lecturenotes2008_000

8/4/2019 LectureNotes2008_000

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LECTURES ON METHODS OF PARTIAL DIFFERENTIAL

EQUATIONS IN SEVERAL COMPLEX VARIABLES 1

MEI-CHI SHAWDepartment of Mathematics, University of Notre Dame

255 Hurley Hall, Notre Dame, IN 46556-4618, (USA), e-mail: [email protected]

Contents

1. The Levi problem in several complex variables 21.1. The Cauchy-Riemann Equation in C 2

1.2. The Cauchy-Riemann equations in Cn 31.3. Domains of holomorphy 51.4. The Levi Problem and the Equation 62. L2 theory for on pseudoconvex domains in Cn 72.1. The -Neumann problem 82.2. L2 existence for and -Neumann operator 102.3. Global Regularity for on pseudoconvex domains 143. Boundary regularity for the -Neumann problem 163.1. -Neumann problem on strictly pseudoconvex Lipschitz

domains 163.2. Regularity for the -Neumann problem on pseudoconvex

domains 204. Strong Okas lemma and the -Neumann problem 214.1. Strong Okas Lemma and bounded plurisubharmonic exhaustion

functions 214.2. Strong Okas lemma and finite type conditions 255. The -Neumann problem on Kahler manifolds with

nonnegative curvature 265.1. The -Neumann problem on domains in CPn 296. Tangential Cauchy-Riemann equations and the Bochner

Martinelli kernel 317. CR manifolds and tangential Cauchy-Riemann equations 317.1. L2 theory for b on strongly pseudoconvex CR manifolds 31

1These notes are results from short courses given by the author in Hongzhou, China,2004, Trento, Italy, 2005 and Luminy, France 2007, Academia Sinica, Taipei, Taiwan,2008 and Fudan, China 2008. She would like to thank all the students and colleagues whoparticipated in the courses. The author would especially like to thank Dr. Donato Ciampawho took notes and put it into the typed form for the summer school on July, 17-29, 2005,at I.T.C.-I.R.S.T. of Povo-Trento on Real PDEs for Complex and CR Geometry. Thesenotes were the bases of the present lecture notes which have been substantially expandedand revised by the author.

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7.2. L2 theory for b on boundary of pseudoconvex domains 337.3. Nonisotropic Sobolev and Holder spaces 36

7.4. Campanato Space 387.5. The Tangential Cauchy-Riemann Complex 397.6. Folland-Stein-Tanaka formula for b 438. Applications of and b 468.1. The Levi problem 478.2. Regularity for the Bergman projection 488.3. The Newlander-Nirenberg theorem 498.4. Embedding strongly pseudoconvex compact CR manifolds 518.5. Nonexistence of Levi-flat hypersurfaces in CPn 539. Open problems 5410. Appendix 5510.1. Preliminaries for Lipschitz domains 55

10.2. Preliminaries for Lipschitz domains 57References 60

1. The Levi problem in several complex variables

1.1. The Cauchy-Riemann Equation in C. Let C be the complex Eu-clidian space with coordinate z = x+iy. We can define the Cauchy-Riemannoperator

z:=

1

2

x+ i

y

,

and we say that a function h is holomorphic in a domain D ofC, and wewrite h O(D), if and only if

(1.1)h

z= 0.

Equation (1.1) is called the homogeneous Cauchy-Riemann equation in C.The behavior of holomorphic function and the study of the homogeneousCauchy-Riemann equation is closely related to the solvability and regularityof the inhomogeneous Cauchy-Riemann equation

(1.2)u

z= f,

with f a given function.

We have the following result.

Theorem 1.1. Let D be a bounded domain in C and let f Ck(D), fork 1. Then the function defined by

u(z) :=1

2i

D

f()

z d d,


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is in Ck(D) and satisfies (1.2). Moreover, if f is only in C(D), then u(z)defined as before satisfies (1.2) in the distribution sense.

Remark 1.2. Observe that the function

u(z) =1

1

z,

is a fundamental solution to (1.1).

This can be derived by differentiating the fundamental solution for .Since

1

2 log |z| = 2

2

zzlog |z| = 0,

where 0 is the Dirac delta function centered at 0, we have

2

z

z

log

|z

|=

1

z

1

z

= 0.

This implies that 1/z is a fundamental solution for /z.Let D be a bounded domain in C with C1 boundary. Let f be a continous

function f on D. We define u = Gf to be the unique solution to the Dirichletproblem with u = f with u = 0 on bD. It follows that 2u/zz = f inC in the distribution sense. Thus we have f = (Gf) and the functionGf = u satisfies u = f.

1.2. The Cauchy-Riemann equations in Cn. Let Cn be the n-dim-ensional complex Euclidian space. We denote coordinates by z = (z1, . . . , zn),where zj = xj +iyj, 1 j n. We can define the Cauchy-Riemann operator

zj :=

1

2 xj + i yj ,for 1 j n. It is easy to see that the foolowing decomposition holds

CTCn = T1,0(Cn) T0,1(Cn),where

T1,0(Cn) = span

zj

, T0,1(Cn) = span

zj

.

Definition 1.3. Let f a C1 function defined on an open subset D ofCn. Wesay that f is holomorphic in D and we write f O(D) if f(z) is holomorphicin each variable zj, when the others are fixed, i.e. f satisfies

fzj

= 0,

for all 1 j n.The Cauchy-Riemann equations are

(1.3)u

zj= fj, j = 1, . . . , n, n 2.


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This system is overdetermined (one unknown function with n equations).In order for (1.3) to be solvable, f must satisfy the following compatibility

conditions(1.4)

fkzj

=fjzk

, 1 j,k n.

As in the 1-dimensional case, we can state the following for the solvabilityof Cauchy Riemann equation.

Theorem 1.4. Letfj C0 (D), 1 j n, where D is a relatively compactsubset ofCn, n 2. Suppose that fj satisfy (1.4). Then there is a functionu C0 (D) which satisfies (1.3).Proof. Set

u(z) =1

2i D f1(, z2, . . . , zn) z1 d d.Then u has compact support and we have

u

z1= f1,

u

zj= fj, j = 2, . . . , n .

One of the most striking difference between the 1-dimensional and then-dimensional case is the so called Hartogs phenomenon, which states that,if a bounded domain D in Cn, n 2, has connected boundary, then anyholomorphic function f defined in an open neighborhood of the boundarybD can be holomorphically extended to all D. This is not true in C, as

one can check easily with the function f(z) = 1/z, which is holomorphic inC \ {0} and cannot be extended in any way as an entire function.More precisely, we can state the following.

Theorem 1.5 (Hartogs). LetD be an open set inCn and letK be a compactsubset in D, such that D \ K is connected. If f O(D \ K), then f can beextended holomorphically to all D.

Proof. Let C0 (D) a cut-off function such that 1 on some openneighborhood ofK. Then

((1 )f) = f,has compact support and satisfies compatibility condition in Cn.

Then, from the previous theorem, there is u with compact support suchthat

u = (f).Finally, the function

h = (1 )f u,is the desired extension.


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1.3. Domains of holomorphy.

Definition 1.6. A domain D in Cn is called a domain of holomorphy if isnot possible to find two nonempty open sets D1, D2 in Cn such that

(i) D1 is connected, D1 D, D2 D D1;(ii) for any f O(D) there is f O(D1) such that f|D2 = f.Equivalently, D is a domain of holomorphy if and only if for any p bD,

there exists a holomorphic function f O(D) such that f is singular at p.An important notion related to domain in Cn is that of pseudoconvexity.

Definition 1.7. Let D be a domain ofCn with C2 boundary, i.e. there isa neighborhood U of D and a function : D R of class C2, such that

D U = {z Cn|(z) < 0}, bD U = {z Cn|(z) = 0},

|(z)| = 0, z bD.We say that D is pseudoconvex if

Lp(; a) : (= |p(a, a)) =n

j,k=1

2

zjzk(p)aj ak 0, p bD,

for all vector a = (a1, . . . , an) Cn (a T0,1p (bD)) such thatn

j=1

aj

zj= 0.

Lp(; a) is called the Levi form of at p. A domain D is called strictlypseudoconvex if the Levi form is positive definite.

A function is called (strictly) plurisubharmonic at p ifLp(; a) 0 (> 0)for all vectors a Cn.

A domain D is pseudoconvex if and only if it is union of strictly pseudocon-vex domains {Dj} with C2 boundary such that D = jDj and Dj Dj+1for each j.

Theorem 1.8. Let D Cn be a strongly pseudoconvex domain with C2boundary bD. Then there exists aC2 defining function forD which is strictly

plurisubharmonic in a neighborhood of bD.It is easy to see that convex domains are domains of holomorphy.It is also easy to show that every strictly pseudoconvex domain is locally

biholomorphic to a strictly convex domain. Thus locally for each p bD,one can find a neighborhood Up and a holomorphic function fp : Up Dso that fp cannot be extended holomorphically past p. Can one find a fpholomorphic in D?


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1.4. The Levi Problem and the Equation. Let D be a pseudoconvexdomain in Cn with n 2. One of the major problems in complex analysisis to show that a pseudoconvex domain D is a domain of holomorphy. Neareach boundary point p bD, one must find a holomorphic function f(z) onD which cannot be continued holomorphically near p. This problem is calledthe Levi problem for D at p. It involves the construction of a holomorphicfunction with certain specific local properties.

If the domain D is strongly pseudoconvex with C boundary bD andp bD, one can construct a local holomorphic function f in an open neigh-borhood U of p, such that f is holomorphic in U D, f C(D U\ {p})and f(z) as z D approaches p. In fact f can be easily obtained asfollows: let r be a strictly plurisubharmonic defining function for D and weassume that p = 0. Let

F(z) = 2n

i=1

r

zi (0)zi

ni,j=1

2r

zizj (0)zizj .

F(z) is holomorphic, and it is called the Levi polynomial of r at 0. UsingTaylors expansion at 0, there exists a sufficiently small neighborhood U of0 and C > 0 such that for any z D U,

ReF(z) = r(z) +n

i,j=1

2r

zizj(0)zizj + O(|z|3) C|z|2.

Thus, F(z) = 0 when z D U\ {0}. Setting

f =1

F

,

it is easily seen that f is locally a holomorphic function which cannot beextended holomorphically across 0.

Global holomorphic functions cannot be obtained simply by employingsmooth cut-off functions to patch together the local holomorphic data, sincethe cut-off functions are no longer holomorphic. Let be a cut-off functionsuch that C0 (U) and = 1 in a neighborhood of 0. We note thatf is not holomorphic in D. However, if f can be corrected by solving a-equation, then the Levi problem will be solved.

Let us consider the (0,1)-form g defined by

g = (f) = ()f.

This form g can obviously be extended smoothly up to the boundary. It iseasy to see that g is a -closed form in D and g C(0,1)(D). If we can finda solution u C(D) such that

u = g in D,

then we define for z D,h(z) = (z)f(z) u(z).


