Bulletin of Mathematical Sciences

, Volume 3, Issue 2, pp 241–285 | Cite as

Cauchy-type integrals in several complex variables

Open Access
Article

Abstract

We present the theory of Cauchy–Fantappié integral operators, with emphasis on the situation when the domain of integration, D , has minimal boundary regularity. Among these operators we focus on those that are more closely related to the classical Cauchy integral for a planar domain, whose kernel is a holomorphic function of the parameter z D . The goal is to prove L p estimates for these operators and, as a consequence, to obtain L p estimates for the canonical Cauchy–Szegö and Bergman projection operators (which are not of Cauchy–Fantappié type).

Mathematics Subject Classification (2000)

32A36 32A50 42B 31B 

1 Introduction

The purpose of this survey is to study Cauchy-type integrals in several complex variables and to announce new results concerning these operators. While this is a broad field with a very wide literature, our exposition will be focused more narrowly on achieving the following goal: the construction of such operators and the establishment of their L p mapping properties under “minimal” conditions of smoothness of the boundary of the domain D in question.

The operators we study are of three interrelated kinds: Cauchy–Fantappié integrals with holomorphic kernels, Cauchy–Szegö projections and Bergman projections. In the case of one complex variable, what happens is by now well-understood. Here the minimal smoothness that can be achieved is “near” C 1 (e.g., the case of a Lipschitz domain). However when the complex dimension is greater than 1 the nature of the Cauchy–Fantappié kernels brings in considerations of pseudo-convexity (in fact strong pseudo-convexity) and these in turn imply that the limit of smoothness should be “near” C 2 .

We establish L p -regularity for one or more of these operators in the following contexts:
  • When D is strongly pseudo-convex and of class C 2 ;

  • When D is strongly C -linearly convex and of class C 1 , 1

with p in the range 1 < p < . See Theorems 1–4 for the precise statements.

This survey is organized as follows. In Sect. 2 we briefly review the situation of one complex variable. Section 3 is devoted to a few generalities about Cauchy-type integrals when n , the complex dimension of the ambient space, is greater than 1. The Cauchy–Fantappié forms are taken up in Sect. 4 and the corresponding Cauchy–Fantappié integral operators are set out in Sect. 5. Here we adapt the standard treatment in [34, Chapt. IV], but our aim is to show that these methods apply when the so-called generating form is merely of class C 1 or even Lipschitz, as is needed in what follows.

The Cauchy–Fantappié integrals constructed up to that point may however lack the basic requirement of producing holomorphic functions, whatever the given data is. In other words, the kernel of the operator may fail to be holomorphic in the free variable z D . To achieve the desired holomorphicity requires that the domain D be pseudo-convex, and two specific forms of this property, strong pseudo-convexity and strong C -linear convexity are discussed in Sect. 6.

There are several approaches to obtain the required holomorphicity when the domain is sufficiently smooth and strongly pseudo-convex. The initial methods are due to Henkin [17, 18] and Ramirez [33]; a later approach is in Kerzman–Stein [21], which is the one we adopt here. It requires to start with a “locally” holomorphic kernel, and then to add a correction term obtained by solving a ¯ -problem. These matters are discussed in Sects. 79. One should note that in the case of strongly C -linearly convex domains, the Cauchy–Leray integral given here requires no correction. So among all the integrals of Cauchy–Fantappié type associated to such domains, the Cauchy–Leray integral is the unique and natural operator that most closely resembles the classical Cauchy integral from one complex variable.

The main L p estimates for the Cauchy–Leray integral and the Szegö and Bergman projections (for C 2 boundaries) are the subject of a series of forthcoming papers; in Sect. 10 we limit ourselves to highlighting the main points of interest in the proofs. For the last two operators, the L p results are consequences of estimates that hold for the corrected Cauchy–Fantappié kernels, denoted C ϵ and B ϵ , that involve also their respective adjoints. Section 11 highlights a further result concerning the Cauchy–Leray integral, also to appear in a separate paper: the corresponding L p theorem under the weaker assumption that the boundary is merely of class C 1 , 1 .

A survey of this kind must by the nature of the subject be far from complete. Among matters not covered here are L p results for the Szegö and Bergman projection and for the Cauchy–Leray integral for other special domains (in particular, with more regularity). For these, see e.g. [2, 3, 4, 6, 7, 8, 12, 13, 15, 23, 29, 30, 32, 37]. It is to be noted that several among these works depend in the main on good estimates or explicit formulas for the Szegö or Bergman kernels. In our situation these are unavailable, and instead we have to proceed via the Cauchy–Fantappié framework.

A few words about notation: Euclidean volume measure for C n R 2 n ( n 1 ) will be denoted dV . The notation b D will indicate the boundary of a domain D C n ( n 1 ) and, for D sufficiently smooth, d σ will denote arc-length ( n = 1 ) or Euclidean surface measure ( n 2 ).

2 The Case n = 1

In the case of one complex dimension the problem of L p estimates has a long and illustrious history. Let us review it briefly. (Some details can be found in [10, 16, 24], which contain further citations to the literature.)

Suppose D is a bounded domain in C whose boundary b D is a rectifiable curve. Then the Cauchy integral is given by
C ( f ) ( z ) = b D f ( w ) C ( w , z ) , for z D
where C ( w , z ) is the Cauchy kernel
C ( w , z ) = 1 2 π i dw w z
When D is the unit disc, then a classical theorem of M. Riesz says that the mapping f C ( f ) | b D , defined initially for f that are (say) smooth, is extendable to a bounded operator on L p ( b D ) , for 1 < p < . Very much the same result holds when the boundary of D is of class C 1 + ϵ , with ϵ > 0 , (proved either by approximating to the result when D is the unit disc, or adapting one of the several methods of proof used in the classical case). However in the limiting case when ϵ = 0 , these ideas break down and new methods are needed. The theorems proved by Calderón, Coifman, McIntosh, Meyers and David (between 1977–1984) showed that the corresponding L p result held in the following list of increasing generality: the boundary is of class C 1 ; it is Lipschitz (the first derivatives are merely bounded and not necessarily continuous); it is an “Ahlfors-regular” curve.
We pass next to the Cauchy–Szegö projection S , the corresponding orthogonal projection with respect to the Hilbert space structure of L 2 ( b D ) . In fact when D is the unit disc, the two operators C and S are identical. Let us now restrict our attention to the case when D is simply connected and when its boundary is Lipschitz. Here a key tool is the conformal map Φ : D D , where D is the unit disc. We consider the induced correspondence τ given by τ ( f ) ( e i θ ) = ( Φ ( e i θ ) ) 1 2 f ( Φ ( e i θ ) ) , and the fact that S = τ 1 S 0 τ , where S 0 is the Cauchy–Szegö projection for the disc D . Using ideas of Calderón, Kenig, Pommerenke and others, one can show that | Φ | r belongs to the Muckenhaupt class A p , with r = 1 p / 2 , from which one gets the following. As a consequence, if we suppose that b D has a Lipschitz bound M , then S is bounded on L p , whenever
  • 1 < p < , if b D is in fact of class C 1 .

