Hartogs theorem for forms : solvability of Cauchy-Riemann operator at critical degree

Hartogs theorem for forms:

solvability of Cauchy-Riemann operator at critical degree

CHIN-HUEICHANG ANDHSUAN-PEILEE

Abstract. The Hartogs Theorem for holomorphic functions is generalized in two settings: a CR version (Theorem 1.2) and a corresponding theorem based on it forC^k ∂-closed forms at the critical degree, 0¯ ≤k≤ ∞(Theorem 1.1). Part of Frenkel’s lemma inC^kcategory is also proved.

Mathematics Subject Classification (2000): 32A26 (primary); 32W10 (sec- ondary).

1.

Introduction

Let P_N denote the unit polydisc in

C

^N,N ≥1. In

C

^m+1,m ≥1, set ω= Pm×

zm+1∈

C

1

2 <|zm+1|<1

, m ≥1.

The classical Hartogs theorem (see [9, p. 55]) states: suppose, for a given holo- morphic function f onω, there is an open setU ⊂ P_m such that f has a holo- morphic extension toU× {z_m+1 ∈

C

^{| |zm}+1| <1|}, then f can be extended holo- morphically to P_m+1. This phenomenon in higher fiber dimension is suggested by Frenkel’s lemma (see [13, p. 15]).

Letnalways be an integer bigger than 1. Forz∈

C

^{m+n we writez=(z,z)} withz =(z₁, . . . ,z_m)andz=(z_m+1, . . . ,z_m+n). Set

= P_m×

P_n\1 2P_n

where ¹₂P_n = {z ∈

C

^{n| 2z ∈ Pn}}. The first part of Frenkel’s lemma says: the Cauchy-Riemann equation

∂¯u= f, f ∈C₍^∞_0,q)(), 1≤q≤m+n, ∂¯f =0 Received July 29, 2005; accepted in revised form January 5, 2006.

always has a solutionu∈C₍^∞_0,q−1)()exceptq =n−1. From now on a(0,n−1) form will be called of the “critical degree”.

Note that in Hartogs theorem the open setU can be replaced by a subset Aof P_msuch that Ais not contained in a subvariety of codimension one inP_m(see [12, p. 16]). The following is our first theorem.

Theorem 1.1. With, A as above. Let f be a C^k ∂¯-closed(0,n−1)form on, 0≤k ≤ ∞. For z ∈ P_m, letγz=∩({z} ×

C

ⁿ⁾be the fiber over zin. For every z∈ A, suppose the Cauchy-Riemann equation onγz,

∂¯_γzv= f, is solvable. Then the Cauchy-Riemann equation on

∂v¯ = f is solvable withv∈C^k().

We explain the notation for the (tangential) Cauchy-Riemann operator used in this paper. In general, if there is no ambiguity, it is denoted by∂¯; if the ground space X is specified, we use∂¯X to denote the Cauchy-Riemann (or tangential Cauchy- Riemann) operator on X. Also in integral representations, we usually use ζ for the dummy variable and z for the resulting variable. In this case, the notation∂¯_ζ (respectively,∂¯z) denotes the (tangential) Cauchy-Riemann operator with respect to ζ (respectively,z) variable.

The proof of Theorem 1.1 depends on Theorem 1.2 which is the CR version of Theorem 1.1. Letρbe aC^k (k ≥3) real valued function in

C

^m+nwhich is strictly plurisubharmonic in a neighborhood of{ρ≤0}. Letσ be aC^kreal valued function in

C

^m, strictly plurisubharmonic in a neighborhood of{σ ≤ 0}. We also assume that{σ <0}is connected and relatively compact in

C

^{m. Set}

M = {ρ=0} ∩({σ <0} ×

C

^n).

Assume thatdρ(respectively,dσ) does not vanish on{ρ =0}(respectively,{σ = 0}) anddρ∧dσ =0 on∂M. Let f be a(0,q)form onM such that∂¯Mf =0 in distribution sense. It is proved in [1] that if f ∈ L₍^p_0,q)(M)(i.e. f is a(0,q)form with coefficients inL^p), 1≤ p≤ ∞, then

∂¯Mu= f (∗)

is solvable on M withu ∈ L₍^p_0,q−1)(M) whenever 1 ≤ q < n−1. In the case q =n−1, the above equation is solvable with f ∈C₍⁰_0,q)(M¯)(i.e. coefficients of f are continuous onM,) if and only if¯ f satisfies the moment condition. Our main result is the following:

Theorem 1.2. Let M be as above and A be a subset of {σ < 0}not contained in any subvariety of codimension one in{σ <0}. Let f be a∂¯-closed(0,n−1)form with f ∈C⁰_(0,n−1)(M¯)∩C^k(M),0≤k≤k−3. The equation(∗)is solvable with u ∈ C₍^k_0,n−2)(M)if and only if all the holomorphic moments of f onz, z ∈ A vanish, in other words, for every z∈ A

z

h(z, ζ)f(z, ζ)dζ =0

for every function h holomorphic nearz, wherez= M∩({z}×

C

ⁿ)is the fiber over zin M.

Remark 1.3.

(a) Whenzis strictly pseudoconvex in{z} ×

C

ⁿ, it is well-known that the solv- ability of the tangential Cauchy-Riemann equation∂¯_zu = f with f of top degree is equivalent to the vanishing of all holomorphic moments of f onz. So ifzis strictly pseudoconvex in{z} ×

C

^{nfor everyz ∈} A, the statement of Theorem 1.2 can be phrased as:

The equation(∗)is solvable withu ∈C⁰(M)if and only if∂¯_zv= f is solvable for everyz∈ A.

(b) The CR function version of Hartogs theorem was proved in [2] where the con- dition for Ais stronger but no convexity condition or boundary regularity is required forM.

We recall the representaion formula for∂¯b-closed form f onM (cf. [1]):

(−1)^qf(z)= ¯∂

M

f(ζ )∧q−1(

r

,

r

^∗)(ζ,z)+(−1)^q

∂M

f(ζ )∧(

r

,

r

^∗,

s

)(ζ,z)

+

∂M

f(ζ )∧q(

r

^∗,

s

)(ζ,z), z∈M (2.7) where(

r

,

r

^∗), (

r

,

r

^∗,

s

), (

r

^∗,

s

)are defined in Section 2. It is clear from this representation that the obstruction to the solvability of(∗)is the last integral which is null whenq <n−1 by type consideration. We therefore in Section 2 define for

f ∈C₍⁰_0,n−1)(M¯),∂¯Mf =0 in distribution sense, the following transform:

T f(z)=(−1)ⁿ⁻¹

∂M

f(ζ )∧(

r

^∗_,

s

_)(ζ,z), (2.8) which may be taken as the global moment of f (cf. ((3.1))). Section 3 consists of the properties of the operatorT needed in this paper. For f in the domain ofT

we show that T f is defined in a set containing M and is∂¯-closed there, so (2.7) becomes a “jump formula” for f (see Remark 3.3(a) for more details). Next, we show thatT can be defined locally over the base set{σ <0}. Finally, the operatorT is proved to be just defined fiberwise over every pointz ∈ {σ <0}. This enables us to defineT onMwith arbitrary base setBin

C

^m. In section 4 we define the moment operator

M

^{h forT f (or f}) with respect to a holomorphic functionh. It turns out that

M

^h(f)is a holomorphic function in the base set. Using this property we prove a more general Theorem 4.4 which implies Theorem 1.2 immediately. Section 5 contains the proof of Theorem 1.1. The interesting thing here is a procedure which improves the method in [11] to produce aC^k solution for 0≤k≤ ∞. The results of this paper hold for(p,n−1)forms, 0 ≤ p ≤ n. For simplicity, we only deal with the case p=0.

Finally, closely related to the topics in this paper, there is another Hartogs theorem (see [9, p. 56, 63]) whose higher dimensional analogue is written in a forthcoming paper.

ACKNOWLEDGEMENTS. We thank the referee for comments that helped to im- prove the presentation of this paper.

2.

Preliminaries

We write ζ ∈

C

^{m+n as (ζ, ζ)where ζ ∈}

C

^{m and ζ ∈}

C

ⁿ. Similarly, for differential forms we writed f =(df,df), and∂f =(∂f, ∂f), whered·,d· denote respectively the differentials with repect to the first 2mvariables and those with respect to the last 2nvariables; likewise for∂·(∂¯·)and∂·(∂¯·). Also we use dζ =dζ1∧. . .∧dζm+n,dζ =dζ1∧. . .∧dζmanddζ =dζm+1∧. . .∧dζm+n; similarly fordζ¯,dζ¯,dζ¯, etc..