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It follows that h is holomorphic in D, h C(D \ {0}) and h is singular at0. Thus one can solve the Levi problem for strongly pseudoconvex domains

provided one can solve equation (3.6.1) with solutions smooth up to theboundary.It has been known that domains of holomorphy are pseudoconvex. The

converse whether a pseudoconvex domain is a domain of holomorphy is theso called Levi problem: given a domain D is it possible, for each pointp bD, to find a holomorphic function f over D such that it cannot becontinued holomorphically near p? The answer to this question is given inthe following theorem.

Theorem 1.9. LetD be a domain inCn. The following are equivalent:(i) D is pseudoconvex;(ii) D is a domain of holomorphy;(iii) for each f

C(D) with f = 0 there is u

C(D) such that

u = f.

It is well known that (ii) implies (i).To show that (i) implies (ii) is the so-called Levi problem.We will show that (i) implies (iii) and (iii) implies (ii).The Levi problem has been solved by(1) (1942) Oka for n = 2 and later for all n 2.(2) (1958) Grauert using Sheaf methods.(3) (1962) Kohn by the -Neumann problem.(4) (1965) Hormander by the weighted -Neumann problem.

2. L2 theory for on pseudoconvex domains in Cn

By the Hilbert space theory of unbounded linear operator, if H1 and H2are two Hilbert spaces, T : H1 H2 is a closed, densely defined linearoperator, then

H2 = R(T) Ker(T),and

H1 = R(T) Ker(T).If we can show that R() is closed and then, if u = f can be solved, wemust have fKer(), because if g is such that g = 0, then

(f, g) = (u,g) = (u, g) = 0.

For a closed densely defined linear operator T, R(T) is closed if and only ifT u Cu, u Dom(T) (Ker(T))or if and only if

Tu Cu, u Dom(T) (Ker(T)).


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2.1. The -Neumann problem. Let D be a domain in Cn. Let f be afunction of class C1 and Let (z1, . . . , zn) be the complex coordinates for C

n,

we getdf =

nj=1

f

zjdzj +

f

zjdzj

.

We define

f :=n

j=1

f

zjdzj, f :=

nj=1

f

zjdzj

so that d = + . This means that the differential of a function can bedecomposed in the sum of a (1, 0)-form f and a (0, 1)-form f. A functionf is holomorphic if and only if f = 0.We define the space of (p,q)-forms p,q(D) as the subspace of (p + q)-forms

which can be written asf =

|I|=p,|J|=q

fIJdzI dzJ,

where I = (i1, . . . , ip) and J = (j1, . . . , jq) are multiindices of length p andq respectively, dzI = dzi1 . . . dzip, dzJ = dzj1 . . . dzjq .

Since d is a complex, it is easy to see that the sequence

. . . p,q1(D) p,q(D) p,q+1(D) . . .

is a complex, i.e. 2 = 0. It is called the Cauchy-Riemann complex.

Let L2(p,q)(D) and C(p,q)(D) denote the (p,q)-form with L

2 or C coeffi-

cients respectively, and let ( , ) denotes the usual L2 inner product. We candefine the formal adjoint of under the L2-norm as

: C(p,q)(D) C(p,q1)(D),(f,g) = (f, g),

for every g C(p,q)(D) with compact support.We denote the (maximal) L2 closure of with the same symbol. Then

we can define its L2 adjoint as Hilbert space adjoint by

: L2(p,q)(D) L2(p,q1)(D),in the following sense: f is in Dom(), the domain of , if there is a

g L2(p,q1)(D) such that, for every Dom() L2(p,q1)(D) we have(f, ) = (g, ).

Then we define f = g.

We have that is an unbounded operator and Dom() L2(p,q)(D). More-

over, if D is a bounded domain, then C(p,q)(D) Dom(). Finally, is a


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closed, densely defined linear operator. The same is true for too.

If f C1

(p,q)(

D) Dom(

), then(f, u) = (f, u), u Dom().

If u C1(p,q1)(D) and D is a C1 domain with defining function , then

(f, u) = (f, u) = (f,u) +

b

f,ud.

Since compactly supported forms in C(p,q)(D) are dense in L2(p,q)(D), it fol-

lows that if f C1(p,q)(D) Dom(), we must have

f = f,

andf = 0on bD.

More explicitly, when f C1(0,1)(D), this condition becomesn

j=1

fj()

j() = 0, bD.

We can now define the Laplacian of the -complex. It is the operator

:= + : L2(p,q)(D) L2(p,q)(D),with the domain of definition

Dom(

) := {f Dom() Dom(

) :f Dom(

),

f Dom()}.

The operator is a linear, closed, densely defined, self adjoint operator.

As in the case to be in Dom(), in order to belong to Dom(), a (p,q)-form f must satisfies the -Neumann boundary conditions

f = 0, f = 0,

on bD. The boundary value problem

u = f in Du = 0 in bDu = 0 in bD

is the -Neumann problem.

Example.

Let D be a bounded domain with 0 bD and assume that there is aneighborhood U of 0 such that

D U = {(z1, . . . , zn) U : (zn) < 0}.


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If we take u C2(0,1)(D), then we can write u =n

j=1 ujdzj and u Dom()if and only if

(1) un = 0, on bD U,

(2)ujzn

= 0, on bD U, j = 1, . . . , n 1.

In fact the condition u = 0 is equivalent to

nj=1

uj

zj= 0 un = 0,

on bD, while the condition u can be rewritten as

ujzn

unzj

= 0 ujzn

= 0, j = 1, . . . , n 1,

on bD.Actually, to obtain L2 existence results, we need to work in the weighted

L2 spaces. If C2(D) we define

L2(D, ) :=

f : f2 :=

D

|f|2edV <

,

the weighted L2 space with weight . Observe that , the Hilbert space

adjoint of in L2(D, ), is related to by

= e

e = + A0(),where A0() is a zeroth order operator which depends on . We set up theweighted -Neumann problem =

+

as before.

2.2. L2 existence for and -Neumann operator. From the prelimi-nary lemma, R() is closed if and only if

u Cu, u Dom() (Ker()).In order to prove the L2 existence both for and -Neumann operator,

we need the following a priori identity.

Proposition 2.1 (Morrey-Kohn). Let D Cn be a domain with C2boundary and defining function . For each f C1(p,q)(D) Dom() wehave

(2.1) f2 + f2 =n

j=1

fzj2 + n

j,k=1

b

2

zjzkfI,jKfJ,kKd.


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Under this assumptions, we can rewrite a more general form for the iden-tity of Morrey-Kohn-Hormander:

f2 + f2 =n

j=1

fzj 2 + nj,k=1b 2

zjzkfI,jKfJ,kKe

d

+

nj,k=1

2

zjzkfI,jKfJ,kKe

dV.(2.2)

Since D is pseudoconvex, we have

(2.3)n

j,k=1

b

2

zjzkfI,jKfJ,kKe

d 0.

If we choose =

|z

|2, the last term becomes

|f|2edV = f2.

Then, if D is pseudoconvex, we get

(f, f) = f2 + f2 f2,which is the a priori estimate.

A crucial fact to prove the actual estimate from the a priori estimate isthe following lemma.

Lemma 2.2 (A Density Lemma). Let D be a bounded domain inCn with

C2 boundary. Then C1(p,q)(D) Dom() is dense in Dom() Dom() inthe graph norm f f + f + f.Proof. We shall give only the main steps of the proof, which is essentially avariation of the Friedrichs Lemma2. By a partition of unity, we may assumethat the domain is star-shaped and 0 D.

(1) The space C(p,q)(D) is dense in Dom() in the graph norm f f +f.

2Friedrichs lemma: Let C0 (RN) a function with support in the unit ball and

such that ZRN

dV = 1,

and define (x) = N(x/). Ifv L2(RN) is with compact support and u is a C1

function in a neighborhood of the support of v, we have

uDj(v ) (uDjv) 0, as 0,

where Dj = /xj and uDjv is defined in the sense of distribution.


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Using convolution, we have that if f L2p,q(D) Dom() and f =f( 11+ z), then

f f(f ) f

0.The first sequence converges since it follows easily from the Youngs inequal-ity:

f f f.The second sequence converges from

(f ) = 11 +

(f)(1

1 + z).

In other words, the weak maximal closure of is equal to the strong maximal

closure.

(2) C(p,q)(D) with compact support in D is dense in Dom().

We assume that = 0. This follows from the fact that is the maximalclosure, its adjoint is minimal . We extend f to f by setting f = 0 outside

D . Then we claim that f = f L2(Cn) in the distribution sense. To seethis, we have for any v C(p,q1)(D),

(f , v)Cn = (f, v)D = (f, v)D = (f,v)D = (f,v)Cn.

From (1), C(p,q1)(D) is dense in Dom(), we have proved the claim.

We can approximate f by f = f( z1 ) and then regularize, we getf C0 (D). Using again Friedrichs lemma,

f f

(f ) f

0.This shows that the Hilbert space adjoint is the same as the strongminimal extension of .

(3) C1(p,q)(D) Dom() is dense in Dom() Dom().This last part is essentially proved using the fact that f

C1(p,q)(D)

Dom() if and only if f = 0 on bD, being a defining function for D.By this and the fact that

(f,g) = (f,g) +

b

f ,gd = (f,g) +

bf, gd.

We first decompose f by f = f + f where f = f is the complexnormal part and f is the complex tangent component. We first approximate


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f by

(2.4) f() = f

+ f

= f

(

1

1 + z) + f

(

1

1 z).

Then we regularize again as in Friederichs lemma, we obtain

(f() ) (f + f ) 0since the complex normal component is not in the Cauchy data of .

On the other hand, we get

(f() ) (f + f ) 0since f Dom(). This gives that

f() f

(f()

) f(f() ) f

0.

This proves the density lemma.

Using the density lemma, we have, for any f Dom() Dom()f2 f2 + f2 = (f, f) f f,

so that

f f,which implies that has closed range and Ker() = 0.

Now we can prove the L2 weighted existence of -Neumann weightedoperator. We have the following.

Proposition 2.3. LetD a bounded pseudoconvex domain withC2 boundaryinCn. For each1 q n there is a bounded operator N : L2(p,q)(D, ) L2(p,q)(D, ) such that:

(1) f = Nf Nf, for any f L2(p,q)(D, );(2) if f = 0 then u :=

Nf is the solution for u = f such that

u 1qf.

Theorem 2.4 (L2 existence theorem for ). LetD Cn a bounded pseu-doconvex domain. For every f L2(p,q)(D), 1 q n, with f = 0, onecan find u L2(p,q1)(D) such that u = f and

u

e

qf,

where = diam(D).


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Choosing = t|z|2, we get

(2.5) te

t2

u2

1

q f2

.

This proves Hormanders theorem with t = 2.Summing up the previous results and passing to the unweighted L2 spaces,

we have the following result due to Hormander.

