Search

A new test for holomorphy in the unit ball of C" in the spirit of Morera's theorem is given. The purpose of this note is to give a new criterion for holomorphy in the unit ball Bn of Cn (n 2 1) in the spirit of the generalization of the classical Morera theorem given by Zalcman in the case of the complex plane [10]. We denote by D1 the group of biholomorphic mappings of Bn. For n = 1 this is just the classical Moebius group; we refer to [8] for a description in the general case. Bn is a rank-one symmetric space and, as such, it has an 9Dhinvariant metric, the Poincare metric and a corresponding 9Dkinvariant measure du = p(z) dA, where dX is the Lebesgue measure and p(z) = (1 I z12)-2'. In order to integrate over hypersurfaces we need differential forms generalizing dz from n = 1 to higher dimensions; these are the (n, n 1) forms Wk = dz A Adzn A dzI A Adzk A Adzn, 1 < k < n. Let S be an open, nonempty, relatively compact subset of Bn, and assume that aui is a piecewise C'-manifold, so that the Stokes theorem can be applied. If f is smooth in Bn, we have (1) f fWk fdf A cok = Cnkf afdA where Cn,k = (2i n(w)2n(fn-1)+k-I THEOREM. Assume that, in addition to the above, B n\Q is connected and that a g is not a real analytic manifold. Then a continuous function f in Bn is holomorphic if and only if for every a E D1 (2) | fWk = ?l k =12..n a(au) PROOF. The necessity of condition (2) follows from (1) above. To prove sufficiency note first that by mollifying the original function using a radial approximate identity we can assume that f E C?(Bn). If f is not holomorphic then, for some k, af/azk is Received by the editors March 4, 1983. 1980 Mathematics Subject Classification. Primary 43A85, 32A10, 30E99.

On demontre un resultat de Hartogs sur la localisation des singularites des fonctions holomorphes a l'aide de la theorie des systemes surdetermines d'equations differentielles lineaires

Introduction. Let 12 C C" be a bounded pseudoconvex domain. Does U have a neighborhood basis consisting of pseudoconvex domains? It is well known that the answer to this question is, in general, "no". But it has been an open problem, at least since 1933 when the fundamental paper [1] of H. Behnke and P. Thullen appeared, whether the answer might be in the affirmative under the additional hypothesis of smoothness of the boundary b£2. The main purpose of this note is to give an example of a bounded pseudoconvex domain £lx C C 2 with smooth boundary that nevertheless does not have a Stein neighborhood basis. Additional hypotheses that guarantee the existence of such a basis are given in [3]. The constructed domain £2X has some more strange properties. In particular, the conjecture of R. 0 . Wells [5, Conjecture 3.1], does not hold for £lx (Theorem 2) and the boundary bQ,1 cannot be described by a smooth function with positive semidefinite Leviform on the whole C at each point p E b£2 r Together with the beautiful example of Kohn and Nirenberg [4] , this domain Q,j shows that bounded smooth domains in C can have quite different analytic properties than strictly pseudoconvex domains. The proofs of the theorems announced in this note will be contained in a later paper of the authors. Definition of £2X. We fix a smooth function X:R —• R + U {0} with the properties X(x) = 0 for x 0 for x > 0, and such that X is "sufficiently" convex. For r > 1 we define a family of smooth functions p : ( C { 0 } ) x C —• R by putting

Nonelementary holomorphic mappings C n → C n , n ≥ 2 {{\mathbf {C}}^n} \to {{\mathbf {C}}^n},n \geq 2 , cause severe geometric distortion near ∞ \infty in sharp contrast to the case n = 1 n = 1 when there is no distortion.

Publisher Summary This chapter presents a glance at holomorphic fraction and uniform holmorphy. It proposes to develop a fairly readable introduction to the closely related subjects of holomorphic factorization and uniform holomorphy. Holomorphic factorization, respectively uniform holomorphy, holds for one of these six situations if and only if it holds for each of the remaining five ones. Holomorphic factorization holds for a given complex locally convex space E, when it holds for its standard projective limit representation or, equivalently for the remaining five situations in proposition 18, particularly. Holomorphic factorization, hence uniform holomorphy, holds for every complex locally convex space E satisfying the openness condition. If a complex locally convex space E satisfies the complete quotient condition, then CCS(E) C COS (E), and thus E satisfies also the openness condition.