The following notations and exterior calculus developed by Harvey and Polk- ing [5] will be used in construting kernels needed in this paper:

LetE¹, . . . ,E^α (which are called sections) be a collection ofN-tuples ofC² functions in(ζ,z)∈

C

^N×

C

^N. Following Harvey-Polking [5] we use

(E¹, . . . ,E^α) = E¹,dζ

E¹, ζ−z∧ · · · ∧ E^α,dζ

E^α, ζ−z (2.1)

∧

λ1+...+λα=N−α

¯∂_ζ,zE¹,dζ E¹, ζ−z

λ1

∧ · · · ∧

¯∂_ζ,zE^α,dζ E^α, ζ−z

λα

where x,y =

xiyi for vectors x,y in

C

^{N and d}ζ here is understood to be the N-vector (dζ1, . . . ,dζN). Then is C¹ away from the singular set

Z = _α

1{(ζ,z)| E^j, ζ − z = 0}. We can rewrite as (E¹, . . . ,E^α) = _N−1

0 q(E¹, . . . ,E^α), whereq is the sum of components ofwhich are of

degreeq ind¯z_j, j = 1, . . . ,N. Outside the singular set Z we have the following identity:

∂¯_ζ,z(E¹, . . . ,E^α)=^α

j=1

(−1)^j(E¹, . . . ,E^j, . . . ,E^α). (2.2)

To construct the sections we use the results of Fornæss [3] which we briefly outline in the following and refer to [3] for details:

We first observe that for any strongly convex domainG ⊂

C

^{N withCk,k≥2} boundary, there exist aC^k functionµwith positive real Hessian, a constantc> 0 such thatG ={µ <0}, anddµ =0 in a neighborhood of∂G. Furthermore, if we define

H

(ξ, η)= N

1

∂µ

∂ξj(ξ)(ξj −ηj), then it satisfies

H

(ξ, η)≥µ(ξ)−µ(η)+c|ξ−η|²

for all ξ,η in a small neighborhood of G. The section¯ (_∂ξ^∂µ₁(ξ),· · ·,_∂ξ^∂µ_N(ξ))will serve the purpose of this paper in case the given domain is strongly convex.

On the other hand, Fornæss proved in [3] that any strongly pseudoconvex do- main X ⊂

C

^{N with Ck, k ≥} 2 boundary, admits an embedding into a bounded strongly convex domain Y ⊂

C

^{N withCk} boundary for some N, such that the boundary of X is mapped into the boundary ofY, and the map is 1-1 holomophic in a neighborhood of X¯ (cf. [3, Theorem 9] for explicit statements).

Now for any strongly pseudoconvex domain X ⊂

C

^{N withCk,k} ≥2 bound- ary, the above oberservation and Fornæss’ embedding theroem together imply the existence of

H

and the section forY ⊂

C

^N. Their pull-backs to

C

^N then give the following resuts (cf. [3, Theorem 16]):

There exist a C^k function ν which is strictly plurisubharmonic in a neigh- borhood of X¯ with X = {ν < 0}, a constant > 0 and a function H(ξ, η) ∈ C^k−1(X×X), where X = {η∈

C

^N, ν(η) < }satisfying

H(ξ,·) is holomorphic in X, (2.3)

∃n_j(ξ, η)∈C^k−1(X×X), j =1, . . . ,N, holomorphic in η, such that (2.4)

H(ξ, η)= N

1

n_j(ξ, η)(ξj −ηj),

∃c>0, such that∀η∈ ¯X,ξ ∈ ¯X (2.5) 2Re H(ξ, η)≥ν(ξ)−ν(η)+c|ξ−η|²,

d_ξH(ξ, η)|_ξ=η =∂ν(ξ). (2.6)

Letρ,σ be as in Section 1. In view of the above discussion, for the strongly pseudoconvex domain{ρ < 0} ⊂

C

^m+n there exist

r

_j’s which correspond to the nj’s in (2.4), and we define the section

r

_(ζ,z) as (

r

_{1, . . . ,}

r

m+n). Similarly for the domain{σ < 0} ⊂

C

^m we define the section

s

_(ζ,z) =(

s

1, . . . ,

s

m). We then use

s

(ζ,z)for the section(

s

1(ζ,z), . . . ,

s

m(ζ,z),0,· · ·,0

n

). Let

r

^∗(ζ,z)= (

r

^∗₁(ζ,z), . . . ,

r

^∗_m+n(ζ,z)), where

r

^∗_j(ζ,z)= −

r

j(z, ζ ). Thus

r

,

r

^∗and

s

areC^k−1 in a neighborhood ofM¯ × ¯M.

