Cauchy-type integrals in several complex variables
We present the theory of Cauchy–Fantappié integral operators, with emphasis on the situation when the domain of integration, , 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 . The goal is to prove estimates for these operators and, as a consequence, to obtain 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
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 mapping properties under “minimal” conditions of smoothness of the boundary of the domain 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” (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” .
When is strongly pseudo-convex and of class ;
When is strongly -linearly convex and of class
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 , 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 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 . To achieve the desired holomorphicity requires that the domain be pseudo-convex, and two specific forms of this property, strong pseudo-convexity and strong-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 ; a later approach is in Kerzman–Stein , 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. 7–9. One should note that in the case of strongly -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 estimates for the Cauchy–Leray integral and the Szegö and Bergman projections (for 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 results are consequences of estimates that hold for the corrected Cauchy–Fantappié kernels, denoted and , 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 theorem under the weaker assumption that the boundary is merely of class .
A survey of this kind must by the nature of the subject be far from complete. Among matters not covered here are 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 () will be denoted . The notation will indicate the boundary of a domain () and, for sufficiently smooth, will denote arc-length () or Euclidean surface measure ().
2 The Case
In the case of one complex dimension the problem of 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.)
, if is in fact of class .
. Here depends on , but , and is the exponent dual to .
3 Cauchy integral in ; some generalities
There is no “universal” holomorphic Cauchy kernel associated to a domain .
Pseudo-convexity of , must, in one form or another, play a role.
Since this condition involves (implicitly) two derivatives, the “best” results are to be expected “near” , (as opposed to near when ).
The kernel should be given by a “natural” or explicit formula (at least up to first approximation) that involves .
The mapping should reproduce holomorphic functions. In particular if is continuous in and holomorphic in then , for .
should be holomorphic in , for any given that is continuous on .
4 Cauchy–Fantappié theory in higher dimension
4.1 Cauchy–Fantappié forms
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 is nowhere holomorphic in : as a result, when 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
Terms that contain : but these do not contribute to because (which follows from since has degree );
The term , which gives the desired conclusion.
Suppose, further, that is generating at relative to . Then the following two properties also hold.
Basic Property 2
In the previous properties was fixed; here it is allowed to vary.
Basic Property 3
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 , 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  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.
Let be given and consider a ball centered at such , .
(We remark in passing that identity (5.1), while valid for the Bochner–Martinelli generating form, is not true for general .)
The conclusion now follows by letting .
We remark that in the case when is the Bochner–Martinelli generating form , 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 , 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 .
The following reproducing formula is inspired by an idea of Ligocka .
6 The role of pseudo-convexity
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-linearly convex domains.
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 to be non-vanishing on ); 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].
If is strictly -linearly convex then for any there is an open set such that and inequality (6.6) holds for any in . Furthermore, if is strongly -linearly convex then the improved inequality (6.5) will hold for any .
If is strongly -linearly convex then the conclusion will follow by considering the function .
We recall that in the classical definition of strong (resp. strict) convexity, the quantity in the left-hand side of (6.5) (resp. (6.6)) is replaced by : it follows that any strongly (resp. strictly) convex domain is indeed strongly (resp. strictly) -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.
Any strongly -linearly convex domain of class is strongly pseudo-convex.
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 , that is for in a neighborhood of each (fixed) : 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 is a bounded, strongly pseudo-convex domain, and we then proceed to prove the reproducing property.
It suffices to choose : the desired inequalities then follow from Lemma 4.
Let be a bounded strongly pseudo-convex domain. Then, there is such that for any and for any , we have that defined as above is generating at relative to with an open set (see Definition 1) that does not depend on . Furthermore, we have that for each (fixed) the coefficients of are in .
Lemma 6 shows that satisfies the hypotheses of Proposition 2; as a consequence we obtain the following results: