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This note is concerned with a generalization of the first identity to the non-linear setting of holomorphic (i. e. complex analytic) functions on E. In fact, our interest in this generalization arose from a study made by the second author [15] of a suitable generalization of the second identity to the holomorphic setting. However, our results and methods have little in common with [15]. In somewhat simplistic terms, our results provide further evidence that, in many situations, one is justified in regarding the space H(E) of holomorphic functions on E as the "holomorphic dual" of E which shares many properties with the standard "linear dual" of E (see, for example, [6]). A brief outline of this paper follows, after which we recall some basic results about holomorphic mappings between infinite dimensional locally convex spaces. In Section 2, we investigate when a holomorphic function f : U-~C on a balanced open subset U of E admits a factorization through ~(U). A characterization of factorizable functions is given, in terms of the derivative o f f Our basic technique is

At the beginning of this century Hartogs [15] directed attention to an interesting property of holomorphic functions of several complex variables, namely, their continuability. For example, if a function is holomorphic on the boundary of a ball in (), then it can be continued to a function holomorphic in the whole ball. Holomorphic maps from into have greater inflexibility than functions, and this is the basis of several interesting facts. A typical situation that illustrates the continuation property of maps is a theorem of Poincare [1]. If in a neighbourhood of a point of the sphere there is defined biholomorphic map sending the sphere into itself, then it can be continued holomorphically to the whole sphere, and moreover it turns out to be a linear-fractional transformation. After Poincare this theorem was stated by other authors ( [11], [44], and [12]). Now let M be a strictly pseudoconvex compact real-analytic hypersurface in , and D the domain bounded by it (a surface is called strictly pseudoconvex if it can be transformed into a strictly convex surface by a biholomorphic change of variables close to any point of it). By Hartog's theorem every automorphism of M can be continued holomorphically to D. Can we expect every automorphism of a surface to be continued holomorphically to some domain lying on the outside of M? For spheres this is not true; for any point lying outside a sphere we can find an automorphism of the sphere for which this point is singular. But it turns out that a sphere is in a certain sense a unique exception; if M is not (locally) equivalent to a sphere, then the answer to our question is positive, that is, all automorphisms of M can be continued holomorphically to a common neighbourhood of M (see Corollary 3 to the theorem in § 8). We note that the part of this neighbourhood lying outside M does not belong to the hull of holomorphy of M, and so functions holomorphic in M cannot, in general, be continued to this part of the neighbourhood. These examples are only fascinating fragments of the trend in modern complex analysis discussed in this paper. We discuss the basic stages in the development of this trend, concentrating attention on recent results. Our subject matter is hypersurfaces in and in other complex manifolds that are non-degenerate in the sense of Levi, the forms of representing these surfaces, their automorphism groups, the geometry of Chern-Moser chains, the continuation of holomorphic maps, questions of classifications, and others.

Concerning completeness with respect to the Carathéodory distance ( c c -completeness), the following theorems are shown. A bounded convex (in geometric sense) domain D D in C n ( R 2 n ) {{\mathbf {C}}^n}({{\mathbf {R}}^{2n}}) is c c -complete, so that it is boundedly holomorphic convex. To preserve c c -completeness in complex spaces, it is sufficient to have a proper local biholomorphic mapping as follows: Let α \alpha be a proper spread map of a c c -hyperbolic complex space ( X , A ) (X,A) onto a c c -hyperbolic complex space ( X ~ , A ~ ) (\tilde X,\tilde A) ; then X X is c c -complete if and only if X ~ \tilde X is c c -complete. We also show the following D D to be domains of bounded holomorphy: let ( X , A ; α ) (X,A;\alpha ) be a Riemann domain and D D a domain in X X with α ( D ) \alpha (D) bounded in C n {{\mathbf {C}}^n} . Let B ( D ) B(D) separate the points of D D . Suppose there is a compact set K K such that for any x ∈ D x \in D there is an analytic automorphism σ ∈ Aut ⁡ ( D ) \sigma \in \operatorname {Aut} (D) and a point a ∈ K a \in K such that σ ( x ) = a \sigma (x) = a . Then D D is a domain of bounded holomorphy.