The kernels(

r

_,

r

^∗_{), (}

r

_,

r

^∗_,

s

_{), (}

r

^∗_,

s

₎etc., are defined according to for- mula (2.1).

For f ∈ C₍⁰_0,q)(M¯), satisfying ∂¯Mf = 0 in distribution sense on M with 1≤q≤n+m−1, we recall the following basic representation formula from [1, p. 543]: forz∈M

(−1)^qf(z)= ¯∂

M

f(ζ )∧q−1(

r

_,

r

^∗_)(ζ,z)+(−1)^q

∂M

f(ζ )∧(

r

_,

r

^∗_,

s

_)(ζ,z)

+

∂M

f(ζ)∧q(

r

^∗,

s

)(ζ,z). (2.7)

The last integral in (2.7) is null whenq < n−1 by type consideration. For f ∈ C₍⁰_0,n−1)(M¯),∂¯M f =0 in distribution sense, we define the following transform:

T f(z)=(−1)ⁿ⁻¹

∂M

f(ζ )∧(

r

^∗,

s

)(ζ,z). (2.8) Remark 2.1. A(0,n−1)form f defined in a subset of

C

^m+ncan be decomposed as follows:

f = s

j=1

f_j where f_j =

|α|+|β|=n−1

|α|=j−1

f_jαdz¯^αdz¯^βands=min(m+1,n). (2.9)

Moreover, when f is∂¯-closed we have:

∂¯zf₁=0, ∂¯zf_j = −¯∂zf_j+1, j =1, . . . ,s−1, and∂¯zf_s =0. (2.10) The definition ofT immediately gives

T f =T f₁=(T f)1. (2.11)

3.

Properties of Tf

In this section we always assume that f ∈C₍⁰_0,n−1)(M¯)and∂¯Mf =0 in distribution sense.

Lemma 3.1. T f =0if there exists u∈C₍⁰_0,n−2)(M¯)such that∂¯Mu= f on M.

Proof. Suppose there existsu ∈C₍⁰_0,n−2)(M¯)satisfying∂¯Mu = f. By the defini- tion ofT f, we have

T f =(−1)ⁿ⁻¹

∂M

∂¯ζu(ζ )∧(

r

^∗_,

s

_)(ζ,z)=

∂M

u∧ ¯∂ζ(

r

^∗_,

s

_)(ζ,z) by Stokes’ theorem. Invoking (2.2) the last integral becomes

∂M

u∧((

r

^∗₎₋₍

s

_{)− ¯∂z(}

r

^∗_,

s

₎₎₌0 by type considerations. This proves the lemma.

Lemma 3.2. There is an open neighborhood

N

^{of M in}{σ <0} ×

C

ⁿ, depending on ρ only, such that T f ∈ C¹(

N

). On the set U =

N

∩ {ρ ≥ 0} we have

∂(¯ T f)=0.

Proof. Observe that in the definition ofT the integration is just over∂M. By (2.3)- (2.6), we see thatT f is well-defined forzin an open set

N

, depending onρonly, containing Min{σ <0} ×

C

^{nand isCk−2there.}

We show that ∂(¯ T f) = 0 in the interior of U. Consider m > 1 first. The identity (2.2) and type considerations imply∂¯z(

r

^∗,

s

)= −¯∂_ζ(

r

^∗,

s

)in this case.

Thus

∂(¯ T f) =(−1)ⁿ⁻¹

∂M

f ∧ ¯∂z(

r

^∗,

s

)=(−1)ⁿ

∂M

f ∧ ¯∂_ζ(

r

^∗,

s

)

= −

∂M

∂¯_ζ(f ∧(

r

^∗,

s

))= −

∂M

d_ζ(f ∧(

r

^∗,

s

))=0

by Stokes’ theorem. Form = 1, the section

s

is the Cauchy kernel which is holo- morphic in bothζandz. Hence (2.2) and type consideration give

∂¯z(T f)=(−1)ⁿ⁻¹

∂M

f ∧(

r

^∗).

Forz ∈U\Mwe can apply Stokes’ theorem to the above integral and get∂¯z(T f)= 0 as∂¯f = ¯∂_ζ

r

^∗=0.

SinceT f ∈C^k−2(

N

), we conclude that∂(¯ T f)=0 onU by continuity. The lemma is proved.

Remark 3.3.

(a) In (2.7) the form inside the parenthesis after∂¯Mis inC¹(M)provided that f is inC¹(M), and so it can be extended to aC¹form in{σ <0} ×

C

^{n. By Lemma} 3.2 the last form in (2.7) is actually inC^k−2(U)and is∂¯-closed. Therefore, in this case, (2.7) becomes a “jump formula” for f. A more general jump formula can be found in [2], but we don’t need it here.

(b) In view of (2.10),(2.11) and Lemma 3.2, we see that all the coefficients ofT f are holomorphic inz.

Letσ˜ be aC¹function defined in a neighborhood of{σ ≤0}such that{ ˜σ <0} ⊂ {σ < 0}. SetM˜ = {ρ = 0} ∩({ ˜σ < 0} ×

C

^{n). Supposedρ∧dσ˜ = 0 on ∂M.˜} Denote by b =(b˜,0,· · ·,0

n

), whereb˜ is the section for the Bochner-Martinelli kernel in

C

^{m. For f} in the domain ofT, define

Tf(z)=(−1)ⁿ⁻¹

∂M˜

f ∧(

r

^∗,b)(ζ,z).

Lemma 3.1 and the proof of Lemma 3.2 still hold forTand consequentlyTf is a

∂¯-closed form inU˜ =

N

∩ {ρ≥0} ∩({ ˜σ <0} ×

C

ⁿ⁾. The next lemma shows that T is “locally” defined over

C

^m.

Lemma 3.4. Let f be in the domain of T . Then T f =Tf onU .˜ Proof. Forz ∈ ˜U, apply (2.2) to get

T f =(−1)ⁿ⁻¹

∂M

f ∧(

r

^∗,

s

)(ζ,z)

=(−1)ⁿ⁻¹

∂M

f ∧((

r

^∗,b)−(

s

,b)− ¯∂_ζ,z(

r

^∗,

s

,b))

=(−1)ⁿ⁻¹

∂M

f ∧(

r

^∗,b) by Stokes’ theorem and type considerations,

=(−1)ⁿ⁻¹

∂M˜

f ∧(

r

^∗,b) +(−1)ⁿ⁻¹

M\ ˜M

d_ζ(f ∧(

r

^∗,b)) by Stokes’ theorem.

In the last integral we use (2.2) again to get

d_ζ(

r

^∗,b)= ¯∂_ζ(

r

^∗,b)= −¯∂z(

r

^∗,b)+(

r

^∗)−(b) and the integral vanishes by type considerations. So

T f =(−1)ⁿ⁻¹

∂M˜

f ∧(

r

^∗,b)=Tf. This completes the proof.

Lemma 3.4 has two implications. First, in Lemma 3.1 the assumption onucan be weakened byu∈C₍⁰_0,n−2)(M). Next, it suggests that the strong pseudoconvexity of the base set can be relaxed when one definesT f. This is seen more precisely in Lemma 3.5.

Fort ≥0, set

M_t = {ρ=t} ∩({σ <0} ×

C

ⁿ⁾ M˜_t = {ρ=t} ∩({ ˜σ <0} ×

C

^{n) and}

z,t = {ρ=t} ∩({z} ×

C

^{n) forz ∈ {σ <0}.} Whent =0 we havez=z,0,M˜ = ˜M₀, andM =M₀.

Now fixz₀ ∈ {σ <0}. As in Lemma 3.4, letσ˜ = |z−z₀|²−², where >0 is chosen so that{ ˜σ <0} ⊂ {σ <0}anddρ∧dσ˜ =0 on∂M. For˜ small enough there existt >0 small such thatz₀,t is strongly pseudoconvex in{z₀} ×

C

^nand

{z| |z−z₀|< } × {z|(z₀,z)∈z₀,t} ⊂ ˜U \M. Fix suchandt. By Lemma 3.4 we have forz₀∈ ˜U satisfyingρ(z₀) >t

T f(z₀)=(−1)ⁿ⁻¹

∂M˜

f(ζ)∧(

r

^∗,b)(ζ,z₀)

=(−1)ⁿ⁻¹

∂M˜

(T f)(ζ )∧(

r

^∗,b)(ζ,z₀) where the last equality follows from (2.7) and Lemma 3.1 forT.