Theorem 2.5 (Hormander). Let D Cn a bounded pseudoconvex do-main. Then has closed range and R() = L2(p,q)(D), 1 q n. More-over, there is a bounded operatore N : L2(p,q)(D) L2(p,q)(D) such that

(i) N = N = I on Dom();(ii) = N N , for any L2(p,q)(D, );(iii) if

L2(p,q)(D) such that = 0 then u = N is the canonical

solution to u = and

u

e

q;

(iv) we have the following estimates for N:

Nf eq

2f,

N f

e

qf,

Nf

e

q

f

,

for anyf L2(p,q)(D).Theorem 2.6. Let D be a pseudoconvex domain in Cn. For any f C(p,q)(D), where 0 p n and 1 q < n, such that f = 0 in D,there exists u C(p,q1)(D) satisfying u = f.

From Hormanders theorem, we have

Corollary 2.7. Let D be a pseudoconvex domain in Cn. Then D is adomain of holomorphy.

2.3. Global Regularity for on pseudoconvex domains. If the bound-

ary bD is smooth, we also have the following global boundary regularityresults for .

Theorem 2.8 (Kohn). Let D Cn be a pseudoconvex domain withsmooth boundary bD. For any f C(p,q)(D), where 0 p n and1 q < n, such that f = 0 in D, there exists u C(p,q1)(D) satisfy-ing u = f.


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Proof. For any s > 0, there exists Ts >> 1 such that Nt Ws(D) for allt > Ts. From the Sobolev embedding theorem, there exists uk Ck(D)for each k N. By a Mittag-Leffler procedure, one can extract a solutionu C(D). Theorem 2.9. Let D be a bounded pseudoconvex domain in Cn, n 2.For each 0 p n, 1 q n and t > 0, there exists a bounded operatorNt : L

2(p,q)(D) L2(p,q)(D), such that

Range(Nt) Dom(t). Ntt = tNt = I on Dom(t).For any f L2(p,q)(D), f =

t Ntf t Ntf.

Nt = Nt on Dom(), 1 q n 1,t Nt = Nt

t on Dom(

t ), 2 q n.

The following estimates hold: For any f L2(p,q)(D),

tq Ntf (t) f (t),

tq Ntf (t) f (t),

tq Ntf (t) f (t) .Iff L2(p,q)(D) andf = 0 inD, then for eacht > 0, there exists a solutionut =

t Ntf satisfying ut = f and the estimate

tq ut 2(t) f 2(t) .We have chosen t = 2, where is the diameter of D, to obtain the best

constant for the bound of the -Neumann operator without weights. Our

next theorem gives the regularity for Nt in the Sobolev spaces when t islarge.

Theorem 2.10. Let D be a smooth bounded pseudoconvex domain in Cn,n 2. For every nonnegative integer k, there exists a constant Sk > 0 suchthat the weighted -Neumann operator Nt maps W

k(p,q)(D) boundedly into

itself whenever t > Sk, where 0 p n, 1 q n.An operator is called exactly regular on Wk(p,q)(D), k 0, if it maps the

Sobolev space Wk(p,q)(D) continuously into forms with Wk(D) coefficients.

The following theorem shows that all the related operators of Nt are alsoexactly regular if Nt is exactly regular.

Theorem 2.11. Let D be a smooth bounded pseudoconvex domain in Cn,n 2. For every nonnegative integer k, there exists a constant Sk > 0such that for every t > Sk the operators Nt,

t Nt,

t Nt and

t Nt are

exactly regular on Wk(p,q)(D), where 0 p n, 1 q n. Furthermore,there exists a constant Sk > 0 such that for t > S

k, the weighted Bergman

projection Pt,(p,0) maps Wk(p,0)(D) boundedly into itself.


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3. Boundary regularity for the -Neumann problem

3.1. -Neumann problem on strictly pseudoconvex Lipschitz do-

mains. We are interested in the following questions:

1. Can one solve equation (6.0.1) with a smooth solution u C(p,q1)(D) iff is in C(p,q)(D)?

2. Does the canonical solution

Nf belong to Ws(p,q1)(D) if f is in

Ws(p,q)(D)?

We derive the 1/2-subelliptic estimate for the -Neumann problem oper-ator when is a strictly pseudoconvex Lipschitz domain. We use Ws() todenote the Sobolev space, s R, and by s its norm.

A bounded pseudoconvex Lipschitz domain in Cn is said to have aplurisubharmonic Lipschitz defining function if is a global defining func-

tion in (or on the boundary of b and is plurisubharmonic in . Recallthat a continuous function is plurisubharmonic if it is subharmonic in everycomplex line. Plurisubharmonicity is well-defined for continuous functionsor even upper semicontinuous functions. We next define strictly (or strongly)pseudoconvex domains with Lipschitz boundaries.

Definition 3.1. A bounded Lipschitz domain in Cn is called strictlypseudoconvex (with Lipschitz boundary b) if there exists an exhaustion{} of such that(1) The sequence {} is an increasing sequence of relatively compact sub-sets of and =

.

(2) Each has a C plurisubharmonic defining function such that

is uniformly bounded in and

ni,j=1

2zizj

aiaj c0|a|2 for z U and a Cn,

where U is a neighborhood of bD and c0 > 0 is a constant independent of.(3) There exist positive constants c1, c2 such that c1 || c2 on U,where c1, c2 are independent of .

We also have the following definition.

Definition 3.2. A bounded Lipschitz domain D in Cn is called strictly

pseudoconvex if it has a strictly plurisubharmonic Lipschitz defining func-tion, i.e., there exists a Lipschitz function : Cn R such that is locallya Lipschitz graph and(1) < 0 in D, > 0 outside D and C1 < |d| < C2 on bD almosteverywhere, where C1, C2 are positive constants,(2) C|z|2 is plurisubharmonic for some C > 0 in U where U is aneighborhood ofb.


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Lemma 3.3. LetD be a bounded Lipschitz domain inCn. ThenD is strictlypseudoconvex in the sense of Definition ***2.1.1 and Definition ***2.1.2 are

equivalent.Proof. We first assume that D is pseudoconvex in the sense of Definition2.1.2. We will construct a sequence of subdomains D and . The proof issimilar to the proof of Lemma 0.3. Let D and be similarly defined asin Lemma 0.3. It remains to prove that satisfies (2) in Definition 2.1.1.But this follows easily from the the fact that suitable regularizations of aplurisubharmonic function are plurisubharmonic.

Let D be strictly pseudoconvex in the sense of Definition 2.1.1. Sincethe sequence is uniformly bounded in

1 on each compact subset of D,we have from the Ascoli Theorem that there is a subsequence convergentto some limit function 1(D). Then is a Lipschitz defining functionwhich satisfies (1) and (2) in Definition 2.1.2. Thus the two definitions areequivalent.

Some examples of strictly pseudoconvex domains with Lipschitz bound-aries are given below.

1. Piecewise smooth strictly pseudoconvex domains.

A bounded domain in Cn is said to have a piecewise smooth strictlypseudoconvex boundary defined by C2-differentiable functions if thereexists a finite open covering U1, . . . , U k of an open neighborhood U of and C2 strictly plurisubharmonic functions j : Uj R, j = 1, . . . , k suchthat

(i) U = {x U| for each 1 i k, either x Ui or i(x) < 0},(ii) for 1 i1 < i2 < < i k, the 1-forms di1 , . . . , d il are linearly

independent over R at every point of

v=1 Ui .Then is a strongly pseudoconvex domain with Lipschitz boundary in

the sense of Definition 2.1.2. To see this, first note that is a Lipschitzdomain from the assumption of transversal intersection. Let i be a C

2

strictly plurisubharmonic defining function for i and = {z Cn|i(z) 0 is sufficientlysmall. Then is a plurisubharmonic function. Let W = {z | 0 0 such that

ni,j=1

2zizj

aiaj c0|a|2 for z U and a Cn, 1 k.


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It follows that r0(z) = (z) c0|z|2 is a plurisubharmonic function on Wsince for each 1 k, c0|z|2 is a plurisubharmonic function on U.Thus (z) c0|z|

2

= i,zUi max{i(z) c0|z|2

} = r0(z) is plurisubharmonicon W.

Since || = |i| for some i on the smooth part of, we have || C2a.e. on . To show that || is bounded from below, we note that is Lipschitz and satisfies the exterior cone condition. There exist a finitecovering {V}1k of , a finite set of unit vectors {}1k and c1 > 0such that the inner product , c1 > 0 a.e. for z V, 1 k.t > 0 are pseudoconvex Thus has a strictly plurisubharmonic Lipschitzdefining function and D is a strongly pseudoconvex Lipschitz domain.

2. Strictly convex domains.

Let be a strictly convex domain. By this we mean that there exists a

Lipschitz defining function such that C|z|2

is convex for some C > 0 andC1 || C2 a.e. on b. Since is convex, has Lipschitz boundary.It is easy to see that is strongly pseudoconvex with Lipschitz boundary.

Let be a bounded Lischitz domain. We observe, first of all, thatthe first order system is elliptic in the interior of . More, iff Dom() Dom(), the f W1(, loc). Then the problem aboutthe regularity of -Neumann operator is only on the boundary.

Theorem 3.4. Let a bounded strictly pseudoconvex domain in Cn withLipschitz boundary. Then there exists C > 0 such that3

(3.1)

f

21/2

Cf

2 +

f

2

, f Dom() Dom()

Now, we can state and prove the main result4.

Theorem 3.5. Let be a bounded strictly pseudoconvex domain with Lips-chitz boundary. For any 0 p n, 1 q n 1, the -Neumann operatorN can be extended as a bounded operator N : W

1/2(p,q) () W

1/2(p,q)() and

we have

(3.2) Nf21/2 Cf21/2, f W 12

(p,q)(),

where C is independent on f. Finally, we have

(3.3) N f1/2 + N f1/2 Cf, f L2(p,q)(),

3In fact, we can prove before that, under the same assumptionsZb

|f|2d C`f2 + f2

.

From this, the result of theorem follows.4A first estimate in this sense was given by Kohn [KO], who showed that N fs+1

Csfs, under the assumption of C boundary.


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Proof. We first prove a priori estimates: Suppose that is a strictly pseu-doconvex domain with C2 boundary. For any f C2(p,q)() Dom(), wewill prove the estimate holds.Let be a C2 strictly plurisubharmonic defining function with || = 1on b. By Greens Theorem5 with u = and v = |f|2/2, we have

|f|2dV +

ff dV +

|f|2

2dV =

b

n

|f|22

d,

where v = (ff) + |f|2 and = 0 on b. Since > 0 by the strictlypseudoconvexity we have

|||f|2dV C

b

|f|2d +

|f|2dV

+

ffdV.

Using the fact that f =

f and the Hardy-Littlewood Lemma

6

which givethe following estimate

f21/2 C

|f|2dV +

|f|2dV

,

we have,

f21/2 CQ(f, f) = C(f, f) Cf1/2f1/2which implies

f1/2 Cf1/2.Here Q(f, f) = (f, f) is the energy functional. Finally, substituting f with

Nf, we getN f1/2 CN f1/2 = Cf1/2.

This proves the a priori estimates.

5Let u, v C2(), thenZ

(uv vu) dV =

Zb

u

v

n v

u

n

d.