N~(f) ={g(z) =z+b2z2+... : ~ a lan -bn l 2 and every ~, Igl < g, a function fn,E(z)=f(z)+ zn~st, then for every ~, I~I < ~, the function f~(z)~ St and hence N~(f)c St. For the converse theorem we only have that if for every ~, l~I < g, f~(z)~ St, then for every @', Igl < ~/n, we have fn,~St.

This Part is devoted to the problem of holomorphic equivalence. This problem consists of the following: given two domains D, G⊂ℂ n , to determine whether there exists a biholomorphic map f: D→G, or not.

1 .1 . The study of holomorphic sections of the Teichmtiller curve was initiated by Hubbard [16] (see also Earle-Kra [7 ] , [8 ] ) . Hubbard proved that for p = 2 the map V,,T, has precisely 6 holomorphic sections, the Weierstrass sections, while for p 3, it has no holomorphic sections. The base space Tp is a contractible domain of holomorphy. Nevertheless, Hubbard's result shows that for p >2 we cannot choose a point on each surface in a way that depends holomorphically on moduli. On the other hand, we can choose on every surface of genus p 2 a divisor class of degree one that depends holomorphically on moduli (see [5]).

Recall ([1], [2], [3]) that Nκ denotes the set of all complex valued functions Q which are meromorphic in the open upper half plane C + and such that the kernel NQ: $${N_Q}\left( {z,\zeta } \right):\left( {Q\left( z \right) - \overline {Q\left( \zeta \right)} } \right)/\left( {z - \overline \zeta } \right)$$ (1.1) for z,ζ e D Q has κ negative squares (here D Q (⊂C +) denotes the domain of holomorphy of Q). This means that for arbitrary n e Z and z1,z2,...,zn e D Q the matrix (NQ(zi,zj)) 1 n has at most κ negative eigenvalues and for at least one choice of n, z1,...,zn it has exactly κ negative eigenvalues. The class No coincides with the Nevanlinna class of all functions which are holomorphic in C + and map C + into C + UR. The following two examples of functions of the class N1 were considered in [2], [4], respectively: $$w\left( z \right):\alpha - z + \int\limits_{ - \infty }^\infty {\left( {{{\left( {t - z} \right)}^{ - 1}} - t{{\left( {1 + {t^2}} \right)}^{ - 1}}} \right)} d{\sigma _O}\left( t \right),v\left( z \right): = \alpha + \left( {1/z} \right) + \int\limits_{ - 8}^\infty {\left( {{{\left( {t - z} \right)}^{ - 1}} - t{{\left( {1 + {t^2}} \right)}^{ - 1}}} \right)} d{\sigma _1}\left( t \right),$$ (1.2) where α e R and σo, σl are nondecreasing functions on R such that $${\int\limits_{ - \infty }^\infty {\left( {1 + {t^2}} \right)} ^{ - 1}}d{\sigma _j}\left( t \right) < \infty ,j = 0,1\,{\sigma _1}\left( {0 + } \right) = {\sigma _1}\left( {0 - } \right).$$

In §1 of this chapter we first extend the fundamental vanishing theorem \( H_{\overline \partial }^q \left( K \right) = 0\,for\,q \geqslant 1 \) for q ≥ 1 on a Stein compactum K proved in Corollary V.2.6 to arbitrary open Stein domains (Theorem 1.4). The proof involves an approximation theorem for holomorphic functions on compact analytic polyhedra which is of independent interest, and which generalizes the classical Runge Approximation Theorem in the complex plane. We also discuss several variations of this approximation theorem. In particular we consider the Runge property for the exhaustion of a pseudoconvex domain D by strictly pseudoconvex domains which arises from the existence of a strictly plurisubharmonic exhaustion function on D. Together with the results of Chapter V this yields the solution of the Levi problem for arbitrary pseudoconvex domains. In §2 we apply these methods to solve the Cauchy-Riemann equations directly on a pseudoconvex domain D, i.e., we show that \( H_{\overline \partial }^q \left( D \right) = 0\,for\,q \geqslant 1 \) for q ≥ 1, and we prove that this property characterizes Stein domains. §3 deals with some topological properties of Stein domains D, for example, we show that if D ⊂ ℂ n , then H r (D, ℂ) = 0 for r > n. This section may be skipped without loss of continuity. Finally, in §4 and §5, the vanishing of \( H_{\overline \partial }^1 \) for Stein domains D is used to generalize to several variables the classical theorems of Mittag-Leffler and Weierstrass on the existence of global meromorphic functions with prescribed poles and zero sets on regions in the complex plane. §5 includes a detailed discussion of the new—strictly higher dimensional—phenomenon of a topological obstruction in the analog of the Weierstrass theorem.

Publisher Summary This chapter discusses the various applications of the existence of well-growing holomorphic functions and examines the question how to construct hulls of holomorphy using only a small family of holomorphic functions. Some special examples are given too. The problem of the boundary behavior of the Bergman kernel function is described. The envelope of holomorphy of a tube domain G = ℝ n + iB is given as its convex hull. The chapter reveals the proof of Bochner's theorem that shows the Fourier-Laplace-transformation plays an important role in the problem how to calculate the hull of holomorphy. To calculate the hull of holomorphy of a given domain G, it's enough to know its extension domain only for the functions of polynomial growth. Such extensions can be constructed using generalized integral representations under the assistance of the Fourier-Laplace-transformation. The chapter describes the existence theorems for square-integrable holomorphic functions.

Publisher Summary This chapter discusses the pseudo-convexity, u-convexity and domains of u-holomorphy. Relationships between these concepts and those of domain of holomorphy and pseudo-convex domains will be obtained. Throughout this chapter, E will denote locally convex Hausdorff space over ¢, c s (E) is the set of all continuous seminorms on E. χ (U) will denote the space of all holomorphic functions from U to ¢. It is clear that if U is sequentially u-holomorphically convex, then U is sequentially holomorphically convex.

In this note a survey on some results and recent developments in the field of Infinite Dimensional Holomorphy is given guided by the following simple and general question: How many holomorphic functions are there on a given open subset Ω of a locally convex Hausdorff space over This question will be considered under various aspects: 1. Bounding Sets 2. Prescribed Radius of Convergence 3. Domains of Holomorphy and Analytic Continuation 4. Holomorphic Completion 5. Compact Holomorphic Mappings and the Approximation Property.

This chapter, which is concerned with domains spread over the complex linear vector space E of a dual pair , consists mainly of extensions of results for the special case (E due to M. Matos [M1, M2] (countable ⋀) and V. Aurich [A4] (arbitrary ⋀). (See also [H3, R6].) The dual pair is fixed throughout and the topology in E is always assumed to be the σ(E, F)-topology. Also, instead of using the rather cumbersome expression, “[E, ℘ ]-domain”, for a domain spread over E, we shall use the simpler term, “ -domain”. Similarly, an -domain of ℘ -holomorphy will be called simply an -domain of holomorphy.

Publisher Summary This chapter discusses some applications of infinite-dimensional holomorphy to mathematical physics. The concepts and results of infinite-dimensional holomorphy are important to mathematical physics whenever the physical systems under consideration involve an infinite number of degrees of freedom. The chapter surveys of the use of holomorphic functions of infinitely many variables for the quantization. The standard particle (or Fock) quantization is reviewed. The chapter describes infinite-dimensional (anti-) holomorphic wave quantizations that are preferable from a conceptual or mathematical point of view. The complex wave quantization of a (flat) Hilbert space has been studied. An interesting generalization to vector-valued Planck constants is related to harmonic analysis on nilpotent Banach Lie groups arising as distinguished boundaries of symmetric Siegel domains. For systems with interaction, the quantization of curved phase spaces of infinite dimension has also been considered, and the chapter provides an introduction to this promising field, based on the theory of symmetric Hilbert manifolds.