Since in the last integral

(

r

^∗,b)= R(ζ,z)∧0(b)

where R(ζ,z)is a form holomorphic in ζ. Applying Stokes’ theorem to the last integral in the above formula, we have

T f(z₀)=(−1)ⁿ⁻¹

S

(T f)(ζ )∧(

r

^∗,b)(ζ,z₀)

by type consideration and the fact that∂¯_ζ0(b)=0, where S = {ζ| |ζ−z₀| = } × {ζ|(z₀, ζ)∈z₀,t}.

Lettend to zero. It follows from Lemma 1.14 of [Ky] that T f(z₀)=(−1)ⁿ⁻¹c_m

ζ∈z 0,t

(T f)(z₀, ζ)∧(˜

r

^∗_z

0,t)(ζ,z₀) wherec_m= ₍⁽_m^2π₋ⁱ₁^)m_)! and

r

˜^∗_z

0,t is the section constructed fromρ(˜ z)=ρ(z₀,z)−t.

We thus have the following lemma:

Lemma 3.5. For any z₀ ∈ {σ <0}, for any z₀ =(z₀,z₀)∈ ˜U and for any t > 0 such thatρ(z₀) >t and{z|ρ(z₀,z) <t}is strictly pseudoconvex inU˜ ∩({z₀}×

C

^{n), we have}

T f(z₀)=(−1)ⁿ⁻¹c_m

ζ∈z 0,t

(T f)(z₀, ζ)∧(˜

r

^∗_z

0,t)(ζ,z₀). (3.1) In particular, if{z|ρ(z₀,z) <0}is strictly pseudoconvex which holds for z₀in a dense open set in{σ <0}, we have

T f(z₀)=(−1)ⁿ⁻¹c_m

ζ∈_z 0

f(z₀, ζ)∧(˜

r

^∗_z

0)(ζ,z₀). (3.1) Proof. It remains to prove(3.1). The denseness of suchz₀follows from Sard’s the- orem. In view of the fact that (3.1) holds for any 0<t<ρ(z₀)with{z|ρ(z₀,z)<

t}strictly pseudoconvex inU˜ ∩ ({z₀}×

C

^n),(3.1)is obtained by taking limit as sucht goes to 0 in (3.1) and by (2.7) and Stokes’theorem.

Remark 3.6. Formula (3.1) (or((3.1))) is of fundamental importance. It shows thatT f can be defined byρonly: the connectivity and the strong pseudoconvexity of the base set{σ <0}can be relaxed. Moreover, it shows thatT f has a continuous extension to the setU₁=

N

∩ {ρ≥0} ∩({σ ≤0} ×

C

^n).

Now instead of{σ < 0}we take the base set to be an arbitrary open set Bin

C

^m. It is easy to see that all the preceding results about T f still holds. Indeed, through (3.1) and((3.1))the properties ofT f become even more transparent.

4.

The moment operator M

h(g)

and the proof of Theorem 1.2

Letρbe the same as before andBbe an arbitrary relatively compact open subset in

C

^{m. Set}

M˜ = {ρ=0} ∩(B×

C

^n).

For an open neighborhoodO ofM, we set˜

U˜ =O∩ {ρ≥0} ∩(B×

C

^n).

Let g be aC¹ ∂¯-closed(0,n−1) form inU˜. Let V be an open set in B andh be a function holomorphic in a neighborhood of{ρ = 0} ∩(V ×

C

ⁿ⁾. Define the moment operator ofgwith respect tohonV by

M

^h(g)(z)=

ζ∈z

h(z, ζ)g(z, ζ)dζ, wheredζ=dζm+1∧ · · · ∧dζm+n. (4.1) Lemma 4.1.

M

h(g)(z)is a holomorphic function in V .