6We give here a modified version of Hardy-Littlewood Lemma for Sobolev spaces. Let a bounded Lipschitz domain in RN and let (x) the distance function from x tothe boundary b. If u L2()W1(, loc) and there is a constant 0 < < 1 such thatZ

(x)22|u|2dV < ,

then u W(). Furthermore, there is a constant C depending only on such that

u2 C

Z

(x)22|u|2dV +

Z

|u|2dV

.


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From the assumption of strong pseudoconvexity for , there exists a se-quence of strongly pseudoconvex smooth subdomains satisfying condi-

tions (1)-(3) in Definition (***). We have for any f C2

(p,q)() Dom(

),b

|f|2dS C( f 2 + f 2 ),

where C is independent of . From the a priori estimate,

f 212()

C( f 2 + f 2 ),where C is independent of .

Since C1(p,q)() Dom() is dense in Dom() Dom() in the graphnorm f + f . The theorem is proved by an approximationargument.

Assume that is a strictly pseudoconvex bounded domain with C

boundary. Then we have the following theorem of Kohn.

Theorem 3.6. Let be a bounded strictly pseudoconvex domain inCn withC boundary b. Then the -Neumann operator N : Ws() Ws+1(),for every s 0.Proof. We only sketch the proof for s = 12 . If T is a tangential operator,

then T f Dom(), and we havef23/2

|||T f|2dV CT f2 + T f2

C

T f1/2

T f1/2

= C

f1/2

f3/2

.

Then N : W1/2() W3/2() is bounded. Furthermore, form the Sobolev embedding theorem7 we have N : C()

C().

3.2. Regularity for the -Neumann problem on pseudoconvex do-mains. When is still with C boundary, but only pseudoconvex, the-Neumann operator is regular for the following type of domains:

1. If there exists a smoth defining function plurisubharmonic on the bound-ary b. The regularity of N from Ws() to itself was proved by Boas andStraube.

2. If is of finite type in the sense of Kohn or DAngelo. The regularity ofN from Ws() to Ws+ for some > 0 was proved by Kohn and Catlin.

When is still with C boundary, but only pseudoconvex, in general,N is not regular from Ws() in itself. This was proved by Barrett for the

7Sobolev Embedding Theorem: If is a bounded domain in RN with Lipschitzboundary, then there is an embedding Wk() Cm() for an integer 0 m < kN/2.


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worm domain of Diederich and Fornss [DF].

4. Strong Okas lemma and the -Neumann problem

Definition 4.1. An open complex manifold is called Stein if there existsa strictly plurisubharmonic exhaustion function : (, ) such that

(1) is strictly plurisubharmonic on ,(2) For each c R, c = < c is relatively compact in .A well known result in several complex variables is the classical Okas

Lemma.

Lemma 4.2 (Oka). Let be a pseudoconvex domain inCn, n 2, then log is plurisubharmonic where is some distance function from z to the boundary. Every pseudoconvex domain inCn is a Stein manifold.

Proof. Let be a distance function from z to the boundary b. Then log is plurisubharmonic (near the boundary). We can modify to make by adding C|z|2 to be strictly plurisubharmonic in . The theorem is provedby regularization.

If is a relatively compact domain in a Stein manifold, H ormanders L2

existence theorem and Kohns results can all be applied to by using asthe weight function.

Definition 4.3. Let M be a complex hermitian manifold with the metricform . Let be relatively compact pseudoconvex domain in M. We saythat a distance function to the boundary b satisfies the strong Okas

condition if it can be extended from a neighborhood of b to such that satisfies

(4.1) i( log ) c0 in for some constant c0 > 0.

4.1. Strong Okas Lemma and bounded plurisubharmonic exhaus-tion functions. We first study the relation between the strong Okas Lemmaand the existence of bounded strictly plurisubharmonic functions on a pseu-doconvex domain in a complex manifold with C2 boundary.

For a bounded pseudoconvex domain with C2 boundary in Cn or ina Stein manifold, a well known result by Diederich-Fornaess [DF] shows

that there exists a Holder continuous strictly plurisubharmonic exhaustionfunction with Holder exponent 0 < < 1. We first examine some equivalentconditions for the existence of such bounded plurisubharmonic functionsbased on the following simple observation.

Proposition 4.4. Let M be a complex hermitian manifold with metric and let M be a pseudoconvex domain with Lipschitz boundary. Thenthe following two conditions are equivalent:


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(1) There exists some 0 < t0 1 such that

(4.2) i( log ) it0

2 .(2) There exists some 0 < t0 1 such that(4.3) i(t0) 0.

Furthermore, suppose that the distance function satisfies the strong Okacondition (4.1). Then (2) implies(3) There exists some 0 < t0 1 such that for any 0 < t < t0, there existssome constant Ct > 0

(4.4) i(t) Ctt( + i 2

).

In particular, t is a Holder continuous strictly plurisubharmonic exhaus-tion function for .

Proof. If (1) holds, we have

(4.5) i( log ) = i ()

+i

2 it0

2.

The above equation is obviously true for C2 distance function. It is also truefor Lipschitz function since the term is well defined. Condition (2)is equivalent to

i()

+ (1 t0) i

2 0.

Comparing this with (4.5), it is easy to see that (1) and (2) are equivalent.Assume that the distance function satisfies the strong Oka condition(4.1). To see that (2) implies (3), we multiply (4.5) by (1 ) and (4.1) by. Adding the two inequalities, we conclude that, for any 0 1, theinequality

i( log ) = i ()

+i

2 c0 + (1 )t0 i

2

holds. Hence, for any 0 < t < t0, we choose = t such that (1 t)t0 > t.Then

i(

t) = itt(()

+ (1

t)

2

)

= itt(( log ) t 2

)

Cttt( + i 2

)(4.6)

where Ct = min(c0t, (1 t)t0 t). This gives that (2) implies (3).


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Theorem 4.5. Let M be a complex hermitian manifold and let Mbe a pseudoconvex domain with C2 boundary b. Let (x) = d(x, b) be

the distance function to b with respect to the hermitian metric such that satisfies the strong Oka condition (4.1). Then there exists 0 < t0 1 suchthat

i( log ) it0 2

.

Proof. Near a boundary point, we choose a special orthonormal basis w1, , wnfor (1, 0)-forms such that wn =

2 . Let L1, , Ln be its dual. Let a

be any (1, 0)-vector. We decompose a = a + a where a = a, Ln is thecomplex normal component and a is the complex tangential component.We have

( log ), a a= ()

, a a + 2()

, a a

+ ()

, a a + |a|2

2.(4.7)

From (4.1), we have

( log ), a a = ()

, a a c0|a|2.If is C2 up to the boundary, we have for any > 0,

|()

, a a| C( 1|a|2 + |a|

2

2).

Also near the boundary when (z) < , we have

|()

, a a| C

|a|2 C 2

|a|2.Choosing sufficiently small, we have

( log ), a a 12

|a|22

K|a|2

for some large constant K. Multiplying (4.1) by Kc0 and adding it to theabove inequality, we have

(K

c0+ 1)

(

log ), a

a

1

2

|a|2

2.

This proves the theorem with t0 =1

2(Kc0+1)

.

Corollary 4.6. Let, (x) andt0 be the same as in Theorem 1.2. Then for

any 0 < t < t0, the function = t is a strictly plurisubharmonic boundedexhaustion function on .


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The corollary follows immediately from the equivalence of (1) and (3) inLemma *** We have the following Diederich-Fornaess Theorem [DF] and

the recent generalization by Harrington [?].Theorem 4.7. Let M be a pseudoconvex domain with Lipschitzboundary in a Stein manifold M. Then there exists a Lipschitz definingfunction and some number 0 < t < 1 such that = ()t is a strictlyplurisubharmonic bounded exhaustion function on .

Proof. We only prove the case when b is of class C2. Since M is Stein,M can be embedded in CN for some large N. Let (x) = d(x, b) be thedistance function to b with respect to metric induced by the Euclideanmetric in CN. From Okas Lemma, we have i( log ) 0 in a neighbor-hood U of b.

Let be a smooth strictly plurisubharmonic function on M. For any

c0 > 0, we can choose some large > 0 such that

i( log(e)) = ilog + c0where is the metric form induced by the Euclidean metric. Thus thestrong Oka condition (4.1) holds. The theorem follows from Corollary ***.

The case for Lipschitz domain is much more involved and we omit thedetails.

Remark. In the proof of Theorem ***, if is in Cn, we can choose = |z|2.Theorem 4.8. Let be a bounded pseudoconvex domain with Lipschitz

boundary in Cn

. Then the-Neumann operator N (as well as

N,

Nand the Bergman projection P) is bounded on Ws() to Ws() for any0 s < 12t0.Remark. 1. If t0 = 1 in Theorem***, the domain has a plurisubhar-monic defining function in . It is proved in Bonami-Charpentier [?] thatwe can take s = 12 . if has smooth boundary and there exists a plurisub-harmonic defining function on b, it follows from Boas-Straube [?] that the-Neumann operator is bounded in the Sobolev space Ws for all s > 0. Wealso mention that for any > 0, there exists a smooth bounded pseudocon-vex domain in C

n such that the -Neumann operator is not bounded onWs (see the paper by Barrett [?]).

When the complex manifold is Kahler with positive curvature, we havealso the following result of Ohsawa-Sibony [?].

Theorem 4.9. (Ohsawa-Sibony) Let CPn be a pseudoconvex domainwith C2 boundary b and let (x) = d(x, b) be the distance function to bwith the Fubini-Study metric . Then there exists t0 = t0() with 0 < t0 1such that

i(t0) 0.


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This follows easily from Takeuchis Theorem (see [?], also [?])

i( log ) c0

where c0 can be chosen to be equal to12 .

Remarks:

1. The theorems above show that if either the complex manifold haspositive curvature or there is a positive line bundle, then the strong Okaslemma holds. Both theorems do not hold without the positivity conditionon the metric (see the counterexample in [DF] and Theorem 1.2 in [ ?]).

2. If is a Lipschitz bounded pseudoconvex domain in a Stein manifold,it is proved in Demailly [?] that there exists a bounded strictly plurisub-harmonic exhaustion function in (see also Kerzman-Rosay [?] for the C1

case). A more refined arguments yield a Holder continuous strictly plurisub-

harmonic exhaustion function in (see Harrington [?]).It is not known if this is true for pseudoconvex domains with Lipschitz

boundary in CPn. We also remark that strictly plurisubharmonic boundedexhaustion functions might not exist if the Lipschitz boundary (as a graph)condition is dropped (see [DF]).

4.2. Strong Okas lemma and finite type conditions. Let be abounded pseudoconvex domain in Cn and let be the Kahler form. TheOkas lemma characterizes pseudoconvex domains with the condition thatthe distance function satisifes i( log )x c(x) where c(x) 0 forevery x . If we can take c(x) as a postive constant, it is called the strongOkas condition. In this section, we will relate the growth rate of c(x) to

more finer properties of psuedoconvexity and the -Neumnann problem.If is a strongly pseudoconvex Lipschitz domain, there exists a strictly

plurisubharmonic defining function such that

(4.8) i( log ) i ()

C1,in the sense of currents. In fact, we can use (4.12) as definition for strongpseudoconvexity.