  • p M < p < p M . Here p M depends on M , but p M > 4 , and p M is the exponent dual to p M .

There is an alternative approach to the second result that relates the Cauchy–Szegö projection S to the Cauchy integral C . It is based on the following identity, used in [21]
S ( I A ) = C , where A = C C .
(2.1)
There are somewhat analogous results for the Bergman projection in the case of one complex dimension. We shall not discuss this further, but refer the reader to the papers cited above.

3 Cauchy integral in C n , n > 1 ; some generalities

We shall see that a very different situation occurs when trying to extend the results of Sect. 2 to higher dimensions. Here are some new issues that arise when n > 1
  1. i

    There is no “universal” holomorphic Cauchy kernel associated to a domain D .

     
  2. ii

    Pseudo-convexity of D , must, in one form or another, play a role.

     
  3. iii

    Since this condition involves (implicitly) two derivatives, the “best” results are to be expected “near” C 2 , (as opposed to near C 1 when n = 1 ).

     
In view of the non-uniqueness of the Cauchy integral (and its problematic existence), it might be worthwhile to set down the minimum conditions that would be required of candidates for the Cauchy integral. We would want such an operator C given in the form
C ( f ) ( z ) = b D f ( w ) C ( w , z ) , z D ,
to satisfy the following conditions:
  1. (a)

    The kernel C ( w , z ) should be given by a “natural” or explicit formula (at least up to first approximation) that involves D .

     
  2. (b)

    The mapping f C ( f ) should reproduce holomorphic functions. In particular if f is continuous in D ¯ and holomorphic in D then C ( f ) ( z ) = f ( z ) , for z D .

     
  3. (c)

    C ( f ) ( z ) should be holomorphic in z D , for any given f that is continuous on b D .

     
Now there is a formalism (the Cauchy–Fantappié formalism of Fantappié (1943), Leray, and Koppleman (1967)), which provides Cauchy integrals satisfying the requirements (a) and (b) in a general setting. Condition (c) however, is more problematic, even when the domain is smooth. Constructing such Cauchy integrals has been carried out only in particular situations, (see below).

4 Cauchy–Fantappié theory in higher dimension

The Cauchy–Fantappié formalism that realizes the kernel C ( w , z ) revolves around the notion of generating form: these are a class of differential forms of type ( 1 , 0 ) in the variable of integration whose coefficients may depend on two sets of variables ( w and z ), and we will accordingly write
η ( w , z ) = j = 1 n η j ( w , z ) dw j with ( w , z ) U × V
to designate such a form. The precise definition is given below, where the notation
η ( w , z ) , ξ = j = 1 n η j ( w , z ) ξ j .
is used to indicate the action of η on the vector ξ C n .

Definition 1

The form η ( w , z ) is generating at z relative to V if there is an open set
U z C n { z }
such that
b V U z
(4.1)
and, furthermore
η ( w , z ) , w z = j = 1 n η j ( w , z ) ( w j z j ) 1 for any w U z .
(4.2)
We say that η is a generating form for V (alternatively, that V supports a generating form η ) if for any z V we have that η is generating at z relative to V .

Example 1

Set
β ( w , z ) = | w z | 2
We define the Bochner–Martinelli generating formto be
η ( w , z ) = w β β ( w , z ) = j = 1 n w ¯ j z ¯ j | w z | 2 dw j
(4.3)
It is clear that η satisfies conditions (4.1) and (4.2) for any domain V and for any z V , with U z : = C n { z } .
The Bochner–Martinelli generating form has several remarkable features. First, it is “universal” in the sense that it is given by a formula (4.3) that does not depend on the choice of domain V ; secondly, in complex dimension n = 1 it agrees (up to a scalar multiple) with the classical Cauchy kernel
1 2 π i dw w z , w U z : = C { z }
and in particular its coefficient ( w z ) 1 is holomorphic as a function of z V for any fixed w b V . On the other hand, it is clear from (4.3) that for n 2 the coefficients of this form are nowhere holomorphic: this failure at holomorphicity is an instance of a crucial, dimension-induced phenomenon which was alluded to in conditions ii. and (c) in Sect. 3 and will be further discussed in Example 2 below and in Sect. 6.

4.1 Cauchy–Fantappié forms

Suppose now that for each fixed z η ( w , z ) is a form of type ( 1 , 0 ) in w with coefficients of class C 1 in each variable. (We are not assuming that η is a generating form). Set
Ω 0 ( η ) ( w , z ) = 1 ( 2 π i ) n η ( ¯ w η ) n 1 ( w , z )
(4.4)
where ( ¯ w η ) n 1 stands for the wedge product: ¯ w η ¯ w η performed ( n 1 ) -times. We call Ω 0 ( η ) the Cauchy-Fantappiè form for η . Note that Ω 0 ( η ) ( w , z ) is of type ( n , n 1 ) in the variable w U while in the variable z V it is just a function.

Example 2

The Cauchy–Fantappié form for the Bochner–Martinelli generating form or, for short, Bochner–Martinelli CF form is
Ω 0 ( w β β ) ( w , z ) = ( n 1 ) ! ( 2 π i | w z | 2 ) n j = 1 n ( w ¯ j z ¯ j ) dw j ( ν j d w ¯ ν dw ν ) .
Now the Bochner–Martinelli integral is the operator
C BM f ( z ) = w b D f ( w ) C BM ( w , z ) , z D , f C ( b D )
where the kernel C BM ( w , z ) is the Bochner–Martinelli CF form restricted to the boundary; more precisely
C BM ( w , z ) = j Ω 0 ( w β β ) ( w , z ) , w b D , z D
where j denotes the pullback of forms under the inclusion
j : b D C n

It is clear that such operator is “natural” in the sense discussed in condition (a) in Sect. 3, and we will see in Sect. 5 that this operator also satisfies condition (b), see Proposition 1 in that section. On the other hand, the kernel C BM ( w , z ) is nowhere holomorphic in z : as a result, when n > 1 the Bochner–Martinelli integral does not satisfy condition (c).