A differential geometric method is introduced to study the modulus of boundary values of holomorphic functions on smoothly bounded pseudoconvex domains D in C n , n ⩾ 2 {{\mathbf {C}}^n},n \geqslant 2 . It is shown that functions in A ( D ) A(D) are determined up to a constant factor by their modulus on an open subset of the Shilov boundary. For the case of H ∞ ( D ) {H^\infty }(D) , it is shown that inner functions which satisfy a certain local condition are constant.

SynopsisSuppose that f: ℝ×ℂN→ℂN is holomorphic in z and continuous in t, and that Φ: ℂN×ℂN→ℂN is holomorphic. Boundary value problems of the formare considered. The particular interest is in the structure and topological properties of the set of solutions. The paper is motivated by the corresponding properties of the set of periodic solutions of ż = f(t, z) when f is periodic in t. Consideration of this complex equation gives information about the periodic solutions of the real equation ẋ = f(t, x).

Let be a domain in , and a solution in of the equation , where in . It is known that, for , is foliated into complex curves on which is harmonic, and and are holomorphic. We show that if and is a complete Hartogs domain with axis of symmetry , then such a foliation exists even for . Bibliography: 10 titles.

The properties of weakly holomorphic functions on analytic sets which are complete intersections are investigated: universal denominators are determined for a system of equations defining the analytic set ; a (residual) current is constructed such that it is -closed if and only if the weakly holomorphic function can be locally extended from ; and integral representations for weakly holomorphic functions are given. These results are applied to the problem of lowering the order of poles of rational differential 2-forms in .Bibliography: 20 titles.

An analogue of the Frobenius theorem is proved for the case of a continuous planar field. This leads to a proof that it is possible to fiber into analytic curves a C1 smooth hypersurface in C2 on both sides of which lie domains of holomorphy. An example constructed of two domains of holomorphy with common boundary which does not contain analytic subsets. Bibliography: 5 titles.

A Siegel domain D D of the second kind (not necessarily affine homogeneous) is shown to be complete with respect to the Carathéodory distance. Thus D D is convex with respect to the bounded holomorphic functions, hence is a domain of holomorphy. A Phragmén-Lindelöf theorem for D D is also given. That is, if a holomorphic function f f in D D is continuous in D ¯ \bar D , bounded on the distinguished boundary S S of D D and not of exponential growth, then f f is bounded in D ¯ \bar D .

A new method of constructing analytic continuations of matrixelements of fieldoperators is presented for the example of the fourpoint-functions whose boundary values generate a double commutator. Writing the Wightman-functions as matrixelements of the translationoperator we get an integral representation of these functions over absolutely integrable complex-valued measures which depend analytically on the variables of the Wightman-functions. By controlling the growth-properties of the analytic continuation of these measures we end up with an integral representation of the analytic continuation of these four Wightman-functions into an ℒ+-invariant domain of holomorphy which connects the original four tubes.

The reproducing kernels of suitably chosen Hilbert spaces provide a simple but powerful way to generate approximations to functions on which linear data is given. These data may consist either of exact values (for the problem of interpolation) or of values subject to experimental error. For applications, it is important to be able to construct reproducing kernels which embody the information which is available from general considerations about the physics of the problem. Explicit methods are given here for construction of reproducing kernels for Hilbert spaces of analytic functions, these functions being characterized by various types of behavior on the boundary of their domain of holomorphy.

Representation formulas for the vertex function are derived which are valid for arbitrary complex values of two of the scalar variables inside the corresponding domain of holomorphy, while the third variable is evaluated in a neighbourhood of positive real values (the corresponding physical-region singularities). The analyticity domains always include at least the corresponding “axiomatic” Kallen-Wightman domain. Two alternative versions are given. In the first formula the occurring weight functions are the boundary values of the vertex function along the real axes of the scalar variables. The values of the arguments are such that the boundary values can be expressed in terms of the causal respectively time-ordered functions, which in a well-known way via reduction technique are related to the different form factors of the underlying field theory. In the second formula the weight functions are just the on-mass-shell matrix elements of the field operators which completely describe the dynamics of the theory and e.g. contain explicit information (in their support properties) on the mass spectrum of the field theory. The assumptions behind the results are the very general physical assumptions of Kallen and Wightman, i.e. essentially Lorentz-covariance, translation-invariance, “reasonable” mass- and energy-spectrum, and local commutativity. We further need some moderate integrability- and boundedness-properties of the vertex function. It is shown that the formalism can be extended to cover all cases of at most polynomial increase of the vertex function in asymptotic directions inside the Kallen-Wightman holomorphy domain. The kernel functions in the representation formulas can be explicitly seen to result in possible singularities only along the physical-region-cuts, i.e. from the contributions of the corresponding physical states, as well as along the more complicated Kallen-Wightman boundary-surfaces of the holomorphy domain, which correspond to the so-called “anomalous cuts”. The kernel functions in the second version above the closely connected to simple perturbation theoretical functions.