Proof. First observe that

M

h(g)(z)=

ζ∈z

h(z, ζ)g₁(z, ζ)dζ

where g₁ is defined by (2.9). Fix z₀ ∈ V. Choose a domain D in

C

^{n with C1} boundary and a neighborhoodW ofz₀ inV such that

z₀⊂ {z₀} ×D, {ρ≤0} ∩(W ×

C

^{n)⊂W ×D andW ×∂D ⊂ ˜U.} For any function h holomorphic in W ×D it follows from Stokes’ theorem and (2.9), (2.10) that

M

^h(g)(z)=

z

h(z, ζ)g₁(z, ζ)dζ =

∂D

h(z, ζ)g₁(z, ζ)dζ wheneverz∈W. Now by (2.10)

∂¯z

M

h(g)(z)=

∂D

h(z, ζ)∂¯zg₁(z, ζ)dζ= −

∂D

h(z, ζ)∂¯_ζg₂(z, ζ)dζ wheneverz∈W and so

∂¯

M

h(g)(z)= −

∂D

d_ζ(h(z, ζ)g₂(z, ζ))dζ=0. This completes the proof.

Remark 4.2. Suppose f is a∂¯-closed(0,n−1)form inC⁰(M¯˜). In view of Remark 3.6, T f is well-defined inU˜ =

N

^{∩ {ρ ≥ 0} ∩(B×}

C

^n)where

N

is given by Lemma 3.2. Locally, for eachz ∈ B there is a small open neighborhoodW ofz contained in Bsuch that forz∈ ˜U ∩(W ×

C

^{n) f} can be represented as in (2.7), we see immediately that

M

h(f)=

M

h(T f)for any holomorphic functionh. In other words, the moment of f is well-defined and is holomorphic inz.

With Remark 4.2 we have:

Corollary 4.3. Fix z ∈ B. All the holomorphic moments of f onz vanish is equivalent to T f(z) = 0 onU˜∩ ({z} ×

C

ⁿ⁾. Moreover, supposez is strictly pseudoconvex, equation∂¯_zu = f is solvable onz if and only if T f(z)= 0for all z∈ ˜U ∩({z} ×

C

ⁿ).

Proof. To prove the first statement, we need only show thatM_h(f)(z)=0 for allh holomorphic nearzimpliesT f(z)=0 onU˜ ∩({z} ×

C

ⁿ⁾. The proof of Lemma 4.1 and Remark 4.2 give

0=

M

^h(f)(z)=

z

h(z, ζ)(T f)1(z, ζ)dζ

=

z,t

h(z, ζ)T f(z, ζ)dζ, t >0,

wheneverz,t ⊂ ˜Uandhis holomorphic nearz,t. The last equality follows from (2.10), (2.11) and Stokes’ theorem. Now suppose z,t is strictly pseudoconvex.

Let h(z, ζ)dζ = (˜

r

^∗_z). By Lemma 3.5 we have T f(z) = 0 for z ∈ ˜U ∩ ({z} ×

C

^{n),ρ(z) ≥ t}. By Sard’s theorem 0 is a limit point of sucht so the first statement is proved. The second statement follows from the first statement and Remark 1.3(a).

Theorem 4.4. Letρbe as in Theorem 1.2. Let B be a connected relatively compact open subset in

C

^{m. Set}

M˜ = {ρ=0} ∩(B×

C

^n).

Let f be a(0,n−1)form in C⁰(M¯˜)with∂¯ f = 0onM in distribution sense. Let˜ A be a subset of B not contained in any subvariety of codimension one in B. If for every z ∈ A all the holomorphic moments of f onz vanish, then T f vanishes identically onU˜ =

N

∩ {ρ≥0} ∩(B×

C

^n)where

N

is defined in Lemma 3.2.

Proof. Step 1. The assumption on the set Aimplies that there is a point p ∈ ¯B such that the intersection of Awith any neighborhood of p is not contained in a subvariety of codimension one in B. LetV be any connected neighborhood of p in B. Let¯ h be any function holomorphic near M˜ ∩ (V ×

C

ⁿ⁾. By Lemma 4.1

M

^h(f)(z)is a holomorphic function inV ∩B. Remark 4.2 and the assumption give that

M

^h(f)(z)=

M

^{h(T f)(z)= 0 for allz ∈V ∩} A. By the choice of p, we must have

M

h(f)(z)identically equal to zero onV.

Step 2. Claim: There is a neighborhoodW of pin B¯ such thatT f vanishes identi- cally onU˜ ∩(W×

C

^n).

Proof of the Claim. By Corollary 4.3 it suffices to show that for every z ∈ W,

M

^h(f)(z)= 0 for allhholomorphic nearz. If there is no such neighborhood, then there exists a sequence{z_j}^∞₁ ⊂ Bsuch thatz_j → pas j → ∞andT f(z_j,·) does not vanish identically for all j =1,2, . . ..