We will now review the properties of Cauchy-Fantappiè forms that are most relevant to us.

Basic Property 1

For any function g C 1 ( U ) we have
Ω 0 ( g ( w ) η ( w , z ) ) = g n ( w ) Ω 0 ( η ) ( w , z ) for any w U .

Proof

The proof is a computation: by the definition (4.4), we have
Ω 0 ( g η ) = g η ( ¯ ( g η ) ) n 1
On the other hand, computing ( ¯ ( g η ) ) n 1 produces two kinds of terms:
  1. (a.)

    Terms that contain ¯ g η : but these do not contribute to Ω 0 ( g η ) because g η ¯ g η = 0 (which follows from η η = 0 since η has degree 1 );

     
  2. (b.)

    The term g n 1 ¯ η , which gives the desired conclusion.

     

Suppose, further, that η ( w , z ) is generating at z relative to V . Then the following two properties also hold.

Basic Property 2

We have that
( ¯ w η ) n ( w , z ) = 0 for any w U z .
(4.5)
Note that if the coefficients of η ( · , z ) are in C 2 ( U z ) , then as a consequence of the fact that ¯ ¯ = 0 , we have that ( ¯ w η ) n ( w , z ) = d w Ω 0 ( η ) ( w , z ) and (4.5) can be formulated equivalently as
d w Ω 0 ( η ) ( w , z ) = 0 , w U z .

Proof

We prove (4.5) in the case: n = 2 and leave the proof for general n as an exercise for the reader. Thus, writing
η = η 1 dw 1 + η 2 dw 2
we obtain
( ¯ w η ) 2 = 2 ¯ w η 1 ¯ w η 2 dw 1 dw 2
(4.6)
Now
η 1 ( w , z ) ( w 1 z 1 ) + η 2 ( w , z ) ( w 2 z 2 ) = 1 for any w U z
because η is generating at z , and applying ¯ w to each side of this identity we obtain
( w 1 z 1 ) ¯ w η 1 ( w , z ) + ( w 2 z 2 ) ¯ w η 2 ( w , z ) = 0 for any w U z
(4.7)
Recall that U z C 2 { z } , see Definition 1, and so
U z U = U z 1 U z 2
where
U z 1 = { w = ( w 1 , w 2 ) U z U , w 1 z 1 }
(4.8)
U z 2 = { w = ( w 1 , w 2 ) U z U , w 2 z 2 }
(4.9)
But for any two sets A and B one has A B = ( A B ) · ( B A ) · ( A B ) where · denotes disjoint union. Now, if w U z 1 U z 2 then (4.7) reads
( w 1 z 1 ) ¯ w η 1 ( w , z ) = 0 , w 1 z 1
but this implies
¯ w η 1 ( w , z ) = 0 for any w U z 1 U z 2 .
One similarly obtains
¯ w η 2 ( w , z ) = 0 for any w U z 2 U z 1 ) .
We are left to consider the case when w U z 1 U z 2 ; note that since
( w 1 z 1 ) ( w 2 z 2 ) 0 for any w U z 1 U z 2
showing that ( ¯ w η ) 2 ( w , z ) = 0 for any w U z 1 U z 2 is now equivalent to showing that
( w 1 z 1 ) ( w 2 z 2 ) ( ¯ w η ) 2 ( w , z ) = 0 for any w U z 1 U z 2
To this end, combining (4.6) with (4.7) we find
( w 1 z 1 ) ( w 2 z 2 ) ( ¯ w η ) 2 ( w , z ) = 2 ( w 1 z 1 ) 2 ¯ w η 1 ( w , z ) ¯ w η 1 ( w , z ) dw 1 dw 2
and indeed
¯ w η 1 ¯ w η 1 = 0
because ¯ w η 1 is a form of degree 1.
Let η ( w , z ) be a form of type ( 1 , 0 ) in the variable w (not necessarily generating for V ) and with coefficients in C 1 ( U × V ) ; set
Ω 1 ( η ) ( w , z ) = ( n 1 ) ( 2 π i ) n ( η ( ¯ w η ) n 2 ¯ z η ) ( w , z )
(4.10)
Note that Ω 1 ( η ) ( w , z ) is of type ( n , n 2 ) in the variable w and of type ( 0 , 1 ) in the variable z . We call Ω 1 ( η ) the Cauchy-Fantappie’ form of order 1 for η , and the previous one, Ω 0 ( η ) , will now be called Cauchy-Fantappie’ form of order 0.

In the previous properties z was fixed; here it is allowed to vary.

Basic Property 3

We have (again for η generating at z )
( 2 π i ) n ¯ z Ω 0 ( η ) ( w , z ) = ( ¯ w η ) n 1 ¯ z η + η ( ¯ w η ) n 2 ¯ z ¯ w η ,
(4.11)
For any w U z U , where U z is as in (4.2). Note that if the coefficients are in fact of class C 2 in each variable, then (4.11) has the equivalent formulation
¯ z Ω 0 ( η ) ( w , z ) = d w Ω 1 ( η ) ( w , z ) .
(4.12)