We consider algebras A of continuous complex valued functions, which are given as the set of global sections of a sheaf S \mathcal {S} on a topological space X. Under the hypothesis that all the higher cohomology groups of the sheaf are zero, we investigate the relationship between ideals in A, kernels of algebra homomorphisms of A into the complex numbers C, and sets of functions vanishing at a point of X. As applications, we obtain some simple proofs of theorems about ideals in certain algebras of holomorphic functions.

As Fuks [3] stated, every domain of holomorphy or meromorphy over Cn is analytically convex in the sense of Hartogs. Oka [6] proved that every domain over Cn analytically convex in the sense of Hartogs is a domain of holomorphy. Therefore a domain of meromorphy over Cn coincides with a domain of holomorphy over Cn.

In the last few years it has been found useful to apply known theorems in the theory of functions of several complex variables to solve problems arising in the quantum theory of fields (11). In particular, in order to derive the dispersion relations of quantum field theory from the general postulates of that theory it appears useful to apply known theorems on holomorphic continuation for functions of several complex variables ((2), (10)). The most important theorems are those which enable a determination to be made of the largest domain to which every function which is holomorphic in a domain D may be continued. This domain is called the envelope of holomorphy of D, and denoted by E(D). If D = E(D) then D is termed a domain of holomorphy. E(D) may be defined as the smallest domain of holomorphy containing D. Only in the special cases that D is a tube, semi-tube, Hartogs, or circular domain has it been possible to determine the envelope of holomorphy E(D) ((3), (7)). An iterative method for the computation of envelopes of holomorphy has recently been given by Bremmerman(4). It is also possible to use the continuity theorem (1) in a direct manner, though in most cases this is exceedingly difficult.

We consider holomorphic m x m-matrices, i.e. m x m-matrices whose matrix elements are holomorphic functions on a connected open set G in the space Cn of n complex variables.After giving some physical motivation for this research we discuss the analytic behavior of the eigenvalues, the construction of a G-adherent transformation matrix and a (physically) useful commutator theory.

This paper is primarily a study of generalized notions of envelope of holomorphy and holomorphic convexity for special (algebraically restricted) subsets of C n {{\mathbf {C}}^n} and in part for arbitrary subsets of C n {{\mathbf {C}}^n} . For any special set S in C n {{\mathbf {C}}^n} , we show that every function holomorphic in a neighborhood of S not only can be holomorphically continued but also holomorphically extended to a neighborhood in C n {{\mathbf {C}}^n} of a maximal set S ~ \tilde {S} , the “envelope of holomorphy” of S, which is also a special set of the same type as S. Formulas are obtained for constructing S ~ \tilde {S} for any special set S. “Holomorphic convexity” is characterized for these special sets. With one exception, the only topological restriction on these special sets is connectivity. Examples are given which illustrate applications of the theorems and help to clarify the concepts of “envelope of holomorphy” and “holomorphic convexity."

Let f f be a holomorphic map of a k k -dimensional polydisk in C n {C^n} into an n n -dimensional complex manifold M M which is regular at the origin. It is shown that f f can be extended to a neighborhood in C n {C^n} of a slightly smaller polydisk so that the extension is also regular. If f f is an embedding or immersion, then the extension may also be taken to be an embedding or immersion.

A domain of bounded holomorphy in a complex analytic manifold is a maximal domain for which every bounded holomorphic function has a bounded analytic continuation. In this paper, we show that this is a local property: if, for each boundary point of a domain, there exists a bounded holomorphic function which cannot be continued to any neighborhood of the point, then there exists a single bounded holomorphic function which cannot be continued to any neighborhood of the boundary points. Introduction.. Let X be a topological space. A subset D of X is said to be a region if it is open and i. is said to be a domain if it is open and connected. We denote by N(p) a fundamental system of open neighborhoods of p, where p E X. 1. DEFINITION. Let X be a topological space and U be an open subset of X. Let C(U) be the family of all continuous complex-valued functions on U, then C(U) is an algebra with 1, and it is equipped with the topology of uniform convergence on compact subsets of U. For a pair of open subsets U and V in X such that Vc U we define 7ru,: C(U) -C(V) by Tuvf=f I V. Let A(U) be a subalgebra of C(U) with I and we assume that ITUVA(U)cA(V); then we call A={A(U), 7Tuv} a presheaf of algebras of functions. A presheaf A has the local belonging property if, for all open sets U of X andf in C(U), for every p E U there is V E N(p), Vc U, such thatfI Vc A(v); thenfc A(U). A sheaf A of algebras of functions is a presheaf of algebras of functions with the local belonging property. A is said to be a ringed structure on X and the pair (X, A) is said to be a ringed space. The functions in A(U) are A-holomorphic functions. We note that A(U) has the relative topology induced by the topology on C(U). A ringed structure A on X is an n-dimensional complex analytic structure on X if for all x E X there are U E N(x) andf1,.. .f,f E A(U) such that F=(fi, ... .,n): U--* Cn Received by the editors February 5, 1970 and, in revised form, September 16, 1970. AMS 1970 subject classifications. Primary 32D15; Secondary 32D05.

This paper deals with the problem of constructing envelopes of holomorphy for Riemann domains over a locally convex space. When this locally convex space is a countable product of complex planes the existence of the envelope of holomorphy is proved and the domains of holomorphy are characterized. For the Riemann domains over the cartesian product CN of a countable number of complex planes, the domains of holomorphy are characterized and the existence of the envelope of holomorphy is proved. Also, for Riemann domains over a complex separated locally convex space E such that the closed convex hull of every compact subset is compact, the existence of the normal envelope of holomorphy is proved. Let (U, p) be a Riemann domain over E. This means that U is a connected separated topological space and 9 is a local homeomorphism from U into E. If a E U and AcE, a+A is defined by a+A = [qp W]-(p(a)+A), W being an open subset of U where p is a homeomorphism, a E W, and cp(a)+A cq(W). When A has only one element h, a+h denotes the unique element of a+{h}. If B is a subset of U, B+A U (b+A) beB where b +A has the meaning just stated. A complex mapping f defined in U is holomorphic if, for every u in U, there is an open convex balanced neighborhood U of zero in E and a sequence of continuous n-homogeneous polynomials in E: (n!) 1 dnf(u), n = O, 1,..., such that u + Uc U and c=o f u + h)(n !) -1 ,n(uh) n = O the series converging uniformly for h in U. In the algebra X6(U) of all complex Received by the editors July 6, 1971. AMS 1970 subject classifications. Primary 46A99; Secondary 58B10.

Any bounded holomorphic function defined on an analytic closed submanifold in general position in a strictly pseudoconvex domain can be continued to a bounded holomorphic function in the entire domain.

The purpose of this note is the announcement of some results in the theory of bounded holomorphic functions on finite open Riemann surfaces. The proofs are too long to be included ; they will be published elsewhere. Special cases of some of the results of this note are to be found in [4]. We consider a fixed finite open Riemann surface R. We thus assume that R is contained as an open set in some compact Riemann surface Ro and that RQ\R consists of finitely many topological bordered surfaces. We assume further that dR, the boundary of R with respect to Ro, consists of the m analytic simple closed curves Ti, • • • , Tm. Denote by H^j^R] the algebra of all functions holomorphic and bounded on R. Given the norm \\'\\u defined by

It is proved that an analyticity domain of the scattering amplitudeA(l, s), which can be obtained by the use of the Mandelstam representation, is the domain of holomorphy, under certain conditions, of an analytic function of the complex angular momentum and of the complex energy variables. The boundary of the analyticity domain is characterized by a sub-harmonic function of s.

The following result has been known for a long time: let 0 α π and let S be the sector { z ≠ 0 and arg z ≠ α (+ 2 kπ )} of the complex plane; let ( u n ) be a given infinite sequence of complex numbers; then there exists a holomorphic function on S which admits the formal power series ∑ +∞ n = 0 u n z n as asymptotic expansion at the origin. A first generalization of this result to the infinite dimensional case is given by the author (A result of existence of holomorphic maps which admit a given asymptotic expansion, in “Advances in Holomorphy” (J. A. Barroso, Ed.), in press). We give here an improvement of this last result, based upon a different proof. Then we give two counterexamples showing that our assumptions on the spaces are essential.

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