Sublaplacians on CR manifolds

Bull. Math. Soc. Sci. Math. RoumanieTome 52(100) No. 1, 2009, 3–32

Sublaplacians on CR manifolds

byElisabetta Barletta and Sorin Dragomir

Abstract

We study the sublaplacian ∆b on a strictly pseudoconvex CR manifoldendowed with a contact form. ∆b is approximated by a continuous family ofsecond order elliptic operators ∆ǫǫ>0. If ∆ǫǫ>0 is uniformly K-positivedefinite (in the sense of W.V. Petryshyn, [14]) then we produce generalizedsolutions to ∆bu = f .

Key Words: CR manifold, Webster metric, sub-Riemannian gradi-ent, sublaplacian.2000 Mathematics Subject Classification: Primary 32V20, Se-condary 35H20, 53C17.

1 Introduction

Any strictly pseudoconvex CR manifold M of CR dimension n carries a naturalformally selfadjoint second order differential operator ∆b associated to a choiceof contact form θ on M (the sublaplacian of (M, θ)). ∆b is similar to the Laplace-Beltrami operator in Riemannian geometry yet its symbol is given by

σ2(∆b)ωv =[‖ω‖2 − ω(Tx)2

]v

hence ellipticity of ∆b fails in the characteristic direction T of dθ. As it turns out∆b is a degenerate elliptic operator (in the sense of J.M. Bony, [4]) i.e. locally

∆bu = −2n+1∑

i,j=1

∂

∂xi

(aij ∂u

∂xj

)+

2n+1∑

j=1

cj ∂u

∂xj

where aij is but positive semi-definite and ∆b =∑2n

a=1 X2a + Y for some smooth

real vector fields Xa , Y : 1 ≤ a ≤ 2n generating a Lie algebra L(X1, · · · ,X2n, Y )of rank 2n. Also ∆b is subelliptic of order ǫ = 1/2 i.e. for each point x ∈ M thereis an open neighborhood U ⊆ M and a constant C > 0 such that

‖u‖21/2 ≤ C

[(∆bu, u)L2(M) + ‖u‖2

L2(M)

]

4 Elisabetta Barletta and Sorin Dragomir

for any u ∈ C∞0 (U). Here ‖ · ‖ǫ is the Sobolev norm of order ǫ. Consequently

(by a classical result of L. Hormander, [10]) ∆b is hypoelliptic i.e. if f is smooththen any distribution solution u to ∆bu = f is smooth as well. More generally∆b satisfies the a priori estimates

‖u‖2s+1 ≤ Cs

(‖∆bu‖2

s + ‖u‖2L2(M)

), u ∈ C∞

0 (U), s ≥ 0.

Recently ∆b was seen to appear as the principal part in nonlinear subellipticsystems of variational origin arising in pseudohermitian geometry (cf. E. Bar-letta & S. Dragomir & H. Urakawa, [1], E. Barletta, [2]-[3]) and extending theharmonic maps theory (cf. e.g. [8]) from the elliptic to the at least hypoellipticsetting. Similar PDE systems were considered within the theory of Hormandersystems of vector fields (cf. J. Jost & C-J. Xu, [11], Z.R. Zhou, [19]-[20]) and inparticular on Carnot groups (cf. C. Wang, [16], L. Capogna & N. Garofalo, [5])where the methods of calculus of variations predominate (while [1]-[3] are devotedmainly to the investigation of differential geometric properties). The theory ofHormander systems of vector fields may be looked at as the local point of view inCR and pseudohermitian geometry (and then nondegeneracy of a CR manifoldis very close to Hormander’s rank condition on the Lie algebra generated by thegiven system of vector fields) yet strictly speaking the two theories interlap onlypartially. Indeed the maximally complex, or Levi, distribution of a nondegen-erate CR manifold admits local frames which are Hormander systems of vectorfields defined on open subsets of R

2n+1 while the general theory of Hormandersystems is built on domains in R

N with arbitrary N and allows for vector fieldswhich aren’t necessarily independent at each point or even smooth. The purposeof this paper is to start a systematic application of the methods of functionalanalysis and calculus of variations to sublaplacians on strictly pseudoconvex CRmanifolds. The material is organized as follows. In Sections 2 to 4 we introduceappropriate (Sobolev type) function spaces and develop a calculus with L2 func-tions on CR manifolds admitting weak horizontal, or sub-Riemannian, gradients.In Section 5 we build an example of sublaplacian (on the lowest dimensionalHeisenberg group) which is strictly positive yet not positive definite. In Section6 we consider regularity fields for sublaplacians and use general results in func-tional analysis to demonstrate that ∆b : D(∆b) ⊂ L2(Ω) → L2(Ω) possesses aselfadjoint extension. Sections 7 and 8 provide a sub-Riemannian approach tothe study of sublaplacians on CR manifolds i.e. ∆b is approximated by the fam-ily ∆ǫǫ>0 of Laplace-Beltrami operators associated to a family of Riemannianmetrics gǫǫ>0 contracting the Levi form of (M, θ) (very much the way Horman-der’s operator H is approximated by the second order elliptic operators Hǫǫ>0

in [11], p. 4635). In particular we may exploit certain ideas of W.V. Petryshyn,[13]-[14], to produce generalized solutions to ∆bu = f as limits of minimizingsequences for the functional

F(u) = (∆bu , Ku)L2(Ω) − (f,Ku)L2(Ω) − (Ku, f)L2(Ω)

Sublaplacians on CR manifolds 5

where K : D(K) ⊂ L2(Ω) → L2(Ω) is a preclosed operator such that D(K) ⊇D(∆b)∩D(T ∗T ). As to the needed material on CR and pseudohermitian geometrywe adopt the notations and conventions in [7]. The beautiful book by G. Dinca,[6], was a source of inspiration for the themes treated in this paper.

Contents

1 Introduction 3

2 Weak horizontal gradients 5

3 Local calculations 8

4 Sublaplacians and conformal transformations 11

5 A strictly positive non positive definite sublaplacian 14

6 Regular points and the defect index 17

7 Sublaplacians and sub-Riemannian geometry 18

8 Generalized solutions to ∆bu = f . 24

2 Weak horizontal gradients

Let π : E → M be a Hermitian vector bundle, with the Hermitian bundle metrich, over a strictly pseudoconvex CR manifold M of CR dimension n. Let θ bea contact form on M and dv = θ ∧ (dθ)n the corresponding volume form. LetΩ ⊆ M be a domain with C2 boundary. Let L2(EΩ) denote the space of allL2 sections in EΩ = π−1(Ω) (the portion of E over Ω) that is s ∈ L2(EΩ) ifh(s, s) ∈ L1(Ω) i.e.

∫Ω

h(s, s) d v < ∞. If Ω × C is the trivial vector bundleover Ω we write simply L2(Ω) = L2(Ω × C). Let H = H(M) be the maximallycomplex distribution on M and Gθ the Levi form (cf. e.g. (1.16) in [7], p. 6). Ifu ∈ C1(Ω) and X ∈ Γ∞

0 (Ω,H ⊗ C) then (by Green’s lemma)

∫

Ω

Gθ(∇Hu,X) d v =

∫

Ω

X(u) d v =

=

∫

∂Ω

u gθ(ν,X) d a −∫

Ω

u div(X) d v = −∫

Ω

u div(X) d v.

Here ∇Hu = ΠH∇u is the horizontal gradient of u that is ΠH : T (M) → H is theprojection associated to the direct sum decomposition T (M) = H⊕RT and ∇u isthe ordinary gradient with respect to the Webster metric gθ (cf. Definition 1.10 in[7], p. 9) i.e. gθ(∇u, Y ) = Y (u) for any Y ∈ X

∞(M). Also ν is the outward unitnormal on ∂Ω and T is the characteristic direction of dθ. The divergence of X


is computed with respect to the volume form d v i.e. LX(d v) = div(X) d v. Thecalculation above suggests the following natural definition. A function u ∈ L2(Ω)is weakly differentiable along H if there is Yu ∈ Γ(HΩ ⊗ C) such that ‖Yu‖ =Gθ(Yu, Y u)1/2 ∈ L1

loc(Ω) and

∫

Ω

Gθ(Yu,X) d v = −∫

Ω

u div(X) d v

for any X ∈ Γ∞0 (HΩ ⊗ C). Such Yu is unique up to a set of measure zero and is

denoted again by Yu = ∇Hu (the weak horizontal gradient of u). Let D(∇H) =W 1,2

H (Ω) consist of all u ∈ L2(Ω) such that u is weakly differentiable along Hand ∇Hu ∈ L2(HΩ ⊗ C). Then we may regard the weak horizontal gradient asa linear operator ∇H : D(∇H) ⊂ L2(Ω) → L2(HΩ ⊗ C). As C∞

0 (Ω) ⊆ D(∇H) itfollows that D(∇H) is a dense subspace of L2(Ω). Let

(∇H

)∗: D

[(∇H

)∗] ⊂ L2(HΩ ⊗ C) → L2(Ω)

be the adjoint of ∇H i.e. i) D[(∇H

)∗]consists of all X ∈ L2(HΩ ⊗C) such that

∫

Ω

Gθ(∇Hu,X) d v =

∫

Ω

u X∗ d v

for some X∗ ∈ L2(Ω) and any u ∈ D(∇H), and ii)(∇H

)∗X = X∗. Then

Γ∞0 (HΩ⊗C) ⊆ D

[(∇H

)∗]and the restriction of

(∇H

)∗to Γ∞

0 (HΩ⊗C) is −div.