Proof

As before, we specialize to the case: n = 2 and leave the proof of the general case as an exercise for the reader. For n = 2 identity (4.11) reads
¯ z ( η ¯ w η ) = ¯ w η ¯ z η + η ¯ z ¯ w η
(4.13)
By the Leibniz rule we have
¯ z ( η ¯ w η ) = ¯ z η ¯ w η + η ¯ z ¯ w η
and so it is clear that (4.13) will follow if we can show that
¯ w η ¯ z η = 0 , for any w U z
for any generating form η with coefficients of class C 1 . Proceeding as in the proof of basic property 2, we decompose
U z U = U z 1 U z 2
where U z 1 and U z 2 are as in (4.8) and (4.9), respectively. Again, we have
η 1 ( w , z ) ( w 1 z 1 ) + η 2 ( w , z ) ( w 2 z 2 ) = 1 for any w U z
because η is generating, and applying ¯ w to each side of this identity we find
0 = { ( ¯ w η 1 ) · ( w 1 z 1 ) + ( ¯ w η 2 ) · ( w 2 z 2 ) , if w U z 1 U z 2 ( ¯ w η 1 ) · ( w 1 z 1 ) , if w U z 1 U z 2 ( ¯ w η 2 ) · ( w 2 z 2 ) , if w U z 2 U z 1
(4.14)
Similarly, applying ¯ z , we have
0 = { ( ¯ z η 1 ) · ( w 1 z 1 ) + ( ¯ z η 2 ) · ( w 2 z 2 ) , if w U z 1 U z 2 ( ¯ z η 1 ) · ( w 1 z 1 ) , if w U z 1 U z 2 ( ¯ z η 2 ) · ( w 2 z 2 ) , if w U z 2 U z 1
(4.15)
Now
¯ w η ¯ z η = ( ¯ w η 1 ¯ z η 2 ¯ w η 2 ¯ z η 1 ) dw 1 dw 2
(4.16)
Note that if w U z 1 U z 2 then w 1 z 1 , and so showing that
¯ w η ¯ z η = 0 for w U z 1 U z 2
is equivalent to showing that
( ¯ w η ¯ z η ) · ( w 1 z 1 ) = 0
that is (using (4.16))
( ¯ w η 1 · ( w 1 z 1 ) ¯ z η 2 ¯ w η 2 ¯ z η 1 · ( w 1 z 1 ) ) dw 1 dw 2 = 0
but this is indeed true by the generating form hypothesis on η as expressed in (4.14) and (4.15). This shows that the desired conclusion is true when w U z 1 U z 2 ; the case: w U z 2 U z 1 is dealt with in a similar fashion. Finally, if w U z 1 U z 2 , then ( w 1 z 1 ) ( w 2 z 2 ) 0 and
( ¯ w η ¯ z η ) · ( w 1 z 1 ) ( w 2 z 2 ) = ( ( ¯ w η 1 ) · ( w 1 z 1 ) ( ¯ z η 2 ) · ( w 2 z 2 ) + ( ¯ w η 2 ) · ( w 2 z 2 ) ( ¯ z η 1 ) · ( w 1 z 1 ) ) dw 1 dw 2
but the two terms in the righthand side of this identity cancel out on account of (4.14) and (4.15).

5 Reproducing formulas: some general facts

In this section we highlight the theory of reproducing formulas for holomorphic functions by means of integral operators that arise from the Cauchy–Fantappié formalism. One of our goals here is to show that the usual reproducing properties of such operators extend to the situation where the data and the generating form have lower regularity. We begin with a rather specific example: the reproducing formula for the Bochner–Martinelli integral, see Proposition 1. The proof of this result is a consequence of a recasting of the classical mean value property for harmonic functions in terms of an identity (5.1) that links the Bochner-artinelli CF form on a sphere with the sphere’s Euclidean surface measure.

Because the Bochner–Martinelli integral of a continuous function is, in general, not holomorphic in z , in fact we need a more general version of Proposition 1 that applies to integral operators whose kernel is allowed to be any Cauchy–Fantappié form: this is done in Proposition 2.

While the operators defined so far are given by surface integrals over the boundary of the ambient domain, following an idea of Ligocka [26] another family of integral operators can be defined (essentially by ifferentiating the kernels of the operators in the statement of Proposition 2) which are realized as ‘solid” integrals over the ambient domain, and we show in Proposition 3 that such operators, too, have the reproducing property.

Lemma 1

Let z C n be given and consider a ball centered at such z , B r ( z ) = { w C n , | w z | < r } .

Then, at the center z and for any w b B r ( z ) we have that the Bochner–Martinelli CF form for the ball B r ( z ) has the following representation
C BM ( w , z ) = d σ ( w ) σ ( b B r ( z ) )
(5.1)
where d σ ( w ) is the element of Euclidean surface measure for b B r ( z ) , and
σ ( b B r ( z ) ) = 2 π n r 2 n 1 ( n 1 ) !
denotes surface measure of the sphere b B r ( z ) .

Proof

We claim that the desired conclusion is a consequence of the following identity
Ω 0 ( w β ) ( w , z ) = ( n 1 ) ! 2 π n w β ( w , z )
(5.2)
where, as usual, we have set β ( w , z ) = | w z | 2 , and denotes the Hodge-star operator mapping forms of type ( p , q ) to forms of type ( n q , n p ) . Let us first prove (5.1) assuming the truth of (5.2). To this end, we first note that from (5.2) and basic property 1 we have
Ω 0 ( w β β ) ( w , z ) = ( n 1 ) ! 2 π n β n w β ( w , z ) , w C n .
But w β ( w , z ) = ρ ( w ) , w C n with ρ ( w ) : = β ( w , z ) r 2 , a defining function for B r ( z ) . Now recall that C BM ( w , z ) = j Ω 0 ( w β / β ) where j is the inclusion: b B r ( z ) C n , see Example 2, so that j β n = r 2 n . Combining these facts we conclude that, for ρ as above
C BM ( w , z ) = ( n 1 ) ! 2 π n r 2 n j ( ρ ) ( w ) , w b B r ( z )
and since d ρ ( w ) = 2 r whenever w b B r ( z ) , we obtain
C BM ( w , z ) = ( n 1 ) ! 2 π n r 2 n 1 2 j ( ρ ) d ρ ( w ) , w b B r ( z ) ;
but
d σ ( w ) = 2 j ( ρ ) d ρ ( w ) , w b B r ( z )
(5.3)
see [34, corollary III.3.5], and this gives (5.1).
We are left to prove (5.2): to this end, we assume n = 2 and leave the case of arbitrary complex dimension as an exercise to for the reader. Since
dw j = 1 2 i 2 dw j d w ¯ j dw j , where j = { 1 , 2 } { j }
and
w β = ( w ¯ 1 z ¯ 1 ) dw 1 + ( w ¯ 2 z ¯ 2 ) dw 2
then
w β = 1 2 i 2 ( w ¯ 1 z ¯ 1 ) dw 1 d w ¯ 2 dw 2 + ( w ¯ 2 z ¯ 2 ) dw 2 d w ¯ 1 dw 1
On the other hand
¯ w w β = d w ¯ 1 dw 1 + d w ¯ 2 dw 2
and so
Ω 0 ( w β ) = 1 ( 2 π i ) 2 w β ¯ w w β = 1 ( 2 π i ) 2 ( ( w ¯ 1 z ¯ 1 ) dw 1 d w ¯ 2 dw 2 + ( w ¯ 2 z ¯ 2 ) dw 2 d w ¯ 1 dw 1 ) = 1 2 π 2 w β .
This shows (5.2) and concludes the proof of the lemma.

(We remark in passing that identity (5.1), while valid for the Bochner–Martinelli generating form, is not true for general η .)