Definition 4.10. Let be a bounded pseudoconvex domain in Cn withLipschitz boundary b. The domain is of finite type if there exist adistance function and c > 0, 0 < 10 such that

i( log ) c2

.Definition 4.11. Let be a bounded pseudoconvex domain in Cn withLipschitz boundary b. The domain is said to satisfy condition (C) (forcompactness) if there exist a distance function and c(z) > 0 such that

i( log ) c(z)where where c(z) + as z b


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Theorem 4.12. Let be a bounded pseudoconvex domain inCn with Lips-chitz boundary b. Suppose that is of finite type, in the sense of *** then

the-Neumann operator N on L

2

(p,q)(), where 1 q n 1, satisfiesN : Ws() Ws+2(), <

andf C(f + f), f Dom() Dom().

If satisfies condition (C), then N is compact.

5. The -Neumann problem on Kahler manifolds withnonnegative curvature

Let M be a complex manifold with a hermitian metric and let Mbe a pseudoconvex domain. If M is Stein, the L2 existence theorems for

and the-Neumann problem follow from Hormanders theory. In the casewhen the manifold M is Kahler with negative curvature, the distance func-

tion to a fixed point is a strictly plurisubharmonic function and hence, M isStein (see [?]). In this section we study the L2 theory when the manifold isnot Stein and there is no strictly plurisubharmonic weight function smoothup to the boundary. One has to modify Hormanders weighted method toestablish the L2 theory for .

Suppose that the manifold has positive curvature, like CPn. Then thereexists some distance function for which satisfies the strong Okas con-dition (0.1). Then we can use = log to be the weight function inHormanders theory and study the weighted -Neumann problem (see [?] or[CS]). However, is not continuous up to the boundary. To establish the

L2 theory without weights, we use an idea by Berndtsson-Charpentier [?]and streamlined in [CSW].

Let t be any real number and C2(). Let L2(t) denote the L2 spacewith respect to the weight function et = t and

f(t) f2L2(t) =

|f|2et =

t|f|2.

We use t to denote the adjoint of with respect to the weighted space.Then t =

tt whenever it is defined, where denotes the formal adjointwith respect to the unweighted L2-norm. The norm .2(t) is equivalent to theSobolev norm on a sub-space ofW

t2 () for harmonic functions or solutions

to elliptic equations.

Let be a distance function which satisfies the strong Oka condition (0.1).Introduce the following two asymmetric weighted norms. These new normswill be used to obtain more refined L2 estimates.

For any (p,q)-form f on , we decompose f into complex normal andtangential parts by setting f = (f()#) and f = f f..

The above decomposition is well-defined for any (p,q)-form f supportednear the boundary and can be extended to the whole domain.


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We define the asymmetric weighted norm

|f|2

A = |f

|2

+ |f

|2

||2and its dual norm

|f|2A = |f|2 + |f|2||2.For any t > 0, let L2A(

t) and L2A(t) denote the weighted L2 spaces on

(p,q)-forms defined by the norm

u2L2A(t) =

t|f|2A =

t(|f|2 + |f|2

||2 )

and

u

2L2A(t) = t|f|2A = t(|f|2 + |f|2||2).

Theorem 5.1. LetM be a complex Kahler manifold with nonnegative sec-tional curvature. Let M be a pseudoconvex domain with C2-smoothboundary b. Let (x) = d(x, b) be the distance function such that satisfies the strong Oka condition (0.1). For any f L2A(p,q)(), where0 p n and1 q n, such thatf = 0 in, there exists u L2(p,q1)()satisfying u = f and

|u|2 C

|f|2A.

Proof. We first show that for any t > 0 and any (p,q)-form f L2A(t),0 p n and 1 q n, such that f = 0 in , there exists u L

2(p,q1)(

t

)satisfying u = f and

(5.1) u2L2(t) C

tf2L2

A(t).

Let = t log , where t > 0. By the Bochner-Hormander-Kohn-Morreyformula with weight function = t log , for any (p,q)-form g Dom() Dom() with q 1 on , we have

g2 + g2 = g2 + (g, g) + ((i)g, g) +

b(i)g, ge,

where

u

2 = nj=1 |DLju|2e, {L1,...,Ln} is a local unitary frame ofT(1,0)() and is a curvature form. From our assumption, we have

(u, u) 0.Since is pseudoconvex, we have that for any (p,q)-form g Dom()

Dom(t ),

g2(t) + t g2(t) ((i)g, g)(t) CtgL2A(t).


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Let (i) denote the dual norm for (p,q)-forms induced by i. Itfollows that for any f L2A(t), there exist u L2(t) satisfying u = fand(5.2)

|u|2t

|u|2(i)t 1

Ct

|u|2At.

This proves (4.12). To get rid of t, we use an argument used in [?]. Letf L2(p,q)(). For any t > 0 , there exists u L2(p,q1)(t) satisfying u = f,such that u is perpendicular to Ker() in L2(t) and u satisfies (4.13).

Consider v = ut. Then v Ker() in L2(2t). It follows from (4.13)that the following holds:

(5.3)

|u|2 =

|v|22t 12C0t

|v |2A

2t.

Since(5.4) |v |2

A2t C(|u|2A + 2t2|u|2),

choosing t sufficiently small and substituting (4.15) into (4.14), one obtains

|u|2 Ct

|u|2A.

This proves the theorem.

Theorem 5.2. LetM be a complex Kahler manifold with nonnegative sec-tional curvature. Let M be a pseudoconvex domain with C2-smoothboundary b. Let (x) = d(x, b) be the distance function such that sat-isfies the strong Oka condition (0.1). Then

(p,q)has closed range and the

-Neumann operatorN(p,q) : L2(p,q)() L2(p,q)() exists for everyp, q such

that 0 p n, 0 q n. Moreover, for any f L2(p,q)(), we havef = N(p,q)f N(p,q)f, 1 q n 1.f = N(p,0)f P f, q = 0,

where P is the orthogonal projection from L2(p,0)() onto L2(p,0)() Ker()

and

N(p,0) = N2(p,1).

Furthermore, there exists 0 < t0 1 such that the -Neumann operatorN, N, N and the Bergman projection P are exactly regular on Ws

(p,q)()

for 0 s < 12t0 with respect to the Ws()-Sobolev norms.The L2-existence theorem for the -Neumann operator N on follows

from the L2-existence of the solution u for the -equation proved in Theorem2.1.

To show that N is regular in Ws for s < 12t0, let t = 2s in the Bochner-Hormander-Kohn-Morrey formula with weight function = t log . After


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rearranging terms, we have for any (p,q)-form g Dom() Dom(),

g

2(t) +

g

2(t)

2

(g, g(t))

= g2(t) + (g, g)(t) ((i(t)g, g).(see Proposition 3.1 in [CSW]). Since for any > 0 we have

|2(g, g(t))| 1

g2(t) + t2

g

2(t),

choosing small, we have from Corollary 4.6 that

g2(t) + g2(t) Ct(g2(t) + g

2(t))

Ctg2(t).

The rest of the proof is similar to the proof of Theorem 2 in [CSW], andwe omit the details. When Cn, the Sobolev regularity for N andthe Bergman projection have been obtained earlier in [?] (see also [?]).

Remarks: It is still not known if Theorems ***2.1 or 2.2 hold for pseudo-convex domains with C1 or Lipschitz boundary.

5.1. The -Neumann problem on domains in CPn. We want to studyregularity and solvability ofon a complex manifold and not only a boundeddomain in Cn. First of all, observe that if we consider all Cn (as a complexmanifold), because it is unbounded we can have a L2 theory as before. How-ever Cn can be covered by balls of radius R, for R which tends to infinity. So

it is possible to have a L2 theory locally on each in Cn sufficiently large.Then, iff L2(p,q)(, loc) and f = 0, for q 1, there is u L2(p,q1)(, loc)solution to u = f in Cn. Moreover, when f C(p,q)(, loc), f = 0 and is pseudoconvex, then there is u C(p,q1)(, loc) solution to u = f (andin particular is a domain of holomorphy).

We racall that X is a Stein manifold if and only if it is a complex sub-manifold ofCN, for some N. This is the same as to have a strictly plurisub-harmonic exhaustion function . Then, on a Stein manifold existence of-Neumann operator is guaranteed by a result of Boas and Straube [BS2].

IfX = CPn, then it is not Stein. In fact, on a compact complex manifold

there arent no constant strictly plurisubhramonic exhaustion functions. Weare interested to study this particular case.

Let CPn be a pseudoconvex domain, i.e. for any p b there isU p such that U is pseudoconvex. Let be a C2 defining function for. For any (0, q)-form f we have

f2 + f2 = f2 + (R0,qf, f) +

bif,fd.


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Here, R0,q is the curvature form. In CPn, this coincides with the Ricci

curvature, so

R0,q = (n + 1)q > 0Then we have

f2 + f2 > (n + 1)qf2.When f is a (n, q)-form, Rn,q 0, then

f2 + f2 = f2 +

bif,fd.

If we set (z) = dist(z, b) we have i( log ) c, where c > 0, theFubini-Study metric ofCPn. Then is Stein.

As in the case of bounded domain in Cn, we want to start from a weighted

L2 theory. Set then t = t log , t > 0. Then it Ct , wherec > 0, t > 0. In particular we can study the L2 weighted space with weightt (in this case e

t = t) and we obtain, for all t > 0 the following estimate

f2t + t f2t 1

Ctf2t .

We have the following result.

Lemma 5.3. Let CPn a bounded doamin with C2 boundary. Thenthere is a 0 < t0 1 such thatt0 is plurisubharmonic, i.e. i(t0) 0.Furthermore, for any 0 < t < t0 we have

(5.5) i(t) Ct t + 2 .Theorem 5.4. Let CPn a bounded doamin with C2 boundary. Thenthere is a bounded operator Np,q : L

2(p,q)() L2(p,q)() for any q 1, such

that:(1) N = N = I on Dom();(2) N, N, N andB = IN, the Bergman projection, are all boundedfrom Ws() to itself, for any 0 < s < t0/2.

Proof. We already know that

t

|f|2 +

|f

|2

dV C t

|f

|2dV.

By the estimate (5.5) we get

t0|f|2dV f2Wt0/2()

.

Applying this two fact with N f and Nf instead of f we obtain thedesired regularities.


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6. Tangential Cauchy-Riemann equations and the BochnerMartinelli kernel

7. CR manifolds and tangential Cauchy-Riemann equations

7.1. L2 theory for b on strongly pseudoconvex CR manifolds. Asan example of CR manifolds, we may consider real orientable hypersurfacesin Cn. Let Cn be a bounded domain with defining function . Itsboundary M = b = {z Cn : (z) = 0} with condition |(z)| = 0,z M, is a CR manifold of dimension 2n 1. Its CR structure is generatedby

Lj :=

nh=1

ahj

zh, j = 1, . . . , n 1,

purely tangential, i.e. Lj = 0, j = 1, . . . , n1. The system overlineLj 1 j n 1 are the tangential Cauchy-Riemann equations on M.If f is a holomoprhic function on and f C1(), then f satisfies(7.1) Ljf = 0, on M, j = 1, , n 1Any function satisfying (5.1) is called a CR function.

