb,εu:= the distributional ∂b ofu ifu∈Domˆ∂b,ε,then

∂ˆb,ε: Dom∂ˆb,ε⊂L²( ˆmε)→L²_(0,1)( ˆmε,θ)ˆ is adenselydefinedclosedlinearoperator.Let

∂ˆ^∗_b,ε: Dom∂ˆ^∗_b,ε⊂L²_(0,1)( ˆmε,θ)ˆ →L²( ˆmε)

be its adjoint.Let 2ˆb,ε be theGaffney extensionof theKohn Laplacian∂ˆ^∗_b,ε∂ˆ_b,ε. Then 2ˆb,ε isalmost aswell-behavedas 2ˆb.Inparticular,2ˆb,ε:Dom ˆ2b,ε⊂L²( ˆmε)→L²( ˆmε)

hasclosedrangeinL²( ˆmε),andifΠˆεdenotestheorthogonalprojectionofL²( ˆmε) onto thekernelof2ˆb,ε, andNˆε:L²( ˆmε)→L²( ˆmε) isthepartial inverseof2ˆb,ε,then

2ˆb,εNˆε+ ˆΠε=I.

Furthermore,

Dom ˜2b= Dom ˆ2b,ε,

andtherewill existsomegε∈ E( ˆρ¹,T^0,1Xˆ) (possiblynon-smoothnear p)suchthat 2bu= ˆ2b,εu+gεˆ∂bu, ∀u∈Dom ˜2b. (1.28) Theupshothereisthatgεwillbecompactlysupportedinthesupportofηε,whereasour previousg maynotbe compactlysupportednear p. Onecanrepeat theproofof (1.18) and(1.19), andshowthat

(I+ ˆR^∗_ε^,m)Π = ˆΠ_ε^∗,m (I+ ˆR^∗,_ε^m)N = ˆN_ε^∗,m(I−Π) onL²(m), where

Rˆ_ε:=g_ε∂ˆ_b,εNˆ_ε. Itfollowsthat

(1−χ)Π = (1 −χ) ˆΠ^∗,_ε^m−(1−χ) ˆR^∗,_ε^mΠ, (1.29) (1−χ)N = (1−χ) ˆN_ε^∗,m(I−Π) −(1−χ) ˆR^∗_ε^,mN . (1.30) Now

(1−χ) ˆR^∗_ε^,m= mˆε

m(1−χ) ˆN_εˆ∂^∗_b,εg^∗_ε m ˆ m_ε,

and if ε is chosen sufficiently small (so that the support of g_ε is disjoint from that of 1−χ), then(1−χ) ˆNεˆ∂^∗_b,εg^∗_ε is aninfinitely smoothingpseudodifferential operator,by pseudolocalityofNˆε.Hencethelasttermof(1.29),andalsothelasttermof(1.30),map E( ˆρ^−4+δ) intoC₀^∞(X).Sinceforevery0< ε<1 andevery0< δ <2,

Πˆ^∗,m_ε :E( ˆρ^−4+δ)→ E( ˆρ^−4+δ), and

Nˆ_ε^∗,m:E( ˆρ^−4+δ)→ E( ˆρ^−2+δ),