Definition 2

Given an integer 1 k and a bounded domain D C n , we say that D is of class C k (alternatively, D is C k -smooth)if there is an open neighborhood U of the boundary of D , and a real-valued function ρ C k ( U ) such that
U D = { w U | ρ ( w ) < 0 }
and
ρ ( w ) 0 for any w U .
Any such function is called a defining function for D .
From this definition it follows that
b D = { w U | ρ ( w ) = 0 } and U D ¯ = { w U | ρ ( w ) > 0 } .

Proposition 1

For any bounded domain V C n with boundary of class C 1 and for any f ϑ ( V ) C ( V ¯ ) , we have
f ( z ) = C BM f ( z ) , z V .

Proof

Given z V , let r > 0 be such that
B r ( z ) ¯ V .
Note that the mean value property for harmonic functions:
f ( z ) = 1 σ ( b B r ( z ) ) b B r ( z ) f ( w ) d σ ( w ) , f Harm ( B r ( z ) ) C ( B r ( z ) ¯ )
and identity (5.1) give
f ( z ) = w b B r ( z ) f ( w ) C BM ( w , z )
(5.4)
To prove the conclusion, we apply Stokes’ theorem on the set
V r ( z ) : = V B r ( z ) ¯
and we obtain
w V r ( z ) d w ( f ( w ) Ω 0 ( w β β ( w , z ) ) ) = w bV r ( z ) f ( w ) C BM ( w , z )
But by basic property 2, and since f is holomorphic, we have
d w ( f ( w ) Ω 0 ( w β β ( w , z ) ) ) = f ( w ) ¯ w Ω 0 ( w β β ( w , z ) ) = 0
and so the previous identity becomes
w bV f ( w ) C BM ( w , z ) = w b B r ( z ) f ( w ) C BM ( w , z )
but the lefthand side is C BM f ( z ) , while (5.4) says that the righthand side equals f ( z ) .

Proposition 2

Let D C n be a bounded domain of class C 1 and let z D be given. Suppose that η ( · , z ) is a generating form at z relative to D . Suppose, furthermore, that the coefficients of η ( · , z ) are in C 1 ( U z ) , where U z is as in Definition 1. Then, we have
f ( z ) = w b D f ( w ) j Ω 0 ( η ) ( w , z ) for any f ϑ ( D ) C ( D ¯ ) .
(5.5)

Proof

Consider a smooth open neighborhood of b D , which we denote U z ( b D ) , such that
U z ( b D ) U z
(5.6)
where U z is as in (4.1) and (4.2).Now fix two neighborhoods U and U of the boundary of D such that
U U U z ( b D )
and let χ 0 ( w , z ) be a smooth cutoff function such that
χ 0 ( w , z ) = { 1 if w U 0 if w C n U ¯
(5.7)
Define
η ( w , z ) = χ 0 ( w , z ) η ( w , z ) + ( 1 χ 0 ( w , z ) ) w β β ( w , z )
and
D = D U z ( b D ) .
Then η is generating at z relative to D (and the open set U z of Definition 1 is the same for η and for η ); furthermore, it follows from (5.6) that
D ¯ U z .
Now let { η } N be a sequence of ( 1 , 0 ) -forms with coefficients in C 2 ( D ¯ ) with the property that
η η ( · , z ) C 1 ( D ¯ ) 0 as .
Suppose first that f ϑ ( U ( D ¯ ) ) . Then by type considerations (and since f is holomorphic in a neighborhood of D ¯ ) for any w D and for any we have
d w ( f ( w ) Ω 0 ( η ) ( w , z ) ) = ¯ w ( f ( w ) Ω 0 ( η ) ( w , z ) ) = f ( w ) ¯ w Ω 0 ( η ) ( w , z ) = f ( w ) ( ¯ w η ) n ( w , z )
Thus, applying Stokes’ theorem on D we find
w D f ( w ) ( ¯ w η ) n ( w , z ) + w b D f ( w ) j Ω 0 ( η ) ( w , z ) = w D b ( U z ( b D ) ) f ( w ) j Ω 0 ( η ) ( w , z )
Letting in the identity above we obtain
w D f ( w ) ( ¯ w η ) n ( w , z ) + w b D f ( w ) j Ω 0 ( η ) ( w , z ) = w D b ( U z ( b D ) ) f ( w ) j Ω 0 ( η ) ( w , z )
Since η is generating at z , by basic property 2 this expression is reduced to
w b D f ( w ) j Ω 0 ( η ) ( w , z ) = w D b ( U z ( b D ) ) f ( w ) j Ω 0 ( η ) ( w , z )
(5.8)
But
η ( w , z ) = { η ( w , z ) , for w in an open neighborhood of b D w β β ( w , z ) , for w in an open neighborhood of b ( U z ( b D ) )
as a result, (5.8) reads
w b D f ( w ) j Ω 0 ( η ) ( w , z ) = w D b ( U z ( b D ) ) f ( w ) C BM ( w , z )
On the other hand, D b ( U z ( b D ) ) = b V for a (smooth) open set V D , and using Proposition 1 we conclude that (5.5) holds in the case when f ϑ ( U ( D ¯ ) ) . To prove the conclusion in the general case: f ϑ ( D ) C ( D ¯ ) , we write D = { ρ ( w ) < 0 } , so that ρ ( z ) < 0 (since z D ) and furthermore
z D k : = { w | ρ ( w ) < 1 k } for any k k ( z ) .
(5.9)
But D k D and so f ϑ ( U ( D ¯ k ) ) ; moreover
b D k U z for k = k ( z ) sufficiently large .
Thus, (5.5) grants
w b D k f ( w ) j k Ω 0 ( η ) ( w , z ) = f ( z ) for any k k ( z )
where j k denotes the pullback under the inclusion j k : b D k C n .

The conclusion now follows by letting k .

We remark that in the case when η is the Bochner–Martinelli generating form η : = w β / β , Proposition 2 is simply a restatement of Proposition 1. However, since the coefficients of the Bochner–Martinelli CF form are nowhere holomorphic in the variable z , Proposition 1 is of limited use in the investigation of the Cauchy–Szegö and Bergman projections, and Proposition 2 will afford the use of more specialized choices of η .

The following reproducing formula is inspired by an idea of Ligocka [26].