For n = 2, let M = {(z1, z2) C2 : z2 = |z|2}, is the boundary of theunbounded ball. If we parametrize M by (z1, t) and z2 = t + i|z|2, we have

L =

z1 iz1

t.

This is the Lewy operator. In [LE] he proved that the tangential Cauchy-Riemann equation Lu = f is not solvable for a large class of functions f.

When n > 2, we have the system

Lju = fj, j = 1, . . . , n 1with the compatibility conditions

Ljfh Lhfj = Lj Lhu LhLj u = [Lj, Lh]u = cjhLu = cjhf,where [Lj, Lh] = c

jhL, which is given by condition (iii) in the definition of

CR manifolds. Defining b = |M the previous system can be rewritten asbu = fbf = 0

.

More precisely, if we denoteb = d|T

0,1(M), then the sequence

. . . 0,q1(M) b 0,q(M) b 0,q+1(M) b . . .is a complex, i.e. 2b = 0. b is the tangential Cauchy-Riemann operator.

We may equip CT M by the induced metric such that T1,0(M)T0,1(M),and define b as the adjoint of b. So we have the b-Laplacian

b := bb + bb : 0,q(M) 0,q(M).


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It is well know that b is not elliptic. IfN(M) denotes the dual ofT1,0(M)

T0,1(M) in CT(M), we have

CTzM = T1,0z (M) T0,1z (M) Nz(M).In particular, dimR N(M) = k, it is spanned by a vector field T which canbe chosen to be purely imaginary, i.e. T = T.

We give an important definition.

Definition 7.1. Let (M, T1,0(M)) be a CR manifold of dimension 2n 1.Let T N(M) and its dual, i.e. (T) = 1, (T1,0(M) T0,1(M)) = 0.Then

z,(L, L) = z(L, L

)() := [L, L], z,for every L, L T1,0(M) is the Levi form in the direction .We say that the Levi form satisfies condition Z(q) (0 q n 1) in thedirection , if z, has at least n

q positive eigenvalues or at least q + 1

negative eigenvalues.We say that the Levi form satisfies condition Y(q) if it satisfies conditionZ(q) for all directions .

Since there are only two directions for , it is easy to see that conditionY(q) is equivalent to the following one:

there are max(n q, q + 1) eigenvalues with the same signor

there are min(n q, q + 1) pairs of eigenvalues with opposite signs.

Furthermore, condition Z(q) is equivalent to the estimate

f21/2 = M

|f|2d Cbf2 + bf2 .Theorem 7.2. Let (M, T1,0(M)) be a compact CR manifold of dimension2n 1. Suppose that M satisfies condition Y(q) for some 0 q n 1.Then we have

f21/2 Cbf2 + bf2 + f2 ,

for everyf C1(0,q)(M). Furthermore, there is an operator Gb : L2(0,q)(M) L2(0,q)(M) such that

(i) bGb = Gbb = I Hb, where Hb : L2(0,q)(M) Hb(0,q) = Ker(b) ={ L2(0,q)(M) : b = b = 0} is the projection;(ii) Hb(0,q) is finite dimensional;(iii) Gb is compact and Gbs+1 Cfs.Proof. We give an idea of the proof only. Suppose that {L1, . . . , Ln, L1, . . . , Ln, T}is an orthonormal basis for CT M. For any f C1(0,q)(M) we have

bf2 + bf2 = f2 + (cjhT fj, fh) + Off + f2 ,


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where f2 = n1j,h=1 Lj fh2 and[Lj , Lh] = cjhT mod T

1,0(M)

T0,1(M).

Now the Y(q) condition assures that, if we put Lj = Xj + iYj, then the setof vector fields {Xj, Yj : 1 j n 1} satisfies a finite type condition oftype 2, i.e. {Xj, Yj, [Xj, Yj ]} generate all the tangent space of M. So wehave

(7.2) Xf2 + f2 = f2 + f2 + f2 Cf21/2,where f2 = n1j,h=1 Ljfh2. Furthermore we may prove that(7.3) Xf2 + |(T f , f )| Cbf2 + bf2 + f2 .This implies that

f1

2 Cbf2 + bf2 + f2 .

7.2. L2 theory for b on boundary of pseudoconvex domains. Let be a bounded pseudoconvex domain in Cn with smooth boundary b. Westudy the global L2 existence theorem for b on b. Let f be a (p,q)-formwith L2 coefficients, where 0 p n and 1 q < n. We study the globalsolvability of

(7.4) bu = f in b.

When q < n 1, it is easy to see that for equation (5.1) to be solvable, fmust satisfy the condition

(7.5) bf = 0 in b.

When q = n 1, condition (5.2) is void and the compatibility condition isgiven by

(7.6)

b

f h = 0, h L2(np,0) Ker(b).

In fact, for all 1 q n 1, f must satisfy the compatibility condition(7.7)

b

f h = 0, h L2(np,n1q) Ker(b).

Lemma 7.3. For 1 q < n 1, if f satisfies (5.4), then f is b-closed.

Proof. From the assumption that q n 2, we set h = v for some smoothv C(np,n2q)() Ker(), then it is easy to see that

bf h =

b

f v = (1)q

bbf v = 0.

Since v is any smooth form in , we have that bf = 0 on b.

We have the following L2 existence theorem.


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Theorem 7.4. Let be a bounded pseudoconvex domain inCn with smoothboundary M. For every Ws(p,q)(M), where 0 p n, 1 q n 2and s 0, such that(7.8) b = 0 on M,

there exists u Ws(p,q1)(M) satisfying bu = on M.When q = n 1, Ws(p,n1)(M) and satisfies

(7.9)

M

= 0, C(np,0)(M) Ker(b),

there exists u Ws(p,n2)(M) satisfying bu = on M.Corollary 7.5. Let be a bounded pseudoconvex domain inCn with smoothboundary M. Thenb : L

2(p,q1)(M) L2(p,q)(M), 0 p n, 1 q n1,

has closed range in L2.

We have the following strong Hodge decomposition theorem for b. Let

S(p,0) : L2(p,0)(M) Ker(b)

S(p,n1) : L2(p,0)(M) Ker(b ).

The projection S(0,0) is the Szego projection operator.

Corollary 7.6. Let M be the boundary of a smooth bounded pseudoconvexdomain in Cn, n 2. Then for any 0 p n, 0 q n 1, thereexists a linear operator Gb : L

2(p,q)(M) L2(p,q)(M) such that

Gb is bounded and R(Gb) Dom(b).For any L2(p,q)(M), we have = b

b Gb b bGb, if 1 q n 2,

= b bGb S(p,0), if q = 0,

= bb Gb S(p,n1), if q = n 1.

If L2(p,q)(M) with b = 0, where 1 q n 2 or L2(p,n1)(M)with S(p,n1) = 0, then = b

b Gb.

The solution u = b Gb in (4) is called the canonical solution, i.e., the

unique solution orthogonal to Ker(b).

Theorem 7.7. Let D be a bounded pseudoconvex Lipschitz domain in Cn

with boundary bD. We assume thatD has a Lipschitz defining function which is plurisubharmonic in D. For every L2(p,q)(bD), where 0 p n, 1 q < n 1, such that

b = 0,

there exists u L2(p,q1)(bD) satisfying bu = in bD.


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Furthermore, there exists a constant C depending only on the diameterand the Lipschitz constant of D but is independent of such that

u L2(p,q1)(bD) C L2(p,q)(bD) .Whenq = n1, we assume thatD has a Lipschitz defining function whichis plurisubharmonic in a neighborhood of D. For every L2(p,n1)(bD)such that satisfies

M = 0, for every C(np,0)(D) Ker(),

the same conclusion holds.

Corollary 7.8. LetD be a bounded Lipschitz domain with a defining func-tion which is plurisubharmonic in a neighborhood of D. The b operatorhas closed range in L2(p,q)(bD) for every 0

p

n, 1

q

n

1.

Since b is a closed, densely defined operator from one Hilbert space toanother, it has a Hilbert space adjoint, denoted by b . The

b operator is

also a closed, densely defined operator (see Riesz-Nagy [?]). Let Ker(b)and Ker(b ) denote the kernels of b and

b respectively. Then Ker(b) and

Ker(b ) are both closed subspaces. Let R(b) and R(b ) denote the range

of b and b respectively. We also have the following consequence from the

Main Theorem. Let the space of harmonic forms Hp,qb (bD) be defined byHp,qb (bD) = { Dom(b) Dom(b ) | b = b = 0}.Corollary 7.9. Corollary 2 (Hodge theory for b)Let D be a bounded Lip-schitz domain with a defining function which is plurisubharmonic in a

neighborhood of D. The harmonic formsHp,qb (bD) = {0} for 0 p n, 1 q < n 1.

We have the following strong Hodge decomposition:

L2(p,0)(bD) = Ker(b) R(b ), 0 p n,L2(p,q)(bD) = R(b) R(b ), 0 p n, 1 q < n 1,L2(p,n1)(bD) = R(b) Ker(b ), 0 p n.

In particular, our results can be applied to the boundary of a stronglypseudoconvex Lipschitz domain or the boundary of any convex Lipschitzdomain. When D is a smooth strongly pseudoconvex CR manifold, the bcomplex has been studied in the work of Kohn [?], Kohn-Rossi [?], Folland-Stein [?] and Rothschild-Stein [?] for 1 q < n 1. Henkin [?] has derivedthe kernel solutions for b on strongly pseudoconvex boundaries in C

n for1 q n 1. There are also numerous other results on b on boundariesof smooth bounded strongly pseudoconvex domains. We refer the reader tothe book by Boggess [?] or Chen-Shaw [?] and the references therein.

When D is the boundary of a pseudoconvex domain with smooth bound-ary, the L2 and Sobolev estimates for b have been obtained by Shaw in


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[?] for 1 q < n 1 and for q = n 1 in Boas-Shaw [?] (See also Kohn[?]). The proof of the L2 existence for b in these papers depends on the (at

least C4

) smoothness of the boundary. Our results are the first for thebcomplex on C1 or Lipschitz boundaries.

The b operator corresponds to a system of first order differential equa-tions with variable coefficients. Since it is not possible to define the mul-tiplication of two distributions in general, one cannot simply define b onnonsmooth manifolds in the sense of distributions. However, is alwayswell defined on any domain D in Cn, regardless of the smoothness of theboundary bD. When D is a bounded Lipschitz domain, the b complex canbe defined via duality by integration by parts and , since Stokes theoremstill holds on any bounded Lipschitz domain. Our method can obviously beextended to the boundaries of Lipschitz domains in complex manifolds. Forthe sake of simplicity and clarity in presentation, we will only stay in Cn.