In particular(∇H

)∗is densely defined. The sublaplacian of (Ω, θ) is the linear

operator

∆b : D(∆b) ⊂ L2(Ω) → L2(Ω)

given by

D(∆b) = u ∈ D(∇H) : ∇Hu ∈ D[(∇H

)∗], (1)

∆b =(∇H

)∗ ∇H . (2)

Note that

(∆bu, u)L2(Ω) = ‖∇Hu‖2L2(HΩ⊗C) ≥ 0

for any u ∈ D(∆b) i.e. the sublaplacian (1)-(2) is positive. In particular (byProposition 1.12 in [6], p. 54) the sublaplacian (1)-(2) is symmetric i.e. i) itsdomain is a dense subspace of L2(Ω) and ii) ∆b ⊂ ∆∗

b [equivalently i) D(∆b) isdense and ii) (∆bu, v)L2(Ω) = (u,∆bv)L2(Ω) for any u, v ∈ D(∆b)]. As customary

the notation A ⊂ B means that D(A) ⊆ D(B) and the restriction of B to D(A)is A.

The restriction (∆b)0 of ∆b to C∞0 (Ω) is referred to as the Lagrange sublapla-

cian of (M, θ).


Proposition 1. The Lagrange sublaplacian is a strictly positive operator i.e.((∆b)0 u, u)L2(Ω) ≥ 0 and ((∆b)0 u, u)L2(Ω) = 0 if and only if u = 0 in Ω.

Proof: Let u ∈ C∞0 (Ω) such that ∇Hu = 0. Let ∇ be the Tanaka-Webster

connection of (M, θ) (cf. e.g. Definition 1.25 in [7], p. 26). Then (by the purityproperty of the torsion of ∇, cf. (1.37) in [7], p. 25)

2iGθ(Z,Z)T (u) =(∇ZZ −∇ZZ − [Z,Z]

)u = 0

for any Z ∈ T1,0(M)\0 hence T (u) = 0 in Ω so that u is constant. Yet u(x) → 0as x → ∂Ω hence u = 0.

Proposition 2. ∇H : D(∇H) ⊂ L2(Ω) → L2(HΩ) is preclosed. In particular∇H admits a closed minimal extension.

Proof: Let (uk)k≥1 be a sequence in D(∇H) converging strongly to 0 ∈ L2(Ω).Let us assume that ∇Huk → Y as k → ∞ for some Y ∈ L2(HΩ ⊗ C). Then forany X ∈ Γ∞

0 (HΩ ⊗ C)

(∇Huk , X

)L2(HΩ⊗C)

= −∫

Ω

uk div(X) d v

hence as k → ∞ one has (Y,X)L2(HΩ⊗C) = 0 so that Y = 0.

As ∇H is preclosed one may consider (cf. Proposition 1.16 in [6], p. 58) itsclosed minimal extension

∇H : D(∇H) ⊂ L2(Ω) → L2(HΩ ⊗ C)

i.e. D(∇H) consists of all u ∈ L2(Ω) such that there is a sequence (uk)k≥1 ⊂D(∇H) with uk → u as k → ∞ and

(∇Huk

)k≥1

is strongly convergent in L2(HΩ⊗C) and ∇Hu = limk→∞ ∇Huk. Then i) ∇H is closed and ii) if A is a closed

extension of ∇H then ∇H ⊂ A. In particular (by Proposition 1.17 in [6], p. 59)(∇H

)∗

=(∇H

)∗.

Proposition 3. W 1,2H (Ω) is a Hilbert space with the inner product

(f, g)1,2 =

∫

Ω

f g d v +

∫

Ω

Gθ(∇Hf , ∇Hg) d v (3)

for any f, g ∈ W 1,2H (Ω). In particular W 1,2

H (Ω) is reflexive.


Proof: Let ‖u‖1,2 = (u, u)1/21,2 be the norm associated to the inner product (3).

Let (uk)k≥1 ⊂ W 1,2H (Ω) be a Cauchy sequence. Then (uk)k≥1 is Cauchy in L2(Ω)

hence uk → u as k → ∞ for some u ∈ L2(Ω). Also (∇Huk)k≥1 is Cauchy inL2(HΩ ⊗ C) hence ∇Huk → Y as k → ∞ for some Y ∈ L2(HΩ ⊗ C). Moreover

−∫

Ω

u div(X) d v = − limk→∞

∫

Ω

uk div(X) d v =

= limk→∞

∫

Ω

Gθ(∇Huk , X) d v =

∫

Ω

Gθ(Y,X) d v

for any X ∈ Γ∞0 (HΩ ⊗ C) hence u ∈ W 1,2

H (Ω).

3 Local calculations

Let Hn = Cn × R ≈ R

2n+1 be the Heisenberg group with coordinates (z, t) =(z1, · · · , zn, t), zα = xα + iyα, 1 ≤ α ≤ n, i =

√−1, and let

Zα =∂

∂zα − izα ∂

∂t, 1 ≤ α ≤ n,

be the Lewy operators (cf. [7], p. 11). Then Hn is a strictly pseudoconvex CRmanifold with the CR structure T0,1(Hn) spanned by Zα : 1 ≤ α ≤ n. The realdifferential 1-form

θ0 = dt + i

n∑

α=1

(zα dzα − zαdzα)

is a contact form on Hn such that the corresponding Levi form Gθ0is positive

definite. Here zα = zα. Let us consider the left invariant frame of H(Hn) givenby

Xα =

√2

2(Zα + Zα) , Xα+n =

i√

2

2(Zα − Zα) ,

so that gθ0(Xa,Xb) = δab. Here Zα = Zα. Let X∗

a be the formal adjoint ofXa = bi

a ∂/∂xi (with xi : 1 ≤ i ≤ 2n + 1 = xα, yα, t : 1 ≤ α ≤ n) i.e.

X∗af = − ∂

∂xi

(biaf

), f ∈ C1

0 (Hn),

and let H be the Hormander operator

Hu =

2n∑

a=1

X∗aXau = − ∂

∂xi

(aij ∂u

∂xj

).

Here aij =∑2n

a=1 biabj

a and

[bia] =

√2

2δβα 0

√2 yα

0

√2

2δβα −

√2 xα


so that

H = −n∑

α=1

(∂2u

∂x2α

+∂2u

∂y2α

)+ 2

∂

∂t

(xα ∂u

∂yα− yα ∂u

∂xα

)− 2|z|2 ∂2u

∂t2

where |z|2 = zαzα. Note that X∗a = −Xa hence H = −∑2n

a=1 X2a so that H is a

degenerate elliptic operator (in the sense of [4]).

Remark 1. If M = Hn and θ = θ0 then ∆b = H.

Let Ω ⊆ Hn be an open subset. We shall need the (Sobolev type) functionspaces

W k,pX (Ω) = f ∈ Lp(Ω) : XJf ∈ Lp(Ω), |J | ≤ k,

J = (a1, · · · , as), 1 ≤ aℓ ≤ 2n, |J | = s, XJf = Xa1· · ·Xas

f.

By a result of C-J. Xu (cf. Theorem 1 in [18], p. 149) if 1 ≤ p < ∞ then W p,kX (Ω)

is a separable Banach space with the norm

‖f‖W k,p

X(Ω) =

∑

|J|≤k

‖XJf‖pLp(Ω)

1/p

.

If 1 < p < ∞ then W k,pX (Ω) is also reflexive. If p = 2 then W k

X(Ω) = W 2,kX (Ω) is

a separable Hilbert space. Let W kX,0(Ω) be the completion of C∞

0 (Ω) in W kX(Ω).