Proposition 3

With same hypotheses as in Proposition 2, we have
f ( z ) = 1 ( 2 π i ) n w D f ( w ) ( ¯ w η ˜ ) n ( w , z ) , f ϑ ( D ) L 1 ( D )
for any ( 1 , 0 ) -form η ˜ ( · , z ) with coefficients in C 1 ( D ¯ ) such that
j Ω 0 ( η ˜ ) ( · , z ) = j Ω 0 ( η ) ( · , z )
(5.10)
where j denotes the pullback under the inclusion j : b D C n .
Note that if one further assumes that the coefficients of η ˜ ( · , z ) are in C 2 ( D ) C 1 ( D ¯ ) then, as a consequence of the fact that ¯ ¯ = 0 , we have
1 ( 2 π i ) n ( ¯ w η ˜ ) n = ¯ w Ω 0 ( η ˜ ) .

Proof

Fix z D arbitrarily and let { η ˜ } N C 1 , 0 2 ( D ¯ ) be such that
η ˜ η ˜ ( · , z ) C 1 ( D ¯ ) 0 as .
(5.11)
Suppose first that f ϑ ( U ( D ¯ ) ) . Applying Stokes’ theorem to the ( n , n 1 ) -form with coefficients in C 1 ( D ¯ )
f · Ω 0 ( η ˜ )
we find
w D f ( w ) ¯ Ω 0 ( η ˜ ) ( w ) = w b D f ( w ) j Ω 0 ( η ˜ ) ( w ) for any .
On the other hand, since the coefficients of η ˜ are in C 2 ( D ) , we have
¯ Ω 0 ( η ˜ ) = 1 ( 2 π i ) n ( ¯ η ˜ ) n for any
and so the previous identity can be written as
1 ( 2 π i ) n w D f ( w ) ( ¯ η ˜ ) n ( w ) = w b D f ( w ) j Ω 0 ( η ˜ ) ( w ) for any .
Letting in the identity above and using (5.11) we obtain
1 ( 2 π i ) n w D f ( w ) ( ¯ η ˜ ) n ( w , z ) = w b D f ( w ) j Ω 0 ( η ˜ ) ( w , z ) .
Combining the latter with the hypothesis (5.10) we obtain
1 ( 2 π i ) n w D f ( w ) ( ¯ w η ˜ ) n ( w , z ) = w b D f ( w ) j Ω 0 ( η ) ( w , z ) = f ( z )
where the last identity is due to Proposition 2.
If f ϑ ( D ) L 1 ( D ) then f ϑ ( U ( D ¯ k ) ) , where D k is as in (5.9); moreover, b D k U z for any k k ( z ) , so by the previous case we have
f ( z ) = w D k f ( w ) ( ¯ w η ˜ ) n ( w , z ) for any k k ( z ) .
The conclusion now follows by observing that
w D k f ( w ) ( ¯ w η ˜ ) n ( w , z ) w D f ( w ) ( ¯ w η ˜ ) n ( w , z )
as k , by the Lebesgue dominated convergence theorem.
Note that the extension η ˜ ( w , z ) : = χ 0 ( w , z ) η ( w , z ) , with χ 0 as in (5.7), satisfies a stronger condition than (5.10), namely the identity
η ˜ ( · , z ) = η ( · , z ) for any w U z ( b D ) .
(5.12)
On the other hand, it will become clear in the sequel that this simple-minded extension is not an adequate tool for the investigation of the Bergman projection, and more ad-hoc constructions are presented in Sects. 7 and 9.

6 The role of pseudo-convexity

In order to obtain operators that satisfy the crucial condition (c) in Sect. 3 one would need generating forms whose coefficients are holomorphic. However, in contrast with the situation for the planar case (where the Cauchy kernel plays the role of a universal generating form with holomorphic coefficient) in higher dimension there is a large class of domains V C n that cannot support generating forms with holomorphic coefficients.1 This dichotomy is related to the notion of domain of holomorphy, that is, the property that for any boundary point w b V there is a holomorphic function f w ϑ ( D ) that cannot be continued holomorphically in a neighborhood of w . (Such notion is in turn equivalent to the notion of pseudo-convexity.) It is clear that every planar domain V C is a domain of holomorphy, because in this case one may take f w ( z ) : = ( w z ) 1 where w b V has been fixed. On the other hand the following
V = { z C 2 | 1 / 2 < | z | < 1 }
is a simple example of a smooth domain in C 2 that is not a domain of holomorphy; other classical examples are discussed e.g., in [34, theorem II.1.1.]. A necessary condition for the existence of a generating form η whose coefficients are holomorphic in the sense described above is then that V be a domain of holomorphy. To prove the necessity of such condition, it suffices to observe that as a consequence of (4.2) and (4.1) one has
j = 1 n η j ( w , z ) ( w j z j ) = 1 for any w b V , z V .
(6.1)
It is now clear that for each fixed w b V , at least one of the η j ( w , z ) ’s blows up as z w (and it is well known that this is strong enough to ensure that V be a domain of holomorphy).

In its current stage of development, the Cauchy–Fantappié framework is most effective in the analysis of two particular categories of pseudo-convex domains: these are the strongly pseudo-convex domains and the related category of strongly C -linearly convex domains.

Definition 3

We say that a domain D C n is strongly pseudo-convex if D is of class C 2 and if any defining function ρ for D satisfies the following inequality
L w ( ρ ) ( ξ ) : = j , k = 1 n 2 ρ ( w ) ζ j ζ ¯ k ξ j ξ ¯ k > 0 for any w b D , ξ T w C ( b D )
(6.2)
where T w C denotes the complex tangent space to b D at w , namely
T w C ( b D ) = { ξ C n | ρ ( w ) , ξ = 0 } ,
see [34, proposition II.2.14].
If D is of class C k with k 1 , and if ρ 1 and ρ 2 are two distinct defining functions for D , it can be shown that there is a positive function h of class C k 1 in a neighborhood U of the boundary of D , such that
ρ 1 ( w ) = h ( w ) ρ 2 ( w ) , w U ,
and
ρ 1 ( w ) = h ( w ) ρ 2 ( w ) for any w U b D ,
(6.3)
see [34, lemma II.2.5]. As a consequence of (6.3), if condition (6.2) is satisfied by one defining function then it will be satisfied by every defining function. The hermitian form L w ( ρ ) defined by (6.2) is called the Levi form, or complex Hessian, of ρ at w . We remark that in fact there is a “special” defining function ρ for D that is strictly plurisubharmonic on a neighborhood U of D ¯ , that is
L w ( ρ ) ( ξ ) > 0 for any w U and any ξ C n { 0 } ,
(6.4)
see [34, proposition II.2.14], and we will assume throughout the sequel that ρ satisfies this stronger condition.