The L2

estimates and existence theorems of b have many applications.The closed range property is also closely related to the embeddability ofCR structures. We note that b does not have closed range locally in thetop degree case (when q = n 1) on any smooth strongly pseudoconvexboundaries, since it corresponds to the Lewy operator. Moreover, it doesnot always have closed range for abstract smooth strongly pseudoconvex CRstructures. Burns [?] has showed that, for Rossis example [?] of a nonem-beddable strongly pseudoconvex CR structure of real dimension three, theb operator does not have closed range in either the L

2 or C sense.

7.3. Nonisotropic Sobolev and Holder spaces. Let (M, T1,0(M)) a CRmanifolds of dimension 2n+k and let

{L1, . . . , Ln, L1, . . . , Ln, T1, . . . , T k

}an

horthonormal frame for CT M. Define

Xf2 :=n

j=1

Ljf2 + Lj f2 .The nonisotropic Sobolev space W1 (M) is defined as the space of functionu L2(M) such that the W1 (M) norm

u2W1

:= Xu2 + u2

is finite. The space W1,p (M) is defined in the same way, only substitutingthe L2 by the Lp norm.

For any integer m > 1, the space Wm,p is defined inductively as the space

of function u such that Lju, Lju Wm1,p (M), for any j = 1, . . . , n. Thus,u Wm,p (M) if and only if u Lp(M) and X1 Xmu Lp(M) for anyXj T1,0(M) T0,1(M), 1 j m. The space W1,p (M) is defined asthe dual space of W1,p

(M), where 1/p + 1/p = 1 or, equivalently, as the

space of function f =N

j=1 Xj uj, Xj T1,0(M) T0,1(M), uj Lp(M).


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The nonisotropic Holder space C (M), 0 < < 1, is defined as follow:u C (M) if and only if

sup(t)

|u((t)) u((0))||t| C < ,

where is any C1 curve in M such that (t) T1,0(M) T0,1(M) and|(t)| 1. We define for any m > 1 the space Cm, (M) inductively asbefore.

As in the previous section, we have the following result.

Theorem 7.10. Let(M, T1,0(M)) be a compact CR manifold of dimension2n + k, k > 0. Suppose that M satisfies condition Y(q) for some 0 q n.Then, there is a compact operator G

b: W1,p

(M)

W1,p

(M) such that

bGb = Gbb = I Hb,where Hb : L

2(0,q)(M) Ker(b) is the projection as before. Furthermore, if

f Cm, (M) andu = Gbf thenu Cm+2, (M), for anym = 1, 0, 1, . . .,0 < < 1. Finally, when f Wm,p (M) then u Wm+2,p (M), 1 < p < .

The main consequence of the previous theorem is the fact that the bproblem is solvable in nonisotropic Sobolev and Holder spaces, as stated bythe following corollary.

Corollary 7.11. Let f

L2(0,q)(M), such that bf = 0 and f

Ker(b).

Then the function u = b Gbf is in L2(0,q+1) and is a solution to bu = f.

Furthermore if f Cm, (M) then the solution u is in Cm+1, (M).What we stated in the previous results, may be written in terms of L2

estimates. In fact, let

Qb(f, f) := bf2 + bf2 + f2 = ((b + I)f, f)denoting the energy functional. Then

|(T f , f )| + Xf2 CQb(f, f),

from which f2W1(M) CQb(f, f) = C((b + I)f, f)

for any fKer(b). Then(7.10) fW1

(M) CbfW1

(M).

In the same way for estimates in the other nonisotropic spaces.


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7.4. Campanato Space. We want to define another important class offunction spaces, the so called Campanatos spaces. Let 0 < < 1, 1 p and x0 M. Let a measure on M and let B(x0) the ball of radius and center in x0. We say that a function f is in the space Lp, (x0) if andonly if there is 0 > 0 such that f Lp(B0(x0)) and there is a constantC > 0 such that

1

(B(x0))

B(x0)

|f f|B(x0)|p1/p

C, 0 < < 0.

This is the pointwise Holder spaces in the Lp sense (for p = , this is thepointwise nonisotropic Holder spaces).

To define the higher order spaces Lp,m+ (x0), m N {1}, we needto proceed in the following way. We may choose coordinates x = (z, t) =

(x1, y1, . . . , xn, yn, t1, . . . , tk) called normal coordinates, such that

Lj =

zj

x0

, Th =

th

x0

, 1 j n, 1 h k.

We say that a polynomial P(z, t) in (z, t) is ofdegree m if

P(z, t) =

|I|+|J|+2|L|m

aIJ LzIzJtL,

where I ,J ,L are multiindices.Under this assumption we say that f Lp,m+ (x0) if and only if f is a

Lp function and there is 0 > 0 and a polynomial of degree m, Px0m , such

that 1

(B(x0))

B(x0)

|f Px0m |p1/p

Cm+, 0 < < 0.

The relation between this space and the ones introduced in the previoussection is stated in the the following lemma.

Lemma 7.12 (Campanato Embedding). For any0 < < 1 and1 p we have

Lp,m+ (M) = Cm, (M).Before proving regularity of b on Campanato spaces, we want first to

scale the equation for the b to the unit ball B1(x0). For any > 0, we setx = (z,2t), the dilation. Let Lj the operator defined by

L

j(f(x)) = (Lj f)(x), x B1(x0).Then we have

Lj L0j =

zj+

k=1

aj

t.


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Now, the b and b associated to the CR structure T

1,0 generated by Lj are

such that

bu

0

bu

bu 0b u as 0.So, ifM satisfies condition Y(q), we can restate all the results of the sectionfor the scaled situation, i.e for the CR manifold M which approaches M,obtain again the same kind of estimates. This leads us to the followingresult.

Proposition 7.13. If f L2, (x0) and u W1 (B0(x0)) is a solution tobu = f in B0(x0)), then u L2,2+ (x0).Proof. We will sketch the idea of the proof.

(1) It sufficees to show that there exists a polynomial P of degree 2 innormal coordinates (z, t) such that

1

B(x0)

B(x0)

|u P|21/2

C2+, 0 < < 0.

Furthermore, we can discretize the estimates, i.e., it suffices to prove for = rm, m = 0, 1, 2, . . ., we get

1

Brm(x0)

Brm(x0)

|u P|21/2

Cr m(2+), 0 < r < r0.

(2) Given > 0, if f is small, i.e. f2 < , for a given , then there is afunction h such that 0b h = 0 and

1B(x0)

B(x0)

|u h|21/2 < , 0 < < 0.Then, h is of the form h = P + O(3), where P is a polynomial of degree 2.

This is proved by the compactness arguments since we already know thatb is subelliptic in the interior. Thus we have the estimates for m = 0. Theestimates for m = 1, 2, are obtained by scaling and induction. 7.5. The Tangential Cauchy-Riemann Complex. Let M be a hyper-surface in a complex manifold. The complex on , when restricted toM, induces the tangential Cauchy-Riemann complex, or the b complex. Infact, the tangential Cauchy-Riemann complex can be formulated on any CRmanifold. There are two different approaches in this setting. One way isto define the tangential Cauchy-Riemann complex intrinsically on any ab-stract CR manifold itself without referring to the ambient space. On theother hand, if the CR manifold is sitting in a complex manifold, the tan-gential Cauchy-Riemann complex can also be defined extrinsically via theambient complex structure.

First, we assume that M is a smooth hypersurface in , and let r bea defining function for M. In some open neighborhood U of M, let Ip,q,


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0 p,q n, be the ideal in p,q() such that at each point z U the fiberIp,qz is generated by r and r, namely, each element in the fiber I

p,qz can be

expressed in the formrH1 + r H2,

where H1 is a smooth (p,q)-form and H2 is a smooth (p,q 1)-form. Denoteby p,q()|M and Ip,q|M the restriction of p,q() and Ip,q respectively toM.

We define the quotient bundle

p,q(M) = p,q()|M/Ip,q|M.We denote by Ep,q the space of smooth sections of p,q(M) over M, i.e.,Ep,q(M) = (M, p,q(M)). Let denote the following map(7.11) : p,q()

p,q(M),

where is obtained by first restricting a (p,q)-form in to M, thenprojecting the restriction to p,q(M). One should note that p,q(M) isnot intrinsic to M, i.e., p,q(M) is not a subspace of the exterior algebragenerated by the complexified cotangent bundle of M. This is due to thefact that r is not orthogonal to the cotangent bundle of M.

The tangential Cauchy-Riemann operator

b : Ep,q(M) Ep,q+1(M)is now defined as follows: For any Ep,q(M), pick a smooth (p,q)-form 1in that satisfies 1 = . Then, b is defined to be 1 in Ep,q+1(M).If 2 is another (p,q)-form in such that 2 = , then

1 2 = rg + r h,for some (p,q)-form g and (p,q 1)-form h. It follows that

(1 2) = rg + r g r h,and hence,

(1 2) = 0.Thus, the definition of b is independent of the choice of 1. Since

2= 0,

we have 2b = 0 and the following boundary complex

0 E

p,0(M)b

Ep,1(M)

b

b

Ep,n1(M)

0.

For the intrinsic approach, we will assume that (M, T1,0(M)) is an ori-entable CR manifold of real dimension 2n 1 with n 2. A real smoothmanifold M is said to be orientable if there exists a nonvanishing top degreeform on M.

We define the quotient bundle

E = CT(M)/T0,1(M).


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Then E is a (holomorphic) vector bundle of complex dimension n. IfM is areal hypersurface in a complex manifold , then E is isomorphic to T1,0()

restricted to M.We define the holomorphic p-forms on E by p. Define the vector bundleCp,q(M), 0 p n, 0 q n 1, by

Cp,q(M) = p qT0,1(M) = Cq(M, p).It follows that Cp,q(M) defined in this way is intrinsic to M. Denoteby Cp,q the space of smooth sections of Cp,q(M) over M, i.e., Cp,q(M) =(M, Cp,q(M)). We define the operator

b : Cp,q(M) Cp,q+1(M)as follows: If Cp,0, b is defined by

b, L

= L

for any section L of T0,1(M). Then b is extended to Cp,q(M) for q > 0 asa derivation. Namely, if Cp,q(M), we define

b, (L1 Lq+1)

=1

q + 1

q+1j=1

(1)j+1Lj, (L1 Lj Lq+1)+

i


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necessary that

bf = 0.

Let L1, , Ln1 be a local basis for smooth sections ofT1,0(M) over someopen subset U M, so L1, , Ln1 is a local basis for T0,1(M) over U.Next we choose a local section T ofCT(M) such that L1, , Ln1, L1, , Ln1and T span CT(M) over U. We may assume that T is purely imaginary.

Definition 7.14. The Hermitian matrix (cij )n1i,j=1 defined by

(7.13) [Li, Lj] = cijT, mod (T1,0(U) T0,1(U))

is called the Levi form associated with the given CR structure.