Chooset > 0 so that p,t is aC^k strictly pseudoconvex real hypersuface in U˜ ∩({p} ×

C

^{n). LetW} be a connected open neighborhood of pin Bsuch thatW

× {z|(p,z)∈p,t} ⊂ ˜U\ ˜M.

For every j = 1,2, . . . there is a functionh_j holomorphic nearz_j such that

M

h_j(f)(z_j) =0. For simplicity we may assume thatz_j is strictly pseudoconvex in{z_j} ×

C

ⁿ (or we do as in the proof of Corollary 4.3).

Fix jso thatz_j ∈W. SetD₁= {z|ρ(p,z) <t}andD₂= {z|ρ(z_j,z) <

0}. By our choice both D₁, D₂ are strictly pseudoconvex domains in

C

^{n and} D₂

^D1 for every j = 1,2, . . .. Since ρ(z_j,·) is plurisubharmonic in D₁ if ρ is plurisubharmonic inU˜ and which can be assumed, it follows from Corollary 5.4.3 of [8] that D₁,D₂form a Runge pair.

Thush_j(z_j,·)can be uniformly approximated onD¯₂by functions holomorphic onD₁. Let{g_k}^∞_k=1be such a sequence of holomorphic functions onD₁. By Step 1,

for allk≥1

M

gk(f)(z)=

M

gk(T f)(z)=0 for allz ∈W. Therefore we must have

M

hj(f)(z_j)=0, contradicting our assumption onz_j andh_j. This completes the proof of the claim.

Step 3. Set

S

≡ {z| z ∈ B, T f(z,·) ≡ 0 onU˜ ∩({z} ×

C

^n)}. By Step 2

S

has non-empty interior. In fact, the argument in Step 2 shows that the interior of

S

is both open and closed in B. Since B is connected, we conclude that

S

= B.

Theorem 4.4 is proved.

From the proof of Theorem 4.4 we have the following:

Corollary 4.5. Under the assumptions of Theorem4.4, the following statements are equivalent:

(a) For all z∈ A, every holomorphic moment of f onzvanishes.

(b) T f(z)≡0onU .˜

(c) T f(z)≡0for all z ∈z, z ∈ A.

(d) ∂¯_z,tu = T f(z,·)is solvable on z,t ⊂ ˜U for every z ∈ A and for every t ≥0, provided thatz,t is strictly pseudoconvex in{z} ×

C

^n.

Corollary 4.6. Under the assumptions of Theorem 4.4, if the set{z|z= ∅, z ∈ B}is not contained in a subvariety of codimension one in B, then T f ≡0onM.˜ In particular, if B = {σ <0}, then∂¯M is solvable at q=n−1.

On the other hand, we have

Remark 4.7. LetM = {ρ=0} ∩({ ˜σ <0} ×

C

^n)whereρis as in the assumption of Theorem 4.4 andσ˜ is anyC¹function satisfyingdρ∧dσ˜ =0 on∂M. Suppose M and{ ˜σ < 0} are connected and m < n, then for f ∈ C₍⁰_0,q)(M¯)satisfying

∂¯Mf = 0 in distribution sense onM withm ≤q ≤n+m −1, considering type, the following representation formula holds forz∈M

(−1)^qf(z)= ¯∂

M

f(ζ )∧q−1(

r

,

r

^∗)(ζ,z)+(−1)^q

∂M

f(ζ )∧(

r

,

r

^∗,b)(ζ,z)

+

∂M

f(ζ)∧q(

r

^∗_,b)(ζ,z).

Note that the last integral vanishes form ≤q≤n−2 by type consideration. In other words,∂¯M is always solvable form ≤q ≤n−2 in this case. Thus the tangential Cauchy-Riemann equation(∗)is solvable atq = n−1 iffTf(z)= T f(z)= 0.

In view of Corollary 4.6, if{ ˜σ <0}is not contained inπ({ρ =0}), the projection of{ρ=0}to the

C

^mplane, then(∗)is solvable atq =n−1.

Proof of Theorem 1.2. The solutionuis obtained from Corollary 4.5 and formula (2.7). The regularity ofucan be proved by routine procedure, see e.g. [14], and we omit the details here.