We should point out that there is another notion of strong pseudo-convexity that includes the domains of Definition 3 as a subclass (this notion does not require the gradient of ρ to be non-vanishing on b D ); within this more general context, the domains of Definition 3 are sometimes referred to as “strongly Levi-pseudo-convex”, see [34, §II.2.6 and II.2.8].

Definition 4

We say that D C n is strongly C -linearly convex if D is of class C 1 and if any defining function for D satisfies this inequality:
| ρ ( w ) , w z | C | w z | 2 for any w b D , z D ¯ .
(6.5)
We we call those domains that satisfy the following, weaker condition
| ρ ( w ) , w z | > 0 for any w b D and any z D ¯ { w }
(6.6)
strictly C -linearly convex. This condition is related to certain separation properties of the domain from its complement by (real or complex) hyperplanes, see [1], [20, IV.4.6]: that this must be so is a consequence of the assertion that, for w and z as in (6.5), the quantity | ρ ( w ) , w z | is comparable to the Euclidean distance of z to the complex tangent space T w C ( b D ) ; we leave the verification of this assertion as an exercise for the reader.
It is not difficult to check that
D : = { z C n | Im z n > ( | z 1 | 2 + + | z n 1 | 2 ) 2 }
is strictly, but not strongly, C -linearly convex.

Lemma 2

If D is strictly C -linearly convex then for any z D there is an open set U z C n { z } such that b D U z and inequality (6.6) holds for any w in U z . Furthermore, if D is strongly C -linearly convex then the improved inequality (6.5) will hold for any w U z .

Proof

Suppose that D is strictly C -linearly convex and fix z D . By the continuity of the function h ( ζ ) : = | ρ ( ζ ) , ζ z | , if (6.6) holds at w b D then there is an open neighborhood U z ( w ) such that h ( ζ ) > 0 for any ζ U z ( w ) and so we have that h ( ζ ) > 0 whenever
ζ U z : = w b D U z ( w ) .
It is clear that b D U z ; furthermore, since h ( z ) = 0 then U z ( w ) C n { z } for any w b D and so U z C n { z } .

If D is strongly C -linearly convex then the conclusion will follow by considering the function h ( ζ ) : = | ρ ( ζ ) , ζ z | C | ζ z | 2 .

Remark 1

We recall that in the classical definition of strong (resp. strict) convexity, the quantity | ρ ( w ) , w z | in the left-hand side of (6.5) (resp. (6.6)) is replaced by Re ρ ( w ) , w z : it follows that any strongly (resp. strictly) convex domain is indeed strongly (resp. strictly) C -linearly convex, but the converse is in general not true. It is clear that strongly (resp. strictly) convex domains satisfy a version of Lemma 2.

Lemma 3

Any strongly C -linearly convex domain of class C 2 is strongly pseudo-convex.

The key point in the proof of this lemma is the observation that, as a consequence of (6.5), the real tangential Hessian of any defining function for a domain as in Lemma 3 is positive definite when restricted to the complex tangent space T w C ( b D ) (viewed as a vector space over the real numbers). The converse of Lemma 3 is not true: we leave as an exercise for the reader to verify that the following (smooth) domain
D : = { z = ( z 1 , z 2 ) C 2 | Im z 2 > 2 ( Re z 1 ) 2 ( Im z 1 ) 2 }
is strongly pseudo-convex but not strongly C -linearly convex.
In closing this section we remark that while the designations “strongly” and “strictly” indicate distinct families of C -linearly convex domains (and of convex domains), for pseudo-convex domains there is no such distinction, and in fact in the literature the terms “strictly pseudo-convex” and “strongly pseudo-convex” are often interchanged: this is because the positivity condition (6.4) implies the seemingly stronger inequality
L w ( ρ , ξ ) c 0 | ξ | 2 for any w U and for any ξ C n .
(6.7)
Indeed, if (6.4) holds then the function γ ( w ) : = min { L w ( ρ , ξ ) | | ξ | = 1 } is positive, and by bilinearity it follows that L w ( ρ , ξ ) γ ( w ) | ξ | 2 for any ξ C n ; since ρ is of class C 2 (and D is bounded) we may further take the minimum of γ ( w ) over, say, w U U and thus obtain (6.7), see [34, II.(2.26)].

7 Locally holomorphic kernels

A first step in the study of the Bergman and Cauchy–Szegö projections is the construction of integral operators with kernels given by Cauchy–Fantappié forms that are (at least) locally holomorphic in z , that is for z in a neighborhood of each (fixed) w : it is at this juncture that the notion of strong pseudo-convexity takes center stage. In this section we show how to construct such operators in the case when D is a bounded, strongly pseudo-convex domain, and we then proceed to prove the reproducing property.

To this end, we fix a strictly plurisubharmonic defining function for D ; that is, we fix
ρ : C n R , ρ C 2 ( C n )
such that D = { ρ < 0 } ; ρ ( w ) 0 for any w b D and
L w ( ρ , w z ) 2 c 0 | w z | 2 , w , z C n
where L w denotes the Levi form for ρ , see (6.2) and (6.7). Consider the Levi polynomial of ρ at w :
Δ ( w , z ) : = ρ ( w ) , w z 1 2 j , k = 1 n 2 ρ ( w ) ζ j ζ k ( w j z j ) ( w k z k )

Lemma 4

Suppose D = { ρ ( w ) < 0 } is bounded and strongly pseudo-convex. Then, there is ϵ ˜ 0 = ϵ ˜ 0 ( c 0 ) > 0 such that
2 R e Δ ( w , z ) ρ ( w ) ρ ( z ) + c 0 | w z | 2
whenever w D c 0 = { w | ρ ( w ) < c 0 } , and z B ϵ ˜ 0 ( w ) ¯ .
Here c 0 is as in (7.1). We leave the proof of this lemma, along with the corollary below, as an exercise for the reader. Now let χ 1 ( w , z ) be a smooth cutoff function such that
χ 1 ( w , z ) = { 1 , if | w z | < ϵ ˜ 0 / 2 0 , if | w z | > ϵ ˜ 0
(7.1)
where ϵ ˜ 0 is as in Lemma 4 and set
g ( w , z ) = χ 1 ( w , z ) Δ ( w , z ) + ( 1 χ 1 ( w , z ) ) | w z | 2 , w , z C n .
(7.2)

Lemma 5

Suppose D = { ρ ( w ) < 0 } is strongly pseudo-convex and of class C 2 . Then, there is δ ˜ 0 = δ ˜ 0 ( ϵ ˜ 0 , c 0 ) > 0 such that
2 R e g ( w , z ) { ρ ( w ) ρ ( z ) + c 0 | w z | 2 , i f | w z | ϵ ˜ 0 / 2 ρ ( w ) + 2 δ ˜ 0 , i f ϵ ˜ 0 / 2 | w z | < ϵ ˜ 0 ϵ ˜ 0 2 , i f | w z | > ϵ ˜ 0
whenever
w D c 0 = { w | ρ ( w ) < c 0 }
(7.3)
and
z D 2 δ ˜ 0 = { w | ρ ( w ) < 2 δ ˜ 0 } .

