Finely holomorphic functions in one and more complex variables-a survey

Equivalence, linearization, and decomposition of holomorphic operator functions

For a proper holomorphic map F from a ball B n into another ball B N , we give a criterion when a rational map F is equivalent to a polynomial one.As application, we show that proper rational holomorphic maps from B 2 into B N of degree two are equivalent to polynomial maps, and we also give an example of a rational holomorphic maps of degree 3 that are 'almost' linear but are not equivalent to holomorphic polynomial maps.

Related to the concept of p-compact operators with p ? [1,?] introduced by Sinha and Karn [20], this paper deals with the space H? Kp (U, F) of all Banach-valued holomorphic mappings on an open subset U of a complex Banach space E whose ranges are relatively p-compact subsets of F. We characterize such holomorphic mappings as those whose Mujica?s linearisations on the canonical predual of H?(U) are p-compact operators. This fact allows us to make a complete study of them. We show that H? Kp is a surjective Banach ideal of bounded holomorphic mappings which is generated by composition with the ideal of p-compact operators and contains the Banach ideal of all right p-nuclear holomorphic mappings. We also characterize holomorphic mappings with relatively p-compact ranges as those bounded holomorphic mappings which factorize through a quotient space of ?p* or as those whose transposes are quasi p-nuclear operators (respectively, factor through a closed subspace of ?p).

We lay the foundations of Fatou theory in one and several complex variables. We describe the main contributions contained in E. M. Stein’s book Boundary Behavior of Holomorphic Functions, published in 1972 and still a source of inspiration. We also give an account of his contributions to the study of the boundary behavior of harmonic functions. The point of this paper is not simply to exposit well-known ideas. Rather, we completely reorganize the subject in order to bring out the profound contributions of E. M. Stein to the study of the boundary behavior both of holomorphic and harmonic functions in one and several variables. In an appendix, we provide a self-contained proof of a new result which is relevant to the differentiation of integrals, a topic which, as witnessed in Stein’s work, and especially by the aforementioned book, has deep connections with the boundary behavior of harmonic and holomorphic functions.

Let C be an open convex cone in ${\bf R}^n $, n-dimensional real space, such that $\bar C$ does not contain any entire straight line. We consider holomorphic functions in tubes $T^C = {\bf R}^n + iC \subset {\bf C}^n $, n-dimensional complex space, which are known to have $\mathfrak{S}'$ distributional boundary values. We prove that if the $\mathfrak{S}'$ boundary value of such a holomorphic function is an $L^p $ function, $1 \leq p \leq \infty $, then the holomorphic function is in the Hardy space $H^p (T^C )$, $1 \leq p \leq \infty $, corresponding to the tube $T^C $ and can be recovered as the Poisson integral of the distributional boundary value. A similar result is proved wih respect to the holomorphic functions in tubes $T^C $ which are known to have boundary values in the distribution spaces $\mathfrak{Z}',K'_r ,(\mathfrak{S}^\alpha )'$, and $(W^\Omega )'$ and which do not necessarily have $S'$ distributional boundary values. In all cases we prove converse theorems. Our basic results are motivated ...

In the classical theory of several complex variables, holomorphic mappings are just n-tuples of holomorphic functions in m variables, with arbitrary n andm, and no relations between these functions are assumed. Some 30 years ago John Ryan introduced complex, or complexified, Clifford analysis which is, in a sense, the study of certain classes of holomorphic mappings where the components are not independent, and instead obey the relations generated by the Cauchy- Riemann and Dirac-type operators. In this paper, we take a closer look at this theory emphasizing some additional properties that holomorphic mappings satisfy in this context. Our attention is mostly restricted to the case of low dimensions where it is possible to identify new and interesting properties and to single out the special role played by bicomplex analysis.

Let E E and F F be complex Banach spaces. We say that a holomorphic mapping f f from E E into F F is compact respectively bounding if f f maps some neighbourhood of every point of E E into a relatively compact respectively bounding subset of F F . Recall that a subset of E E is bounding if it is mapped onto a bounded set by every complex valued holomorphic mapping on E E . Compact holomorphic mappings have been studied by R. Aron and M. Schottenloher in [1]. Since every relatively compact subset of a Banach space is trivially bounding it is clear that every compact holomorphic mapping is bounding. We show that the product of three bounding holomorphic mappings is compact.

Invariants for Riemann surfaces covered by the disc and for hyperbolic manifolds in general involving minimizing the measure of the image over the homotopy and homology classes of closed curves and maps of the k -sphere into the manifold are investigated. The invariants are monotonic under holomorphic mappings and strictly monotonic under certain circumstances. Applications to holomorphic maps of annular regions in ℂ and tubular neighborhoods of compact totally real submanifolds in general in ℂ n , n ≥ 2 , are given. The contractibility of a hyperbolic domain with contracting holomorphic mapping is explained.