Note that Xa : 1 ≤ a ≤ 2n is a system of vector fields on Hn satisfyingHormander’s condition (i.e. the vector fields Xa and their commutators up tolength r = 2 span the tangent space Tx(Hn) at any point x ∈ Hn). ConsequentlyW k

X,0(Ω) ⊂ Hk/2(Ω) where Hj(Ω) are the ordinary Sobolev spaces (cf. also (3.2)in [11], p. 4635). Similar considerations hold locally on an arbitrary strictlypseudoconvex CR manifold. Let Xa : 1 ≤ a ≤ 2n be a local Gθ-orthonormalframe in H defined on the open subset U ⊆ M . If u ∈ C1(U) then (by Green’slemma) ∫

U

Xa(u)ϕ d v =

∫

U

Xa(uϕ) − uXa(ϕ) d v =

= −∫

U

u ϕ div(Xa) + Xa(ϕ) d v = −∫

U

u div(ϕXa) d v

for any ϕ ∈ C∞0 (U). We may then adopt the following definition. Let Ω ⊆ M be

a domain such that Ω∩U 6= ∅. A function u ∈ L2(Ω∩U) is weakly differentiablein the direction Xa if there is va ∈ L1

loc(Ω ∩ U) such that

∫

Ω∩U

va ϕ d v = −∫

Ω∩U

u div(ϕXa) d v


for any ϕ ∈ C∞0 (Ω ∩ U). Such va is unique up to a set of measure zero and

denoted by va = Xa(u). Let D(Xa) consist of all u ∈ L2(Ω ∩ U) such that u isweakly differentiable in the direction Xa and Xa(u) ∈ L2(Ω ∩ U). We set

W 1,2X (Ω ∩ U) =

2n⋂

a=1

D(Xa).

We wish to investigate the relationship among the spaces W 1,2H and W 1,2

X . Let

u ∈ W 1,2H (Ω ∩ U) and let us set va = Gθ(∇Hu , Xa). Then

∫

Ω∩U

|va|2 d v ≤∫

Ω∩U

‖∇Hu‖2‖Xa‖2 d v = ‖∇Hu‖2L2(Ω∩U)

so that va ∈ L2(Ω ∩ U). Next

∫

Ω∩U

va ϕ d v =

∫

Ω∩U

Gθ(∇Hu , ϕXa) d v = −∫

Ω∩U

u div(ϕXa) d v

for any ϕ ∈ C∞0 (Ω ∩ U) hence u ∈ D(Xa) for any 1 ≤ a ≤ 2n. Viceversa,

let u ∈ W 1,2X (Ω ∩ U) and let us set Y =

∑2na=1 Xa(u)Xa. For arbitrary X ∈

Γ∞0 (HΩ∩U ⊗ C) let ϕa = Gθ(Xa,X) ∈ C∞

0 (Ω ∩ U). Then

∫

Ω∩U

Gθ(Y,X) d v =∑

a

∫

Ω∩U

Xa(u)ϕa d v =

= −∑

a

∫

Ω∩U

u div(ϕaXa) d v = −∫

Ω∩U

u div(X) d v

hence u ∈ W 1,2H (Ω ∩ U) and ∇Hu =

∑a Xa(u)Xa. We may state

Proposition 4. The restriction to Ω ∩ U induces a natural continuous mapW 1,2

H (Ω) → W 1,2H (Ω ∩ U) and W 1,2

H (Ω ∩ U) = W 1,2X (Ω ∩ U).

Proof: Only the first statement needs to be checked. The restriction to Ω ∩ Uinduces natural maps L2(Ω) → L2(Ω ∩ U) and L2(HΩ ⊗ C) → L2(HΩ∩U ⊗ C).Let u ∈ W 1,2

H (Ω) and X ∈ Γ∞0 (HΩ∩U ⊗ C). Let us set X(x) = X(x) for any

x ∈ Ω ∩ U and X(x) = 0 for any x ∈ Ω \ U so that X ∈ Γ∞0 (HΩ ⊗ C). Let

Y ∈ L2(HΩ∩U ⊗ C) be the restriction of ∇Hu to Ω ∩ U . Then

∫

Ω∩U

Gθ(Y,X) d v =

∫

Ω

Gθ(∇Hu , X) d v =

= −∫

Ω

u div(X) d v = −∫

Ω∩U

u div(X) d v

hence u|Ω∩U ∈ W 1,2H (Ω ∩ U) and ∇H (u|Ω∩U

)= Y .


4 Sublaplacians and conformal transformations

Let M be a strictly pseudoconvex CR manifold of CR dimension n. Let θ =efθ, with f ∈ C∞(Ω) real valued, be another contact form on M and ∆b thecorresponding sublaplacian.

Proposition 5. Let Ω ⊆ M be a domain and f ∈ C∞(Ω) a real valued function.Let us assume that either Ω is bounded or f ∈ L∞(Ω) and ∇Hf ∈ L∞(HΩ ⊗ C).Then D(∆b) = D(∆b) and

ef ∆bu = ∆bu − n(∇Hu)f (4)

for any u ∈ D(∆b).

We need the following

Lemma 1. Let Xa : 1 ≤ a ≤ 2n be a local Gθ-orthonormal frame of H, definedon the open subset U ⊆ M , such that Xα+n = JXα for any 1 ≤ α ≤ n. Then

ef T = T +1

2

n∑

α=1

fα+nXα − fαXα+n (5)

on U ∩ Ω, where fa = Xa(f). In particular efJT = 12 ∇Hf and

ΠH = ΠH +1

2θ ⊗ J∇Hf (6)

where ΠH : T (M) → H is the projection associated to the decomposition T (M) =

H ⊕ RT . Also T is the characteristic direction of dθ and J : H → H is thecomplex structure along H given by J(Z + Z) = i(Z − Z) for any Z ∈ T1,0(M).Moreover

div(X) = div(X) + (n + 1)X(f) (7)

for any X ∈ X∞(Ω).

Proof: The identity (5) follows from θ(T ) = 1 and T ⌋ dθ = 0 while (7) is a

consequence of d v = θ ∧ (dθ)n = e(n+1)f d v.Let u ∈ D(∇H) = W 1,2

H (Ω). Then (by (7) in Lemma 1)

∫

Ω

Gθ(∇Hu , X) d v = −∫

Ω

u div(X) d v =

= −∫

Ω

u[div(X) − (n + 1)X(f)

]e−(n+1)f d v =

= −∫

Ω

u div(e−(n+1)fX

)d v


for any X ∈ Γ∞0 (HΩ ⊗ C). Therefore u ∈ D(∇H) and

∇Hu = e−f∇Hu. (8)

Moreover it is elementary that ϕu ∈ D(∇H) for any ϕ ∈ C∞(Ω) ∩ L∞(Ω) with∇Hϕ ∈ L∞(HΩ ⊗ C) and any u ∈ D(∇H). Indeed

−∫

Ω

ϕu div(X) d v = −∫

Ω

u[div(ϕX) − X(ϕ)

]d v =

=

∫

Ω

Gθ(∇Hu, ϕX) d v +

∫

Ω

uGθ(∇Hϕ,X) d v

for any X ∈ Γ∞0 (HΩ⊗C). In particular ∇H(ϕu) = ϕ∇Hu+u∇Hϕ ∈ L2(HΩ⊗C).

Let u ∈ D(∆b). Then u ∈ D(∇H) and ∇Hu ∈ D[(∇H)∗] i.e. there is w ∈L2(Ω) (here w = ∆bu) such that

∫

Ω

ψw d v =

∫

Ω

Gθ(∇Hψ , ∇Hu) d v =

∫

Ω

e−nfGθ(∇Hψ , ∇Hu) d v

for any ψ ∈ D(∇H). Let us set ϕ = e−nfψ ∈ D(∇H). Then

∫

Ω

ϕ[e−fw − nGθ(∇Hf , ∇Hu)

]−d v =

∫

Ω

Gθ(∇Hϕ , ∇Hu) d v

so that ∇Hu ∈ D[(∇H)∗] and

(∇H)∗∇Hu = e−fw − n(∇Hu)f

which (by (8)) yields (4). Proposition 5 is proved.

Remark 2. It may be easily shown that (4) holds for any u ∈ C2(Ω) as well.Indeed let x0 ∈ Ω and let Xa : 1 ≤ a ≤ 2n be a local Gθ-orthonormal frameof H defined on an open neighborhood U ⊆ M of x0. Then1 (cf. (2.6) in [7], p.112)

∆bu = −2n∑

a=1

Xa(Xau) − (∇XaXa)u

on Ω ∩ U . We set Xa = e−f/2Xa for any 1 ≤ a ≤ 2n. Then

Xa(Xau) = e−fXa(Xau) − 1

2Xa(f)Xa(u),

(∇XaXa)u = e−f(∇Xa

Xa)u − 1

2Xa(f)Xa(u),

1The sublaplacian in this work and in reference [7] differ by a sign.


where ∇ is the Tanaka-Webster connection of (M, θ). Then

ef ∆bu = ∆bu + Lfu (9)

where Lf is the first order differential operator given by

Lf =

2n∑

a=1

∇XaXa −∇Xa

Xa.

Since ∇gθ = 0 one has

X(gθ(Y,Z)) + Y (gθ(Z,X)) − Z(gθ(X,Y )) = 2gθ(∇XY,Z)+

+gθ(Z, [Y,X]) − gθ(Y, [Z,X]) − gθ(X, [Z, Y ])+

+gθ(Z, T∇(Y,X)) − gθ(Y, T∇(Z,X)) − gθ(X,T∇(Z, Y ))

for any X,Y,Z ∈ X∞(M), where T∇ is the torsion tensor field of ∇. Recall that

T∇(X,Y ) = (dθ)(X,Y )T for any X,Y ∈ Γ∞(H) (cf. Lemma 1.3 in [7], p. 37).Let X,Z ∈ Γ∞(H) and Y = X. Then

2X(gθ(X,Z)) − Z(‖X‖2) = 2gθ(∇XX,Z) − 2gθ(X, [Z,X]).

Consequently

gθ(∇XaXa,Xb) = gθ(Xa, [Xb,Xa]), 1 ≤ a, b ≤ 2n. (10)

Next (as H is parallel with respect to ∇)

2efgθ(∇XaXa,Xb) = 2δabXa(ef ) − Xb(e

f ) + 2gθ(Xa, [Xb,Xa])

or

gθ(∇XaXa,Xb) = δabXa(f) − 1

2Xb(f) + gθ(Xa, ΠH [Xb,Xa]). (11)

Subtracting in (10) from (11)

gθ(Lf , Xb) = −(n − 1)Xb(f) +

2n∑

a=1

θ([Xa,Xb])gθ(Xa, T ). (12)

By (5) in Lemma 1

∑

a

θ([Xa,Xb])gθ(Xa, T ) = 2efgθ(JXb, T ) = −fb

hence (12) becomes gθ(Lf , Xb) = −nfb so that Lf = −n∇Hf i.e. (9) implies(4) for any smooth u.


5 A strictly positive non positive definite sublaplacian

Let H1 = C × R be the lowest dimensional Heisenberg group endowed with thecanonical contact form θ = dt + i (zdz − zdz). Let ∆b be the correspondingsublaplacian. Let B(0, R) = x ∈ H1 : |x| ≤ R where |x| = (|z|4 + t2)1/4 is theHeisenberg norm of x = (z, t) ∈ H1. Let Ω = H1 \ B(0, 1). Let M consist of allu ∈ L2(Ω) such that i) u ∈ C2(Ω), ii) u|Σ(0,1) = 0, iii)

∫Σ(0,R)

u gθ(∇Hu , ν) d a →0 as R → ∞, and iv) ∆bu ∈ L2(Ω), where Σ(0, R) = x ∈ H1 : |x| = R. Also νis the outward unit normal field (with respect to the Webster metric of (H1, θ))on the boundary of B(0, R). Let ΩR = x ∈ H1 : 1 < |x| < R with R > 1. Forany u ∈ M (by Green’s lemma)

∫

ΩR

(∆bu)u d v = −∫

ΩR

div(∇Hu)u d v =

= −∫

ΩR

div(u∇Hu

)− (∇Hu)(u) d v =

= −∫

Σ(0,R)

u gθ(∇Hu , ν) d a +

∫

ΩR

gθ(∇Hu, ∇Hu) d v.

Let R → ∞. By (iii) above

(∆bu , u)L2(Ω) = ‖∇Hu‖2L2(Ω) ≥ 0, u ∈ M,

with equality if and only if u = constant on Ω (as H1 is a nondegenerate CRmanifold) and then u = 0 (by (ii) above). Hence ∆b : M ⊂ L2(Ω) → L2(Ω) is astrictly positive operator. Next let

uα(x) = (|x| − 1)|x|4e−α |x| , α > 0.

Clearly uα ∈ C2(Ω) and uα|∂Ω = 0. To compute the limit in (iii) we need to studythe geometry of a Heisenberg sphere Σ(0, R) in the ambient space (H1, gθ). Let usconsider the half spaces H

+1 = (z, t) ∈ H1 : t > 0 and H

−1 = (z, t) ∈ H1 : t < 0

and set U± = Σ(0, R)∩H±1 . Let us consider the parametrizations ψ± : D(0, R) →

U± given by ψ±(z) = (z,±√

R4 − |z|4) where D(0, R) = z ∈ C : |z| < R. Let

X =∂

∂x+ 2yT, Y =

∂

∂y− 2xT, (13)

be the left invariant vector fields spanning the Levi distribution (a Hormandersystem on R

3) where z = x + iy and T = ∂/∂t. Then

X = X − 2

(y +

x|z|2t

)T, Y = Y + 2

(x − y|z|2

t

)T,

span the portion of T (Σ(0, R)) over U = U+ ∪ U−. As

gθ(X,X) = gθ(Y, Y ) = 2, gθ(X,Y ) = 0,


gθ(X,T ) = gθ(Y, T ) = 0, gθ(T, T ) = 1,

a unit normal field ν on U (i.e. gθ(X, ν) = gθ(Y , ν) = 0 and ‖ν‖ = 1) is given by

ν = c

T +

(x|z|2

t+ y

)X +

(y|z|2

t− x

)Y

,

c ∈

±t√t2 + 2|x|4|z|2

.

The components gij of the first fundamental form of ι : U+ → H1 are given by

gθ

(∂

∂x,

∂

∂x

)= 2(1 + 2y2), gθ

(∂

∂y,

∂

∂y

)= 2(1 + 2x2),

gθ

(∂

∂x,

∂

∂y

)= −4xy,

hence det[gij ] = 4(1+2|z|2) and then d a = 2√

1 + 2|z|2 dx∧dy. We set ϕ(z, t) =(|z|4 + t2

)1/4for simplicity. Then

X(ϕ) = ϕ−3(x|z|2 + yt

), Y (ϕ) = ϕ−3

(y|z|2 − xt

),

X(uα) = −ϕ3e−αϕQα(ϕ)X(ϕ), Y (uα) = −ϕ3e−αϕQα(ϕ)Y (ϕ),

where Qα(R) = αR2 − (α + 5)R + 4. Consequently

‖∇Huα‖2 =1

2

[X(uα)2 + Y (uα)2

]=

1

2|z|2ϕ4e−2αϕQα(ϕ)2.

It follows that ‖∇Huα‖2 ψ+ = 12 |z|2R4e−2αRQα(R)2 so that for any R > 1

∣∣∣∣∣

∫

U+

uα gθ(∇Huα , ν) d a

∣∣∣∣∣ ≤∫

U+

|uα|‖∇Huα‖ d a =

= 2

∫

D(0,R)

(uα ψ+)(‖∇Huα‖ ψ+

) √1 + 2|z|2 dx ∧ dy =

=√

2(R − 1)R6e−2αR |Qα(R)|∫

D(0,R)

|z|√

1 + 2|z|2 dx ∧ dy.

Hence there is a function fα(R) such that 0 ≤ fα(R) ≤ CR13 for some constantC > 0 and then

∣∣∣∣∣

∫

U+

uα gθ(∇Huα , ν) d a

∣∣∣∣∣ ≤ fα(R)e−2αR → 0


as R → ∞. By a similar calculation limR→∞

∫U

−

uα gθ(∇Huα , ν) d a = 0 as well.

Finally ‖∇Hϕ‖2 = 12ϕ−2|z|2 and ∆bϕ = − 3

2ϕ−3|z|2 hence

∆buα = −1

2|z|2e−αϕAα(ϕ) ∈ L2(Ω)

for some2 polynomial Aα so that ∆buα ∈ L2(Ω). Summing up the informationgot so far uα ∈ M for any α > 0. At this point we may show that ∆b : M ⊂L2(Ω) → L2(Ω) is not positive definite. Note first that the area of the Heisenbergsphere Σ(0, R) is ∫

Σ(0,R)

d a =4π

3

[(1 + 2R2

)3/2 − 1].

Moreover

‖uα‖2L2(Ω) =

∫

Ω

(|x| − 1)2|x|8e−2α|x| d v(x) =

=

∫ ∞

1

dR

∫

|x|=R

(R − 1)2R8e−2αR d a(x) ≥

≥ 4π

3

∫ ∞

1

R8(R − 1)2(R3 − 1)e−2αR dR =

=4π

3[F (2α) −

∫ 1

0

f(t)e−2αt dt]

where F (s) is the Laplace transform of f(t) = (t − 1)2(t3 − 1)t8. Similarly

(∆buα, uα)L2(Ω) =1

2

∫

Ω

|z|2|x|4e−2α|x|Qα(|x|)2 d v(x) ≤

≤∫

Ω

|x|6Qα(|x|)2e−2α|x| d v(x) =

=

∫ ∞

1

dR

∫

|x|=R

R6Qα(R)2e−2αR d a(x) ≤

≤ 8π

3

∫ ∞

1

R7(2R2 + R + 1)Qα(R)2e−2αRdR =

=8π

3[G(2α) −

∫ 1

0

g(t)e−2αt dt]

where G(s) is the Laplace transform of g(t) = t7(2t2 + t + 1)Qα(t)2. There is apolynomial p(α) such that p(0) = 13! and

(∆buα , uα)L2(Ω)

‖uα‖2L2(Ω)

≤ O(α) − (2α)14∫ 1

0g(t)e−2αtdt

p(α) − (2α)14∫ 1

0f(t)e−2αtdt

,

2An uninspiring calculation shows that Aα(R) = α2R3 − α(α + 13)R2 + (11α + 35)R − 24.


i.e.(∆buα , uα)L2(Ω)

‖uα‖2L2(Ω)

→ 0, α → 0.

Then for any γ > 0 there is α = α(γ) such that (∆buα, uα)L2(Ω) < γ2‖uα‖2L2(Ω).

6 Regular points and the defect index

Let Reg(∆b) ⊆ C be the regularity field of the sublaplacian (1)-(2) i.e. if λ ∈Reg(∆b) then there is γ = γ(λ) > 0 such that

‖(∆b − λI)u‖L2(Ω) ≥ γ‖u‖L2(Ω)

for any u ∈ D(∆b). Let λ = a + ib be the real and imaginary parts of λ ∈ C. AsA = ∆b − aI is a symmetric operator

‖ (∆b − λI)u‖2L2(Ω) = ‖ (A − ibI) u‖2

L2(Ω) = ‖Au‖2L2(Ω) + b2‖u‖2

L2(Ω)

hence ‖(∆b − λI)u‖L2(Ω) ≥ |Im(λ)| ‖u‖L2(Ω) for any u ∈ D(∆b). ConsequentlyC± ⊂ Reg(∆b) where C+ = z ∈ C : Im(z) > 0 and C− = z ∈ C : Im(z) < 0.Let Γ ⊂ Reg(∆b) be a connected component. By Theorem 1.22 in [6], p. 76, thespaces

L2(Ω) ⊖R(∆b − λI), λ ∈ Γ,

have the same dimension (as Hilbert spaces). The defect index of ∆b correspond-ing to the connected component Γ is

defΓ(∆b) = dim[L2(Ω) ⊖R(∆b − λI)

], λ ∈ Γ.

Note that ∆b : D(∆b) ⊂ L2(Ω) → L2(Ω) is a real operator i.e. u ∈ D(∆b) =⇒u ∈ D(∆b) and ∆bu = ∆bu for any u ∈ D(∆b). Then (by Theorem 1.31 in [6], p.89)

defC−

(∆b) = defC+(∆b). (14)

At this point one may use the von Neumann theorem (cf. e.g. Theorem 1.28 in[6], p. 84) to conclude that

Theorem 1. The sublaplacian ∆b : D(∆b) ⊂ L2(Ω) → L2(Ω) admits a selfad-joint extension.

As ∆b is symmetric its Cayley transform

C∆b: R(∆b + iI) ⊂ L2(Ω) → L2(Ω), C∆b

= (∆b − iI) (∆b + iI)−1,

(an isometry) is well defined. The defect indices of ∆b coincide with those of itsCayley transform (cf. Theorem 1.27 in [6], p. 83). Then (by (14) and Theorem1.26 in [6], p. 82) C∆b

admits a unitary (i.e. maximal isometric) extension C∆b

and

∆b = i(I + C∆b

)

(I − C∆b

)−1


is a maximal symmetric extension of ∆b.Let S1 → C(M)

π→ M be the canonical circle bundle over M and Fθ theFefferman metric associated to a contact form θ (a Lorentzian metric on C(M),cf. Definition 2.15 in [7], p. 128). Let µ be the canonical measure on C(M)associated to Fθ. Let ¤ be the Laplace-Beltrami operator of (C(M), Fθ) (thewave operator). If D ⊆ C(M) is an open set we regard the wave operator as alinear operator ¤ : D(¤) ⊂ L2(D,µ) → L2(D,µ) with domain D(¤) = C∞

0 (D).As S1 ⊂ Isom(C(M), Fθ) the wave operator is S1-invariant (see also Proposition2.7 in [7], p. 140). Consequently the pushforward of ¤

(π∗¤)(u) π = ¤(u π), u ∈ C2(M),

is well defined. By a result of J.M. Lee, [12], π∗¤ = ∆b on C2(M) (here we adoptthe conventions3 in [7]).

Proposition 6. Let Ω ⊆ M be a domain and D = π−1(Ω). If (∆b)0 is theLagrange sublaplacian of (M, θ) then Reg(¤) ⊆ Reg [(∆b)0].

Proof: If U ⊆ M is an open set then4

∫

π−1(U)

(f π) dµ = 2π

∫

U

f d v (15)

for any function f ∈ C∞(U) supported in U (by integration along the fibre). Letu ∈ C∞

0 (Ω). Then Supp(u π) is contained in π−1(Supp(u)) (a compact set, aseach fibre π−1(x) ≈ S1 is compact). Therefore u π ∈ C∞

0 (D). Let λ ∈ Reg(¤).Then there is γ = γ(λ) such that (by (15))

γ√

π‖u‖L2(Ω) = γ ‖u π‖L2(D) ≤ ‖(¤ − λI)(u π)‖L2(D) =

=√

π‖ [(∆b)0 − λI]u‖L2(Ω)

hence λ ∈ Reg [(∆b)0].

Remark 3. The behavior of Reg(∆b) under a transformation θ = efθ is un-known.

7 Sublaplacians and sub-Riemannian geometry

Let M be a strictly pseudoconvex CR manifold and θ a contact form on M withthe Levi form Gθ positive definite. As H possesses the bracket generating pro-perty (M,H,Gθ) is a sub-Riemannian manifold (in the sense of R.S. Strichartz,[15]). Let Xa : 1 ≤ a ≤ 2n be a local gθ-orthonormal frame of H defined

3Under the conventions in [12] π∗¤ = 1

2∆b.

4The symbol π in the right hand side of (15) is the irrational number π ∈ R \ Q.


on the open set U . Let T be the characteristic direction of dθ. Let (U, xi) bea local coordinate system on M . We set Xa = bi

a∂i and T = T i∂i for somebia, T i ∈ C∞(U), where ∂i = ∂/∂xi. Let us consider the positive semi-definite

matrix

aij =

2n∑

a=1

biabj

a ∈ C∞(U), 1 ≤ i, j ≤ 2n + 1.

If gij = gθ(∂i, ∂j) and [gij ] = [gij ]−1 then (cf. [7], p. 113)

aij = gij − T iT j .

For each ǫ > 0 let gǫ be the Riemannian metric on M locally given by gǫ(∂i, ∂j) =gǫ

ij where

[gǫij ] = [gij

ǫ ]−1, gijǫ = aij + ǫgij .

Lemma 2. The Riemannian metric gǫ and the Webster metric are related by

gǫ(X,Y ) =1

1 + ǫ

[gθ(X,Y ) +

1

ǫθ(X)θ(Y )

](16)

for any X,Y ∈ X∞(M). Consequently

∇ǫu = ∇Hu + ǫ∇u (17)

for any u ∈ C1(M).

Here ∇ǫu is the gradient of u with respect to gǫ i.e. gǫ(∇ǫu,X) = X(u) forany X ∈ X

∞(M). As a consequence of (17) ∇ǫu approaches the horizontal, orsub-Riemannian, gradient of u as ǫ → 0. The identity (16) follows from

gǫ(Xa,Xb) =1

1 + ǫδab , gǫ(Xa, T ) = 0, gǫ(T, T ) =

1

ǫ, (18)

for any 1 ≤ a, b ≤ 2n.

Proposition 7. Let ∇ǫ be the Levi-Civita connection of (M, gǫ). Then

∇ǫXY = ∇XY + Ω(X,Y ) − ǫ

1 + ǫA(X,Y )T, (19)

∇ǫXT = τ(X) +

(1 +

1

ǫ

)JX, (20)

∇ǫT X = ∇T X +

(1 +

1

ǫ

)JX, (21)

∇ǫT T = 0, (22)

for any X,Y ∈ H. Consequently the Laplace-Beltrami operator of (M, gǫ) is givenby

∆ǫu = (1 + ǫ)∆bu − ǫ T (T (u)), (23)

for any u ∈ C2(M).


Here Ω = −dθ, τ is the pseudohermitian torsion of ∇ (cf. [7], p. 36-37) andA(X,Y ) = gθ(τX, Y ) for any X,Y ∈ X

∞(M). As a consequence of (23) thesublaplacian of (M, θ) is approximated by the second order elliptic operator ∆ǫ.

Remark 4. If M = Hn then ∆ǫ = (1 + ǫ)H − ǫ ∂2/∂t2.

Proof of Proposition 7.

Proof: Using (16) and ∇ǫgǫ = ∇gθ = 0 one derives

2gǫ(∇ǫXY,Z) − 2

ǫ(1 + ǫ)X(θ(Y ))θ(Z) =

1

1 + ǫ2gθ(∇XY,Z)− (24)

−gθ(Z, T∇(X,Y )) + gθ(Y, T∇(X,Z)) + gθ(X,T∇(Y,Z))+

+2

ǫ[θ(X)(dθ)(Y,Z) + θ(Y )(dθ)(X,Z) − θ(Z)(dθ)(X,Y )]

for any X,Y,Z ∈ X∞(M). Next the identity (24) and T∇ = 2(θ ∧ τ −Ω⊗ T ) (cf.

[7], p. 37) yield (19)-(22). Finally we set

Xǫa =

√1 + ǫ Xa , 1 ≤ a ≤ 2n, Xǫ

2n+1 =√

ǫ T,

so that Xǫi : 1 ≤ i ≤ 2n + 1 is a local gǫ-orthonormal frame of T (M). Then

∆ǫu = −2n+1∑

i=1

Xǫi (X

ǫi u) − (∇ǫ

XǫiXǫ

i )u

and (19)-(22) lead to (23).

Let divǫ be the divergence operator with respect to gǫ i.e. for any C1 vectorfield Y ∈ X(M)

divǫ(Y ) =

2n+1∑

i=1

gǫ(∇ǫXǫ

iY,Xǫ

i ) =

= (1 + ǫ)

2n∑

a=1

gǫ(∇ǫXa

Y,Xa) + ǫ gǫ(∇ǫT Y, T )

on U . Using the decomposition Y = ΠHY + θ(Y )T and the formulae (19)-(22)one may easily show that

∇ǫXY = ∇XY + θ(Y )τX +

1 + ǫ

ǫJX+

+Ω(X,Y ) − ǫ

1 + ǫA(X,Y )T,

∇ǫT Y = ∇T Y +

1 + ǫ

ǫφY,


for any X ∈ Γ∞(H). Here J : H → H was extended to a bundle endomorphismφ : T (M) → T (M) by requesting that φ = J on H and φT = 0. Consequently(by (16))

gǫ(∇ǫXY,X) =

1

1 + ǫgθ(∇XY,X) + θ(Y )A(X,X),

gǫ(∇ǫT Y, T ) =

1

ǫgθ(∇T Y, T ),

so that (by traceGθ(A) = 0)

divǫ(Y ) = div(Y ). (25)

Our considerations require that we dissociate the underlying integration measurefrom the domain metric (cf. e.g. [11], p. 4638). Precisely (by taking into account(25)) we say that u ∈ L2(Ω) admits a weak ∇ǫ-gradient if there is Xǫ ∈ X(Ω)⊗C

such that gθ(Xǫ,Xǫ)1/2 ∈ L1

loc(Ω) and

∫

Ω

gǫ(Xǫ, Y ) dv = −∫

Ω

u div(Y ) dv (26)

for any Y ∈ X∞0 (Ω). The vector field Xǫ is uniquely determined up to a set of

measure zero and we adopt the notation ∇ǫu = Xǫ. Let now D(∇ǫ) consist of allu ∈ L2(Ω) admitting a weak ∇ǫ-gradient such that ∇ǫu ∈ L2(T (Ω) ⊗ C). Alsolet D

[(∇ǫ)

∗]consist of all X ∈ L2(T (Ω) ⊗ C) such that

∫

Ω

gǫ(∇ǫu , X) dv =

∫

Ω

u uX dv

for some uX ∈ L2(Ω) and any u ∈ D(∇ǫ). The adjoint (∇ǫ)∗

: D[(∇ǫ)

∗] ⊂L2(T (Ω) ⊗ C) → L2(Ω) is given by (∇ǫ)

∗X = uX . Finally we set D(∆ǫ) = u ∈

D(∇ǫ) : ∇ǫu ∈ D[(∇ǫ)

∗]. We shall use the operator ∆ǫ : D(∆ǫ) ⊂ L2(Ω) →L2(Ω) given by ∆ǫu = (∇ǫ)

∗ ∇ǫ.Let u ∈ C1(Ω). For any ϕ ∈ C∞

0 (Ω)

∫

Ω

T (u)ϕdv =

∫

Ω

T (uϕ) − uT (ϕ) dv =

=

∫

Ω

div(uϕT ) − uϕ div(T ) − uT (ϕ) dv = −∫

Ω

uT (ϕ) dv

by Green’s lemma and div(T ) = 0 (as T is ∇-parallel). Therefore we may adoptthe following definition. A function u ∈ L2(Ω) is weakly differentiable in theT -direction if ∫

Ω

u1 ϕdv = −∫

Ω

uT (ϕ) dv

for some u1 ∈ L1loc(Ω) and any ϕ ∈ C∞

0 (Ω). Such u1 is unique up to a setof measure zero and we adopt the customary notation u1 = T (u). Next let


D(T ) consist of all u ∈ L2(Ω) such that u is weakly differentiable in the T -direction and T (u) ∈ L2(Ω). We may then regard T as a (densely defined) linearoperator T : D(T ) ⊂ L2(Ω) → L2(Ω). Let T ∗ : D(T ∗) ⊂ L2(Ω) → L2(Ω) bethe adjoint of T . Then C∞

0 (Ω) ⊆ D(T ∗) and the restriction of T ∗ to C∞0 (Ω) is

−T . We shall need the operator T ∗T : D(T ∗T ) ⊂ L2(Ω) → L2(Ω) defined onD(T ∗T ) = u ∈ D(T ) : T (u) ∈ D(T ∗).

Remark 5. We adopt the notations and conventions in [17], p. 113-116. Letσ2(∆ǫ) ∈ Smbl2(E,E) be the symbol of ∆ǫ where E = Ω×C is the trivial bundleover Ω. Let T ′(Ω) = T ∗(Ω) \ (0) and π : T ′(Ω) → Ω the projection. We wish tocompute

σ2(∆ǫ)ω :(π−1E

)ω→

(π−1E

)ω

, ω ∈ T ′(Ω).

Let v ∈ Ex =(π−1E

)ω

where x = π(ω) and let us consider f ∈ C∞(Ω) ands ∈ Γ∞(E) such that (df)x = ω and s(x) = v. Then

σ2(∆ǫ)ω(v) = ∆ǫ

[−1

2(f − f(x))2s

](x) =

=[(1 + ǫ)‖ω‖2 − ω(Tx)2

]v

so that σ2(∆ǫ)ω : Ex → Ex is a C-linear isomorphism (accounting for the el-lipticity of ∆ǫ). As ǫ → 0 this gives the formula for σ2(∆b) reported in theIntroduction. In particular σ2(∆ǫ)θx

(v) = ǫ v.

Remark 6. The coefficients of ∆ǫ (with respect to a local coordinate system(U, xi) on M) are jointly continuous in x ∈ U and ǫ > 0. Consequently (by The-orem 4.13 in [17], p. 142) the function ǫ 7→ dimC Ker(∆ǫ) is upper semicontinu-ous. Also for each ǫ0 > 0 there is δ > 0 such that dimC Ker(∆ǫ) ≤ dimC Ker(∆ǫ0)for any |ǫ − ǫ0| < δ.

The family of second order differential operators ∆ǫǫ>0 is said to be uni-formly K-positive definite (in a neighborhood of the origin) if there exist ǫ0 > 0,a preclosed operator K : D(K) ⊂ L2(Ω) → L2(Ω), and two constants α > 0 andβ > 0 such that for any 0 < ǫ ≤ ǫ0 the following properties are satisfied

i) D(∆ǫ) ⊆ D(K), K(D(∆ǫ)) = L2(Ω),

ii) (∆ǫu , Ku)L2(Ω) ≥ α2‖u‖2L2(Ω),

iii) ‖Ku‖2L2(Ω) ≤ β2 (∆ǫu , Ku)L2(Ω) ,

for any u ∈ D(∆ǫ). As we shall shortly see if ∆ǫǫ>0 is uniformly K-positivedefinite then one may produce generalized solutions to the equation ∆bu = f .Note that (together with W.V. Petryshyn, [13]-[14]) for each ǫ > 0 and f ∈ L2(Ω)we may consider the functional Fǫ : D(∆ǫ) → R given by

Fǫ(u) = (∆ǫu , Ku)L2(Ω) − (f , Ku)L2(Ω) − (Ku , f)L2(Ω) .


Then (by a result in [14]) the minimum points of Fǫ are (under the assumptions(i)-(ii) above) solutions to ∆ǫu = f . In general however these may depend on ǫ(and one may not employ the identity (30) below to produce solutions to ∆bu =f). In turn we base our discussion on the functional F (see (28) below) got asFǫ(u) → F(u) for ǫ → 0.

We need to establish the following weak version of the identity (17) in Lemma2.

Lemma 3. D(∇ǫ) = D(∇H) ∩ D(T ) and

∇ǫu = (1 + ǫ)∇Hu + ǫ T (u)T (27)

for any u ∈ D(∇ǫ).

Proof: Let u ∈ D(∇ǫ). Then (26) for Y ∈ Γ∞0 (H) may be written

1

1 + ǫ

∫

Ω

gθ(ΠH∇ǫu, Y ) dv = −∫

Ω

u div(Y ) dv

hence u is weakly differentiable along H and its weak horizontal gradient is givenby ∇Hu = (1/(1 + ǫ))ΠH∇ǫu. Moreover

∫

Ω

‖∇Hu‖2 dv =1

(1 + ǫ)2

∫

Ω

‖ΠH∇ǫu‖2 dv ≤

≤ 1

(1 + ǫ)2

∫

Ω

‖∇ǫu‖2 dv ≤ ‖∇ǫu‖2L2(T (Ω)⊗C) < ∞

hence u ∈ D(∇H). Similarly let ϕ ∈ C∞0 (Ω) and let us use (26) for Y = ϕT i.e.

∫

Ω

ϕgǫ(∇ǫu, T ) dv = −∫

Ω

u div(ϕT ) dv

or (by div(ϕT ) = T (ϕ))

1

ǫ

∫

Ω

ϕθ(∇ǫu) dv = −∫

Ω

uT (ϕ) dv

hence u is weakly differentiable in the T -direction and its weak T -derivative isgiven by T (u) = (1/ǫ) θ(∇ǫu). Also

∫

Ω

|T (u)|2 dv =1

ǫ2

∫

Ω

|gθ(∇ǫu, T )|2 dv ≤

≤ 1

ǫ2

∫

Ω

‖∇ǫu‖2‖T‖2 dv =1

ǫ2‖∇ǫu‖2

L2(T (Ω)⊗C)

hence u ∈ D(T ). Summing up the information got so far D(∇ǫ) ⊆ D(∇H)∩D(T )and (17) weak holds for any u ∈ D(∇ǫ). Viceversa let u ∈ D(∇H) ∩ D(T ). Let


Y ∈ X∞0 (Ω) be an arbitrary test vector field decomposed as Y = YH + ϕT where

YH = ΠHY and ϕ = θ(Y ). Clearly YH ∈ Γ∞0 (HΩ) and ϕ ∈ C∞

0 (Ω). Then

∫

Ω

u div(Y ) dv =

∫

Ω

u div(YH) + T (ϕ) dv =

= −∫

Ω

gθ(∇Hu, YH) dv −∫

Ω

T (u)ϕdv =

= −(1 + ǫ)

∫

Ω

gǫ(∇Hu, Y ) dv − ǫ

∫

Ω

T (u)gǫ(T, Y )

hence u admits a weak ∇ǫ-gradient and ∇ǫu = (1 + ǫ)∇Hu + ǫ T (u)T . Finally

1

2

∫

Ω

‖∇ǫu‖2 dv ≤∫

Ω

(1 + ǫ)2‖∇Hu‖2 + ǫ2|T (u)|2‖T‖2

dv =

= (1 + ǫ)2‖∇Hu‖2L2(HΩ⊗C) + ǫ2‖T (u)‖2

L2(Ω) < ∞

hence u ∈ D(∇ǫ).

8 Generalized solutions to ∆bu = f .

Let (M, θ) be a strictly pseudoconvex CR manifold and Ω ⊂ M a domain. Letθ be a contact for on M and ∆b the sublaplacian of (M, θ). Let K : D(K) ⊂L2(Ω) → L2(Ω) be a preclosed operator such that D(∆b) ⊆ D(K). Let f ∈ L2(Ω)and let F be the functional given by

F(u) =

∫

Ω

(∆bu)Kud v − 2Re

∫

Ω

f Kud v, u ∈ D(∆b). (28)

We shall establish the following

Theorem 2. If the family ∆ǫǫ>0 of Laplace-Beltrami operators of the Rieman-nian metrics gǫǫ>0 is uniformly K-positive definite then any solution u0 ∈ S =D(∆b) ∩ D(T ∗T ) to the equation

∆bu = f (29)

is a minimum point of the functional F : S → R. Viceversa if i) K(S) is densein L2(Ω) and ii) u0 ∈ S is a minimum point of F : S → R then u0 satisfies (29).

The proof of Theorem 2 relies on the weak version of the identity (23) inProposition 7 above i.e.


Lemma 4. For each ǫ > 0 one has D(∆b)∩D(T ∗T ) ⊆ D(∆ǫ) and the followingidentity holds

∆ǫf = (1 + ǫ)∆bf + ǫ T ∗[T (f)] (30)

for any f ∈ D(∆b) ∩ D(T ∗T ).

Proof: . We set S = D(∆b) ∩ D(T ∗T ) for simplicity. If f ∈ S then f ∈D(∇H) ∩ D(T ) i.e. f ∈ D(∇ǫ) and

∇Hf ∈ D[(∇H

)∗], T (f) ∈ D(T ∗).

Then for each u ∈ D(∇ǫ) (by (27))

∫

Ω

gǫ(∇ǫu , ∇ǫf) dv =

= (1 + ǫ)

∫

Ω

gθ(∇Hu , ∇Hf) dv + ǫ

∫

Ω

T (u)T (f) dv =

= (1 + ǫ)

∫

Ω

u ∆bf dv + ǫ

∫

Ω

u T ∗(T (f)) dv

and (1+ǫ)∆bf +ǫT ∗(T (f)) ∈ L2(Ω) hence ∇ǫf ∈ D[(∇ǫ)

∗]and (30) holds good.

Now let us assume that the family ∆ǫǫ>0 is uniformly K-positive definitefor some preclosed operator K : D(K) ⊂ L2(Ω) → L2(Ω). Then S ⊆ D(K) andthere is a constant α > 0 such that for each u ∈ S (by (30))

α2‖u‖2L2(Ω) ≤ (∆ǫu , Ku)L2(Ω) =

= (1 + ǫ) (∆bu , Ku)L2(Ω) + ǫ (T ∗Tu , Ku)L2(Ω)

and then for ǫ → 0α2‖u‖2

L2(Ω) ≤ (∆bu , Ku)L2(Ω) . (31)

Similarly‖Ku‖2

L2(Ω) ≤ β2 (∆bu , Ku)L2(Ω) , (32)

for any u ∈ S.To prove Theorem 2 let us observe first that the sublaplacian ∆b is K-

symmetric on S i.e.

(∆bf,Kg)L2(Ω) = (Kf , ∆bg)L2(Ω) , f, g ∈ S.

Indeed

(∆b(f + g) , K(f + g))L2(Ω) − (∆b(f − g) , K(f − g))L2(Ω) +


+i(∆b(f + ig) , K(f + ig))L2(Ω) − (∆b(f − ig) , K(f − ig))L2(Ω) =

= 4 (∆bf , Kg)L2(Ω) .

At this point one may interchange f and g to get another identity of the sort andexploit (∆bu , Ku)L2(Ω) ∈ R for any u ∈ S (a consequence of (31)) to conclude.Next let u0 ∈ S such that ∆bu0 = f . Then

(∆b(u − u0) , K(u − u0))L2(Ω) − (∆bu0 , Ku0)L2(Ω) = F(u)

so that (again by (31))

F(u0) = − (∆bu0 , Ku0)L2(Ω) = minu∈S

F(u).

Viceversa let u0 be a minimum point of the functional F : S → R. As ∆b isK-symmetric on S

F(u0 + tu) = F(u0) + 2t Re (∆bu0 − f , Ku) + O(t2)

hence

Re

∫

Ω

(∆bu0 − f)Ku dv = 0

for any u ∈ S. Let us replace u by i u so that to obtain

Im

∫

Ω

(∆bu0 − f)Ku dv = 0.

Therefore ∆bu0 − f is orthogonal on K(S) and K(S) = L2(Ω). Theorem 2 isproved.

Theorem 3. Under the assumptions of Theorem 2 one hasi) The functional F is bounded from below on S.ii) For any g, h ∈ S and any 0 ≤ t ≤ 1

tF(g) + (1 − t)F(h) −F [tg + (1 − t)h] ≥ α2t(1 − t)‖g − h‖2L2(Ω). (33)

iii) All minimizing sequences of F : S → R converge in L2(Ω) and have thesame limit.

iv) Let u be a generalized solution to ∆bu = f . If u ∈ D(∆b) ∩ D(T ∗T ) thenu is a classical solution.

Proof: The proof of (i)-(iii) in Theorem 3 is similar to that of Theorem 4.3 in[6], p. 247. Statement (iv) is an analog of Proposition 4.3 in [6], p. 248, andfollows from Theorem 2 above (while (i)-(iii) are independent statements). Letu ∈ S. Then (by (32) and the Cauchy-Schwartz inequality)

F(u) = (∆bu , Ku)L2(Ω) − 2Re (Ku, f)L2(Ω) ≥


≥ 1

β2‖Ku‖2

L2(Ω) − 2‖Ku‖L2(Ω)‖f‖L2(Ω) ≥ −β2‖f‖L2(Ω).

Statement (ii) follows (by (31)) from

tF(g) + (1 − t)F(h) −F [tg + (1 − t)h] =

= t(1 − t) (∆b(g − h) , K(g − h))L2(Ω) ≥

≥ α2t(1 − t)‖g − h‖2L2(Ω).