Proof

It suffices to choose 0 < δ ˜ 0 < c 0 ϵ ˜ 0 2 / 16 : the desired inequalities then follow from Lemma 4.

Corollary 1

Let D = { ρ ( w ) < 0 } be a bounded, strongly pseudo-convex domain. Let
Δ j ( w , z ) : = ρ ζ j ( w ) 1 2 k = 1 n 2 ρ ( w ) ζ j ζ k ( w k z k ) , j = 1 , , n ,
Define
η j ( w , z ) : = 1 g ( w , z ) ( χ 1 ( w , z ) Δ j ( w , z ) + ( 1 χ 1 ( w , z ) ) ( w ¯ j z ¯ j ) )
where χ 1 and g are as in (7.1) and (7.2), and set
η ( w , z ) : = j = 1 n η j ( w , z ) dw j for ( w , z ) D c 0 × D
with D c 0 as in (7.3). Then we have that η ( w , z ) is a generating form for D , and one may take for U z in Definition 1 the set
U z : = { w | max { ρ ( z ) , δ ˜ 0 } < ρ ( w ) < min { | ρ ( z ) | , c 0 } } .
(7.4)
Note, however, that the coefficients of η in this construction are only continuous in the variable w and so the Cauchy–Fantappié form Ω 0 ( η ) cannot be defined for such η because doing so would require differentiating the coefficients of η with respect to w , see (4.4). For this reason, proceeding as in [34], we refine the previous construction as follows. For ϵ ˜ 0 as in Lemma 4 and for any 0 < ϵ < ϵ ˜ 0 , we let τ j , k ϵ C ( C n ) be such that
max w D ¯ | 2 ρ ( w ) ζ j ζ k τ j , k ϵ ( w ) | < ϵ , j , k = 1 , , n
We now define the following quantities:
Δ j ϵ ( w , z ) : = ρ ζ j ( w ) 1 2 k = 1 n τ j , k ϵ ( w ) ( w k z k ) , j = 1 , , n ;
(7.5)
Δ ϵ ( w , z ) : = j = 1 n Δ j ϵ ( w , z ) ( w j z j ) ;
and, for χ 1 as in (7.1):
g ϵ ( w , z ) : = χ 1 ( w , z ) Δ ϵ ( w , z ) + ( 1 χ 1 ( w , z ) ) | w z | 2 ;
(7.6)
η j ϵ ( w , z ) : = 1 g ϵ ( w , z ) ( χ 1 ( w , z ) Δ j ϵ ( w , z ) + ( 1 χ 1 ( w , z ) ) ( w ¯ j z ¯ j ) )
and finally
η ϵ ( w , z ) : = j = 1 n η j ϵ ( w , z ) dw j .

Lemma 6

Let D = { ρ ( w ) < 0 } be a bounded strongly pseudo-convex domain. Then, there is ϵ 0 = ϵ 0 ( c 0 ) > 0 such that for any 0 < ϵ < ϵ 0 and for any z D , we have that η ϵ ( w , z ) defined as above is generating at z relative to D with an open set U z (see Definition 1) that does not depend on ϵ . Furthermore, we have that for each (fixed) z D the coefficients of η ϵ ( · , z ) are in C 1 ( U z ) .

Proof

We first observe that Δ ϵ can be expressed in terms of the Levi polynomial Δ , as follows
Δ ϵ ( w , z ) : = Δ ( w , z ) + 1 2 j , k = 1 n ( 2 ρ ( w ) ζ j ζ k τ j , k ϵ ( w ) ) ( w j z j ) ( w k z k )
and so by Lemma 4 we have
2 R e Δ ϵ ( w , z ) ρ ( w ) ρ ( z ) + c 0 | w z | 2
for any
0 < ϵ < ϵ 0 : = min { ϵ ˜ 0 , 2 c 0 / n 2 }
whenever w D c 0 = { ρ ( w ) < c 0 } and z B ϵ 0 ( w ) ¯ . Proceeding as in the proof of Lemma 5 we then find that
2 R e g ϵ ( w , z ) { ρ ( w ) ρ ( z ) + c 0 | w z | 2 , i f | w z | ϵ 0 / 2 ρ ( w ) + μ 0 , i f ϵ 0 / 2 | w z | < ϵ ˜ 0 ϵ ˜ 0 2 , i f | w z | ϵ ˜ 0
for any 0 < ϵ < ϵ 0 whenever
w D c 0 = { w | ρ ( w ) < c 0 }
and
z D μ 0 = { w | ρ ( w ) < μ 0 }
as soon as we choose
0 < μ 0 < c 0 ϵ 0 2 / 8 .
(7.7)
We then define the open set U z C n { z } as in (7.4) but now with δ 0 in place of δ ˜ 0 (note that U z does not depend on ϵ ). Then, proceeding as in the proof of corollary 1 we find that
inf w U z Re g ϵ ( w , z ) > 0 for any 0 < ϵ < ϵ 0 .
From this it follows that η ϵ is a generating form for D ; it is clear from (7.5) that the coefficients of η ϵ are in C 1 ( U z ) .

Lemma 6 shows that η ϵ satisfies the hypotheses of Proposition 2; as a consequence we obtain the following results:

Proposition 4

Let D be a bounded strongly pseudo-convex domain. Then, for any 0 < ϵ < ϵ 0 we have
f ( z ) = w b D f ( w ) j Ω 0 ( η ϵ ) ( w , z ) for any f ϑ ( D ) C ( D ¯ ) , z D
where ϵ 0 and η ϵ are as in Lemma 6.

Proposition 5

Let D = { ρ ( w ) < 0 } be a bounded strongly pseudo-convex domain. Let
η ˜ ϵ ( w , z ) : = g ϵ ( w , z ) g ϵ ( w , z ) ρ ( w ) η ϵ ( w , z ) , w D ¯ , z D .
where η ϵ is as in Lemma 6. Then, for any 0 < ϵ < ϵ