The problem of constructing a holomorphic function of one complex variable with a given zero set was solved by Weierstrass in the middle of the nineteenth century. He showed that if Ω is a domain in the complex plane, if {a v } is a sequence of points without limit point in Ω, and if {m v } is a sequence of positive integers, then there exists a function f(z)∈ ℋ(Ω) which has a zero of order exactly m v at every point a v . The equivalent problem for several complex variables is Cousin’s Second Problem, which we state as follows: if Ω is a domain in ℂ n , then for every zero set defined locally in Ω, does there exist a global holomorphic function which defines the same zero set? More specifically, if U i is an open covering of Ω and f i ∈ ℋ(U i ) are such f i f j −1 ∈ ℋ (U i ⋂U j ) and f j f i −1 ∈ ℋ(U i ⋂U j ) for all pairs i,j, does there exist f ∈ ℋ(Ω) such that f f i −1 ∈ ℋ(U i ) and f−1 f i ∈ ℋ(U i ) for all i? The answer is in general negative, even when Ω is a domain of holomorphy, and depends upon the topological as well as the complex analytic properties of Ω (cf. [A, B]); however, when Ω is a simply connected domain of holomorphy (as in the case of ℂ n ), the answer is always affirmative.

Let E be a complex convex linear topological space (CLTS) and denote by Em the cartesian product E ×⋯× E (m factors). A function P: E → ℂ is called a homogeneous polynomial of degree m on E if there exists a nonzero, symmetric, m-linear functional $$ \beta :{E^m} \to \mathbb{C},\quad ({x_1},...,{x_m}) \to \beta ({x_1},...,{x_m}) $$ such that $$ P(x) = \beta (x,...,x),\quad x \in E $$ . A constant function is a homogeneous polynomial of degree zero. A function which is a finite sum of homogeneous polynomials is called a generalized polynomial. The collection of all generalized polynomials is obviously a point separating algebra of functions on E. Each of its elements P has a unique representation of the form P = P0 +…+ Pm, where Pk is either zero or a homogeneous polynomial of degree k, for 0 ≤ k ≤ m, and Pm ≠ 0. The integer m is called the degree of P and the polynomials Pk are called its homogeneous components. If x0 is a fixed element of E then Q(x) = P(x - x0), x ∈ E, defines a generalized polynomial with degree equal to that of P. Observe that a linear functional on E is a polynomial of degree one. Elements of the algebra generated by the linear functionals are called finite polynomials.

Let E and F denote locally convex spaces over C and let U denote an open subset of E. A function f: U→F is called holomorphic if a it is continuous, b. for each a ∈ U, υ ∈ E and \( \phi \in F' \) the mapping \( \lambda \in \mathbb{C} \to \phi o f(a + \lambda v) \) is holomorphic as a function of one complex variable on its domain of definition.

In §1 and §2 of this chapter we present the standard local properties of holomorphic functions and maps which are obtained by combining basic one complex variable theory with the calculus of several (real) variables. The reader should go through this material rapidly, with the goal of familiarizing himself with the results, notation, and terminology, and return to the appropriate sections later on, as needed. The inclusion at this stage of holomorphic maps and of complex submanifolds, i.e., the level sets of nonsingular holomorphic maps, is quite natural in several variables. In particular, it allows us to present elementary proofs of two results which distinguish complex analysis from real analysis, namely: (i) the only compact complex submanifolds of ℂ n are finite sets, and (ii) the Jacobian determinant of an injective holomorphic map from an open set in ℂ n into ℂ n is nowhere zero. Section 3, which gives an introduction to analytic sets, may be omitted without loss of continuity. We have included it mainly to familiarize the reader with a topic which is fundamental for many aspects of the general theory of several complex variables, and in order to show, by means of the Weierstrass Preparation Theorem, how algebraic methods become indispensable for a thorough understanding of the deeper local properties of holomorphic functions and their zero sets.

Let D1, D2 be bounded domains in ℂn and let o be â biholomorphic mapping between the two. Under what conditions is it possible to conclude that o extends (with some regularity) to the closure \({\bar D_1}\) of D1?

Let L be a simple contour, S + and S − the finite and infinite parts of the plane, bounded by L; let the positive direction on L be such that S + remains on the left. The contour L will not be included in S + or S −.

Publisher Summary This chapter discusses the fact that among the subjects studied in infinite dimensional holomorphy, the one of analytic continuation is perhaps the most developed. The Levi problem is positively answered in the case of Riemann domains over Banach spaces with Banach approximation property (B.A.P.). This result follows from investigations about properties of permanence for domains of existence under elementary set operations. It is observed that the behavior of the space of polynomials is directly related to the answer of the Levi problem. Holomorphic functions; morphism between Riemann domains over the same basic space; the concept of A-domain of holomorphy; etc., are all defined in the obvious way. A Banach space E is said to have the B.A.P., if E is separable and there exists a sequence of operators of finite rank.