We recall that uνν≥1 ⊂ S is a minimizing sequence for F : S → R if F(uν) →a = infu∈S F(u) as ν → ∞. Let t = 1/2 and g = uµ, h = uν in (33) so that toobtain

α2

2‖uµ − uν‖2

L2(Ω) ≤ F(uµ) + F(uν) − 2F(

uµ + uν

2

)≤

≤ F(uµ) + F(uν) − 2a

hence uνν≥1 is a Cauchy sequence in L2(Ω). Given two minimizing sequencesgνν≥1 and hνν≥1 with the limits g0, h0 ∈ L2(Ω) we set t = 1/2 and g = gν ,h = hν in (33) so that to obtain

α2

2‖gν − hν‖2

L2(Ω) ≤ F(gν) + F(hν) − 2a → 0, ν → ∞,

where from g0 = h0.It remains that we prove statement (iv) in Theorem 3. We recall that a

generalized solution to ∆bu = f is the limit in L2(Ω) of a minimizing sequencefor the functional F : S → R.

The functional F : S → R is G-differentiable and its Gateaux differential isgiven by

(VF)(u)h = 2Re

∫

Ω

(∆bu − f)K(h) dv

for any u, h ∈ S. By the mean value formula (cf. e.g. Theorem 7.7 in [6], p. 322)for any u, v ∈ S there is 0 < τ < 1 such that

F(v) −F(u) = (VF) [u + τ(v − u)] (v − u) =

= 2Re (∆b[u + τ(v − u)] − f , K(v − u))L2(Ω) ≥

≥ 2Re (∆bu − f , K(v − u))L2(Ω) +2τ

β2‖K(v − u)‖2

L2(Ω)

as a consequence of (32). We conclude that

F(v) −F(u) ≥ 2Re (∆bu − f , K(v − u))L2(Ω) (34)

for any u, v ∈ S. An analog of the inequality (34) is claimed in [6], p. 248, as amere consequence of the convexity of F (the proof depends however on (32)). Let


uνν≥1 be a minimizing sequence for F on S. Then K(uν)ν≥1 is convergentin L2(Ω). Indeed for any 0 ≤ t ≤ 1 and any g, h ∈ S (by (32))

tF(g) + (1 − t)F(h) −F [tg + (1 − t)h] =

= t(1 − t) (∆b(g − h) , K(g − h))L2(Ω) ≥

≥ 1

β2t(1 − t)‖K(g − h)‖2

L2(Ω) .

Let us set t = 1/2 and g = uν , h = uµ so that to obtain

1

2β2‖K(uν − uµ)‖2

L2(Ω) ≤

≤ F(uν) + F(uµ) − 2F(

uν + uµ

2

)≤ F(uν) + F(uµ) − 2a

hence K(uν)ν≥1 is a Cauchy sequence in L2(Ω). Let now u0 be a generalizedsolution to ∆bu = f belonging to S. Let uνν≥1 be a minimizing sequence forF on S. Then (by part (iii) in Theorem 3) uν → u0 in L2(Ω). On the other handif g = limν→∞ K(uν) ∈ L2(Ω) then K(uν − u0) → g − K(u0) as ν → ∞. As Kis a preclosed operator it must be that g − K(u0) = 0 hence K(uν − u0) → 0 asν → ∞. Going back to (34) (for v = uν and u = u0)

F(uν) −F(u0) ≥ 2Re (∆bu0 − f , K(uν − u0))

hence for ν → ∞ we get a ≥ F(u0). On the other hand (as u0 ∈ S) a ≤ F(u0).We conclude that u0 is a minimum point for S and then (by Theorem 2) ∆bu0 = f .

Let K = X + Y where X,Y are given by (13). Let Ω ⊂ H1 be a boundeddomain with C2 boundary. We close this section with an example of an operatorBp : Dxy(Bp) ⊂ L2(Ω) → L2(Ω) arising on H1 and satisfying the inequalities

(Bpu,Ku)L2(Ω) ≥ α2‖u‖2L2(Ω), ‖Ku‖2

L2(Ω) ≤ β2 (Bpu,Ku)L2(Ω) , (35)

for some constants α > 0, β > 0, and any u ∈ Dxy(Bp). Let p ∈ Z, p ≥ 2, andlet us set

Bpu =

p−1∑

i=0

p∑

j=0

(−1)p−1Xp−1−iY i(aijY

jXp−ju)

+ a(Xu + Y u), (36)

where a ∈ C0(Ω), a ≥ 0, and aij ∈ Cp−1(Ω) for 1 ≤ i ≤ p− 1 and 0 ≤ j ≤ p. LetD(B2) consist of all u ∈ C3(Ω) such that Xu = Y u = 0 on ∂Ω while if p ≥ 3 thedomain D(Bp) of Bp consists of all u ∈ C2p−1(Ω) such that

Xk+1u = 0, XkY u = 0, 0 ≤ k ≤ p − 2 − i, 0 ≤ i ≤ p − 2,


Y ℓXu = 0, Y ℓ+1u = 0, 0 ≤ ℓ ≤ p − 2,

Y ℓXp−iu = 0, Y ℓXp−1−iY u = 0, 0 ≤ ℓ ≤ i − 1, 1 ≤ i ≤ p − 2,

on ∂Ω. We shall establish

Theorem 4. If D(K) = W 1,2(X,Y )(Ω) then K : D(K) ⊂ L2(Ω) → L2(Ω) is a

preclosed operator. Let us set

bij =

a0j , if i = 0,

ai−1,j + aij , if 1 ≤ i ≤ p − 1,

ap−1,j , if i = p,

and assume that

p∑

i,j=0

bijξiξj ≥ C2

[p∑

i=0

(pi

)ξi

]2

on Ω (37)

for some constant C > 0 and any (ξ0, · · · , ξp) ∈ Rp+1. Let D ⊂ C be a bounded

domain with C2 boundary and Ω = D × (0, c), c > 0. Then the inequalities (35)hold for any u ∈ Dxy(Bp) = D(Bp) ∩ C2p−1(D).

If aij ∈ Cp−1(D) the restriction of Bp to Dxy(Bp) is a weakly elliptic differ-ential operator (in the sense of [6], p. 241). It should be observed that the weakellipticity requirement (37) is similar to the X-ellipticity condition in [9] (i.e. one

requires that∑

i,j bijξiξj ≥ C2〈X, ξ〉 with X =

(pi

)∂/∂xi).

Lemma 5. Let L ∈ X,Y . If Lig = 0 on ∂Ω for any 0 ≤ i ≤ m − 1 then

∫

Ω

Lm(f)g d v = (−1)m

∫

Ω

f Lm(g) d v.

Let us take the L2 inner product of (36) with Ku, u ∈ D(Bp), and take intoaccount Lemma 5 in order to integrate by parts. We obtain

(Bpu , Ku)L2(Ω) =

∫

Ω

a (Ku)2 d v+

+

p∑

j=0

∫

Ω

(Y jXp−ju)

p−1∑

i=0

aij(YiXp−iu + Y iXp−1−iY u) d v.

On the other hand [X,Y ] = −4T and [X,T ] = [Y, T ] = 0 (where T = ∂/∂t)hence

XmY = Y Xm − 4mXm−1T, m ≥ 1.


Consequently

(Bpu , Ku)L2(Ω) =

p∑

i,j=0

∫

Ω

bij

(Y iXp−iu

) (Y jXp−ju

)d v+ (38)

+

∫

Ω

a (Ku)2 d v − 4

p∑

j=0

p−2∑

i=0

(p − i − 1)

∫

Ω

aij(YjXp−ju)Y iXp−i−2Tu d v.

If u ∈ Dxy(Bp) then (by (37))

(Bpu , Ku)L2(Ω) ≥ C2

∫

Ω

(Kpu)2 d v.

It should be observed that the restriction of K to Dxy(Bp) is the first orderoperator δ = ∂/∂x+∂/∂y in [6], p. 242. Then (by Proposition 4.1 in [6], p. 239)

∫

Ω

(Ku)2 d v ≥ c γ2

∫

D

u2 dx dy

for some constant γ > 0. An iterative argument now leads to (35). Theorem 4 isproved.

The choice p = 2 and a00 = a10 = a02 = a12 = 1, a01 = a11 = 0 leads to thethird order operator B2u = K∆bu+ aKu. However B2 is not weakly elliptic andintegration by parts as in the proof of Theorem 4 gives but (B2u,Ku)L2(Ω) =∫Ω

a(Ku)2dv − (∆bu,K2u)L2(Ω) for any u ∈ D(B2).

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Received: 12.11.2008.

Universita degli Studi della Basilicata,Dipartimento di Matematica e Informatica,

Via Dell’Ateneo Lucano 10,Contrada Macchia Romana,

85100 Potenza, Italy,E-mail: [email protected]

[email protected]

Sublaplacians on CR manifolds

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