Let Δ n be the ball |x| < 1 in the complex vector space ℂ n , let f: Δ n → ℂ n be a holomorphic mapping and let M be a positive integer. Assume that the origin 0 = (0, ..., 0) is an isolated fixed point of both f and the M-th iteration f M of f. Then the (local) Dold index P M (f, 0) at the origin is well defined, which can be interpreted to be the number of virtual periodic points of period M of f hidden at the origin: any holomorphic mapping f 1: Δ n → ℂ n sufficiently close to f has exactly P M (f, 0) distinct periodic points of period M near the origin, provided that all the fixed points of f 1 near the origin are simple. Therefore, the number O M (f, 0) = P M (f, 0)/M can be understood to be the number of virtual periodic orbits of period M hidden at the fixed point. According to the works of Shub-Sullivan and Chow-Mallet-Paret-Yorke, a necessary condition so that there exists at least one virtual periodic orbit of period M hidden at the fixed point, i.e., O M (f, 0) ⩾ 1, is that the linear part of f at the origin has a periodic point of period M. It is proved by the author recently that the converse holds true. In this paper, we will study the condition for the linear part of f at 0 so that O M (f, 0) ⩾ 2. For a 2 × 2 matrix A that is arbitrarily given, the goal of this paper is to give a necessary and sufficient condition for A, such that O M (f, 0) ⩾ 2 for all holomorphic mappings f: Δ2 → ℂ2 such that f(0) = 0, Df(0) = A and that the origin 0 is an isolated fixed point of f M .

The main objects of study in this chapter are holomorphic functions h: U→ V, with U and V open in ℂ, that are one-to-one and onto. Such a holomorphic function is called a conformal (or biholomorphic) mapping. The fact that h is supposed to be one-to-one implies that h’ is nowhere zero on U [remember that if h’ vanishes to order k ≥ 0 at a point P ∈ U, then h is (k+1)-to-1 in a small neighborhood of P—see §§5.2.1]. As a result, h-1: V →U is also holomorphic—as we discussed in §§5.2.1. A conformal map h: U → V from one open set to another can be used to transfer holomorphic functions on U to V and vice versa: that is, f : V → ℂ is holomorphic if and only if f o h is holomorphic on U; and g : U → ℂ is holomorphic if and only if g o h-1 is holomorphic on V.

We construct codimension-2 BPS brane solutions in $D=9,8,7$ maximal supergravities by solving Killing spinor equations. We assume the Poincare invariance along the worldvolume and vanishing gauge fields, and determine the metric and the scalar fields. The solution in $D=9$ is essentially the same as the ten-dimensional one, which is specified by a holomorphic function in the transverse space. For $D=8$, the solution is specified by two holomorphic functions, and regarded as $T^2\times T^2$ compactification of F-theory. For $D=7$, we find that the solution can be interpreted as M-theory on Calabi-Yau, and under an additional assumption a solution is specified by two holomorphic functions.

We study the relationship between integrable systems with a position-dependent mass (PDM) and complex holomorphic functions and the potential applications of the latter to deduce the former. For a prescribed mass term the associated complex function is derived. The complex function and related plane transformation are used to generate the PDM systems of three integrable Hénon–Heiles systems and a Holt system as well. We also figure out a holomorphic function, which ensures separability of the corresponding PDM systems in the polar-like coordinates. The holomorphic function together with Jacobi method have yielded a variety of generalized separable systems. At last we put forward an example of a family of separable systems to show that not all PDM systems can be deduced through some holomorphic function.

Given a holomorphic mapping of bounded type g ∈ Hb(U, F), where U ⊆ E is a balanced open subset, and E, F are complex Banach spaces, let A : Hb(F) ∈ Hb(U) be the homomorphism defined by A(f) = fog for all f ∈ Hb(F). We prove that: (a) for F having the Dunford-Pettis property, A is weakly compact if and only if g is weakly compact; (b) A is completely continuous if and only if g(W) is a Dunford-Pettis set for every U-bounded subset W ⊂ U. To obtain these results, we prove that the class of Dunford - Pettis sets is stable under projecti ve tensor products. Moreover, we diaracterize the reflexivity of the space Hb(U,F) and prove that E' and F have the Schur property if and only if Hb(U, F) has the Schur property. As an application, we obtain some results on linearization of holomorphic mappings.

On finite regular and holomorphic mappings

For given 2n+1 hyperplanes with moving targets in pointwise general position and any holomorphic map f from into a n-dimensional complex projective space omitting those 2n+1 hyperplanes, we show that there exist finitely many (n+1)×(n+1) everywhere invertible matrices with entire holomorphic functions as entries such that the holomorphic map f multiplied by one of the matrices becomes constant. In addition, this set of matrices depends only on the hyperplanes, and can be determined effectively.