In Section 1.1 we define holomorphic functions of several complex variables and prove simple properties of these functions. In Section 1.2, by means of a simple extension of the Cauchy-Green formula to several variables, we solve the inhomogeneous Cauchy-Riemann equations for some special cases. As a consequence we obtain a theorem of Hartogs which gives examples of open sets in ℂ n (n ≥ 2) where all holomorphic functions can be continued holomorphically to a larger open set. Open sets for which this is not possible are called domains of holomorphy. These are investigated in Section 1.3. Section 1.4 is devoted to continuous plurisubharmonic and strictly plurisubharmonic C 2 -functions. In Section 1.5, by means of these functions, pseudoconvex and strictly pseudoconvex open sets are introduced. We prove that every domain of holomorphy is pseudoconvex, but the converse (Levi’s problem) is left to Section 2.7. Sections 1.6–1.12 are devoted to integral representation formulas for functions as well as for differential forms n ℂ n . These formulas form the basic tool for the methods developed in the present book.

In this paper, we shall be concerned with the finiteness property of certain holomorphic mappings into a compact quotient of symmetric bounded domain.Let be a symmetric bounded domain in n-dimensional complex Euclidean space Cn and Γ\ be a compact quotient of S by a torsion free discrete subgroup Γ of automorphism group of . Further, we denote by l() the maximum value of dimension of proper boundary component of , which is less than n (=dim).

Suppose that f: D1 → D2 is a proper holomorphic mapping between smooth bounded domains D1 and D2 contained in ℂn. There are two fundamental problems in the theory of functions of several complex variables concerning the boundary behavior of f.

Publisher Summary This chapter presents a lecture on some of the problems in applying functional analysis to holomorphy. In the discussion greater generality in this exposition is avoided. It is assumed that E, F denote Hausdorff complex locally convex spaces, and U to be a nonvoid open substet of E. The chapter also presents several related definitions and mentions one among them as: E is holomorphically bornological (hbo) if, for every F and U, whenever f: U → F is finitely holomorphic and f is bounded on every compact subset of U, then it follows that f is holomorphic.

An n × n-matrix function Q belongs to the class N κ n×n if it is defined and meromorphic in the upper half plane C +, and for an arbitrary k e Z, z1,z2,…, zk e D Q (the domain of holomorphy of Q) and ξ1,ξ2,…,ξk e C n the matrix $${\{ [\frac{{Q({z_i}) - Q({z_j})*}}{{{z_{\text{i}}} - {{\overline z }_{\text{j}}}}}{\xi _j},{\xi _j}]\} _{i,j = 1,2,...,k}}$$ has at most κ negative eigenvalues and, for at least one choice of k, z1,z2,…,zk, it has exactly κ negative eigenvalues. We always assume that Q e N κ n×n has been extended to the lower half plane as follows: \(Q(\overline z ) = Q(z)*\) (z e D Q). For the basic properties of these functions see [5], [2].

Let M, N be connected complex submanifolds of a neighborhood of the origin 0 ∈ Cd, the space of d complex variables, such that 0 ∈ M ∩ N. We shall suppose throughout that M ⊄ N and N ⊄ M in any neighborhood of 0. Let X = M ∪ N. X is an analytic subvariety with the irreducible branches M and N. Let Δ be a neighborhood of 0 in Cd.

Let D be a bounded domain of the complex n-space Cn(n≥2), or more generally a pair (M,D) a finite manifold (cf. Definition 2.1), and we assume the boundary ∂D is a smooth and connected submanifold. It is well known by Hartogs-Osgood’s theorem that every holomorphic function on a neighbourhood of ∂D can be continued holomorphically to D. Generalizing the above theorem we shall prove that if a differentiable function on ∂D satisfies certain conditions which are satisfied for the trace of a holomorphic function on a neighbourhood of ∂D, then it can be continued holomorphically to D (Theorem 2-5). The above conditions will be called the tangential Cauchy Riemann equations.