The Levi matrix (cij ) clearly depends on the choices of L1, , Ln1 andT. However, the number of nonzero eigenvalues and the absolute value of thesignature of (cij ) at each point are independent of the choices ofL1, , Ln1and T. Hence, after changing T to T, it makes sense to consider positivedefiniteness of the matrix (cij ).

Definition 7.15. The CR structure is called pseudoconvex at p M if thematrix (cij (p)) is positive semidefinite after an appropriate choice of T. Itis called strictly pseudoconvex at p

M if the matrix (cij(p)) is positive

definite. If the CR structure is (strictly) pseudoconvex at every point ofM, then M is called a (strictly) pseudoconvex CR manifold. If the Leviform vanishes completely on an open set U M, i.e., cij = 0 on U for1 i, j n 1, M is called Levi flat.

Theorem 7.16. LetD Cn, n 2, be a bounded domain with C bound-ary. Then D is (strictly) pseudoconvex if and only if M = bD is a (strictly)pseudoconvex CR manifold.

Proof. Let r be a C defining function for D, and let p bD. We mayassume that (r/zn)(p) = 0. Hence,

Lk =r

zn

zk r

zk

zn, for k = 1, , n 1,

form a local basis for the tangential type (1, 0) vector fields near p on theboundary. IfL =

nj=1 aj (/zj) is a tangential type (1, 0) vector field near

p, then we haven

j=1 aj(r/zj) = 0 on bD and L = (r/zn)1n1

j=1 ajLj


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on bD. Hence, if we let = r r, we obtainn1

i,j=1cij aiaj =

n1i,j=1

, [Li, Lj ]aiaj

=n1

i,j=1

(Li, Lj Lj, Li 2d,Li Lj)aiaj

=

n1i,j=1

4r,Li Ljaiaj

= 4

r

zn

2

r,L L

= 4 rzn 2 n

i,j=1

2r

zizj aiaj ,

which gives the desired equivalence between these two definitions. Thisproves the theorem.

We note that, locally, a CR manifold in Cn is pseudoconvex if and onlyif it is the boundary of a smooth pseudoconvex domain from one side.

Lemma 7.17. Any compact strongly pseudoconvexCR manifold(M, T1,0(M))is orientable.

Proof. Locally, let , 1, , n1 be the one forms dual to T, L1, , Ln1which are defined as above. The vector field T is chosen so that the Leviform is positive definite. Then we consider the following 2n 1 form

1 1 n1 n1.It is not hard to see that the 2n1 form generated by other bases will differonly by a positive function. Hence, a partition of unity argument will givethe desired nowhere vanishing 2n 1 form on M, and the lemma is proved.

7.6. Folland-Stein-Tanaka formula for b. Let M be a strongly pseudo-convex CR manifold with the pseudohermitian Levi metric. The b complex

b : C0,q(M) C0,q+1(M), 0 q n 1.The formal adjoint of b under the L

2 norm, denoted by b, is defined bythe requirement that

(bf, g) = (f, bg) 1for compactly supported smooth forms.


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Proposition 7.18. Lete1, , en1 be an orthonormal frame field forT1,0(M)in an open neighborhood in M and let w1, , wn1 be its dual. Then

b =n1j=1

wj ej . 2

Proposition 7.19. Let M be a strongly pseudoconvex CR manifold withthe pseudohermitian Levi metric. Let e1, , en1 be an orthonormal framefield forT1,0(M) in an open neighborhood in M and let w1, , wn1 be itsdual. Then

b = n1j=1

(ej)ej . 3

Proof. Let be a (0, q)-form and be a (0, q + 1)-form on M. We have

b, = , b +

4where is a (0,1)-form defined by = j wj with (ej) = , ej and is the function defined to be

=n1

j

ejj.

Let be a (2n 2)-form defined by

=n1

j

j ej dV .

We first prove the following claim:

d() = dV. 5To see this, let X =

j j ej be the dual of . Let AX be the (1,1) tensor

on M defined byAX = LX X .

Since dV = 0, we haveLX dV = AX dV.

Let 1, , 2n1 be a basis for TxM. Since AX is a derivative which mapsevery function into zero, we have

LX dV(1, , 2n1) = AX dV(1, , 2n1)= AX (dV(1,

, 2n1))

j dV(1, , AX j, , 2n1)=

j

dV(1, , AX j, , 2n1)

= TraceAX dV(1, , 2n1)Thus we have

LX dV = TraceAX dV. 6


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Recall the torsion *** formula

AX Y = LX Y

X Y =

YX

T(X, Y).

7

where T is the torsion tensor of . Now we choose a special frame forCTx(M) with ej = j + in+j1, ej , where j = 1, , n 1 and 2n1 = ,the basic vector field,. Then we have

T(ej , ek) = (ei, ek) = 1i

jkand

T(Y, ) T1,0 + T0,1for all Y T1,0 + T0,1. Thus the torsion term in (7) has no contributionwhen taking trace of AX . Also note that X Y preserves types for X andY in T1,0 + T0,1 and = 0. We have

TraceAX = ejX, ej =

and

TraceAX dV = dV. 8

For each vector field X, the divergence formula is defined to be

(div)XdV = LX dV = d(XdV), 9we have proved from (6)-(9) that

dV = LX dV = d().

Integrating with respect to the volume element dV in (4), we have provedthe proposition.

We define the b-Laplacian

b = bb + bb.

Theorem 7.20. Folland-Stein-Tanaka formula for b Let (M, T1,0(M) be

a strongly pseudoconvex CR manifold and ba a basic field. Let e1, , en1be an orthonormal frame field for T1,0(M) in an open neighborhood in Mand let w1, , wn1 be its dual.

b = j

2ej ej+

q

i

i,j

wk(ej)Kej ek i,j

wk(ej )Rej ek , 4

b = j 2ejej (n q 1)i i,j wk(ej)Kej ek , 5

b = n q 1n 1 j

2ej ej

q

n 1 j

2ejej

i,j

wk(ej)Kej ek

n q 1n 1 i,j

wk(ej)Rej ek

6

where2XY = XY XY


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is the second covariant differential and

RXY =

X

Y +

Y

X

[X,Y] =

2

Y X 2

XY

is the curvature tensor, both extended C-linearly to (p,q)-forms, K is thecurvature for the holomorphic vector bundle p.

Proof.

bb = j

(ej)ej (

k

wk ek)

= j,k

(ej)w

k ejek=

j

ejej+

j,k

wk(ej)ejek .

bb = k wk ek(j (ej)ej) = j,k wk(ej )(ekej)

since ek(ej) =(ej )ek .From the Riccis formula, we have

[ej , ek ] = Rej ek Kej ek DT(ei,ek)where T is the torsion tensor of and

T(ej, ek) = (ei, ek) = 1i

jk.

Thus we obtain

b = jejej+

j,k wk(ej )[ej , ek ]

= j

ejej +

1

iq

i,j

wk(ej)Kej ek i,j

wk(ej )Rej ek

which proves (4). (5) follows from (4) and the Ricci formula and

j Kej ej =

0. (6) follows from (4) and (5).

Pseudo-hermitian manifolds with constant coefficients

1. The Heisenberg group, the flat case.2. The sphere in Cn, positive curvature.

3. The hyperbolic case.

8. Applications of and b

There are many applications from the solutions of the Cauchy-Riemannequations and the tangential Cauchy-Riemann equations. We will give ap-plications in the following areas.


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8.1. The Levi problem. Let D be a pseudoconvex domain in Cn withn 2. One of the major problems in complex analysis is to show that apseudoconvex domain D is a domain of holomorphy. Near each boundarypoint p bD, one must find a holomorphic function f(z) on D which cannotbe continued holomorphically near p. This problem is called the Levi prob-lem for D at p. It involves the construction of a holomorphic function withcertain specific local properties. From a well-known results of Benhke andStein, it suffices to show that a strongly pseudoconvex domain is a domainof holomorphy.

If the domain D is strongly pseudoconvex with C boundary bD andp bD, one can construct a local holomorphic function f in an open neigh-borhood U of p, such that f is holomorphic in U D, f C(D U\ {p})and f(z) as z D approaches p. In fact f can be easily obtained asfollows: let r be a strictly plurisubharmonic defining function for D and we

assume that p = 0. Let

F(z) = 2n

i=1

r

zi(0)zi

ni,j=1

2r

zizj(0)zizj .

F(z) is holomorphic, and it is called the Levi polynomial of r at 0. UsingTaylors expansion at 0, there exists a sufficiently small neighborhood U of0 and C > 0 such that for any z D U,

ReF(z) = r(z) +n

i,j=1

2r

zizj(0)zizj + O(|z|3) C|z|2.

Thus, F(z)

= 0 when z

D

U

\ {0

}. Setting

f =1

FN

for some positive integer N, it is easily seen that f is locally a holomorphicfunction which cannot be extended holomorphically across 0. We can requirethat f is not locally L2 near 0 by choosing N large.

Global holomorphic functions cannot be obtained simply by employingsmooth cut-off functions to patch together the local holomorphic data, sincethe cut-off functions are no longer holomorphic. Let be a cut-off functionsuch that C0 (U) and = 1 in a neighborhood of 0. We note thatf is not holomorphic in D. However, if f can be corrected by solving a-equation, then the Levi problem will be solved.

Let us consider the (0,1)-form g defined byg = (f) = ()f.

This form g can obviously be extended smoothly up to the boundary. It iseasy to see that g is a -closed form in D and g C(0,1)(D). If we can finda solution u C(D) such that(8.1) u = g in D,


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then we define for z D,h(z) = (z)f(z)

u(z).

It follows that h is holomorphic in D, h C(D \ {0}) and h is singularat 0. Thus one can solve the Levi problem for strongly pseudoconvex do-mains provided one can solve equation (6.1) with solutions smooth up to theboundary. From Kohns solution (or Hormanders L2 solution), we can takeu = N(f) and u C(D) and h = f u is a holomorphic functionwith singularity at 0. Notice that h = f N(f) = P f where P is theBergman projection on D.

This gives that any strictly pseudoconvex domain is a domain of holo-morphy. From a well known results of Benhke and Stein, any pseudoconvexdomain is a domain of holomorphy since it can be exhausted by strictlypseudoconvex domains. This gives solution of the Levi problem. Notice

that Kohns method works even for pseudoconvex domains in complex man-ifolds.

8.2. Regularity for the Bergman projection. Let D be a bounded pseu-doconvex domain in Cn, we consider the Bergman projection

(8.2) P : L2(D) L2(D) Ker().From the Hormanders L2 theory for , we have the strong Hodge decom-

postion for for L2(0,1)(D). For any f L2(0,1)(D), we havef = Nf

where N is the -Neumann operator.

This gives that =

on functions has closed range also. Thus we havefor any f L2(D), Notice that Ker() is the same as Ker() for functions.We have

f = N0f P f.Since

f = N0f

and

N1f = N1()N0f = N1(

+ )N0f,

we have

N1f = N0f

Thus N0 =

N0 =

N1, this give the Kohns formula for the Bergmanprojection :

(8.3) P f = f N1f. f L2(D).S

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