One complex variable from the several variables point of view

The theory of analytic functions of several complex variables enjoyed a period of remarkable development in the middle part of the twentieth century. After initial successes by Poincare and others in the late 19th and early 20th centuries, the theory encountered obstacles that prevented it from growing quickly into an analogue of the theory for functions of one complex variable. Beginning in the 1930s, initially through the work of Oka, then H. Cartan, and continuing with the work of Grauert, Remmert, and others, new tools were introduced into the theory of several complex variables that resolved many of the open problems and fundamentally changed the landscape of the subject. These tools included a central role for sheaf theory and increased uses of topology and algebra. The book by Gunning and Rossi was the first of the modern era of the theory of several complex variables, which is distinguished by the use of these methods. The intention of Gunning and Rossi's book is to provide an extensive introduction to the Oka-Cartan theory and some of its applications, and to the general theory of analytic spaces. Fundamental concepts and techniques are discussed as early as possible. The first chapter covers material suitable for a one-semester graduate course, presenting many of the central problems and techniques, often in special cases. The later chapters give more detailed expositions of sheaf theory for analytic functions and the theory of complex analytic spaces. Since its original publication, this book has become a classic resource for the modern approach to functions of several complex variables and the theory of analytic spaces.

concept has become familiar, the model is regarded as a realization of the concept. The most important disadvantage of diagrams is that they are not able to describe the manifold completely, e.g. to indicate whether a circle is to be taken with its boundary or without it. Because of this during the transition from the diagram to the abstract notion we often commit mistakes by attributing to the manifold properties which are possessed only by the model. A means of checking our conclusions would be to repeat our reasoning but now operating exclusively with abstract symbols, e.g. of the type indicated earlier in this section. This situation necessitates a certain kind of training and particular flexibility of mind, enabling one to pass from an intuitive model to an abstract formulation of notions and back. These difficulties arise naturally in operating with models of four-dimensional space; in addition to this there occur further difficulties, which we shall indicate in what follows. 4. The advantage of models in teaching and research. Despite the difficulties mentioned before it seems that in many cases the use of models for a geometrically minded individual is of great value in research. Two important reasons may perhaps be indicated: First, our interest in a thing is determined to a considerable extent by the amount of mental energy which is required for working with it. We involuntarily prefer to think in terms of familiar things which can be conceived without special effort. In other words, the extent of our interest in any subject depends to a certain degree on our previous associations-and here again we the advantage of being able to use intuitive geometry to help in reasoning about abstract relations. Second, if a relationship is known, it is not too important how we formulate it, provided only that our formulation is sufficiently precise. Research, on the other hand, is the search for new relations and in that case it becomes highly significant in which way our problems are formulated. If we can succeed in stating our problem in terms of a class of objects with which we have gained great familiarity, then we may be able to exploit our previous experience with these things. The form of the problem in the new representation may suggest comparisons and hypotheses and further questions, and our previous experience will enable us to new relationships all the more readily. Thus, the moment that complex numbers are regarded as points in a plane, the analogy with integrals of real functions leads to the formulation of curvilinear integrals in the complex plane-the most fruitful approach in the theory of functions. Furthermore, even after some fact has been discovered and it has been reduced to abstract form which completely and rigorously describes it, even then, the intuitive processes by means of which we have worked are of importance in further research because of the ready associations and the many problems which in this way arise almost spontaneously. This content downloaded from 157.55.39.180 on Tue, 27 Sep 2016 05:23:13 UTC All use subject to http://about.jstor.org/terms 1946] MODELS IN THE THEORY OF SEVERAL COMPLEX VARIABLES 499 5. Models of four-dimensional manifolds. It would, perhaps, be of interest to illustrate the foregoing by a description of a type of model used for visualization of four-dimensional space.* These models are moving three-dimensional pictures, that is to say time is chosen as one of the coordinates. A four-dimensional domain is then represented as a moving picture. The continuous process is thus recorded by a finite number of frames, and this last is in turn often replaced by a number of stills. In order to have a better idea as to how intuitive these models are, let us put ourselves in the position of beings (Flatlanders) who are able to see two-dimensional domains but who do not possess intuition for three-dimensional space. xs

The Caratheodory and Kobayashi metrics have proved to be important tools in the function theory of several complex variables. But they are less familiar in the context of one complex variable. Our purpose here is to gather in one place the basic ideas about these important invariant metrics for domains in the plane and to provide some illuminating examples and applications.

The aim of this article is to extend the theory of several complex variables to the non-commutative realm. Some basic results, such as the Bochner-Martinelli formula, the existence theorem of the solutions to the non-homogeneous Cauchy-Riemann equations, and the Hartogs theorem, are generalized from complex analysis in several variables to Clifford analysis in several paravector variables. In particular, the Bochner-Martinelli formula in several paravector variables unifies the corresponding formulas in the theory of one complex variable, several complex variables, and several quaternionic variables with suitable modifications.

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.

An introduction to the subject Some integral formulas Subharmonicity and its applications Convexity Hormander's solution of the $\bar\partial$ equation Solution of the Levi problem and other applications of $\bar\partial$ techniques Cousin problems, cohomology, and sheaves The zero set of a holomorphic function Some harmonic analysis Constructive methods Integral formulas for solutions to the $\bar\partial$ problem and norm estimates Holomorphic mappings and invariant metrics Manifolds Area measures Exterior algebra Vectors, covectors, and differential forms List of notation Bibliography Index.

The paper is concerned with the differential equation\(V_{\bar z} = C(z,\bar z)\bar V\) with\(m^2 (\log C)_{z\bar z} + + \varepsilon C\bar C = 0,m > 0,\varepsilon = \pm 1\). The Vekua resolvents are determined by means of an associated second-order differential equation. Applications are given to pseudo-analytic functions, to a differential equation in the theory of several complex variables and to the Ernst equation in general relativity.

After the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the ∂ ¯ {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are L p ⁢ ( ℝ n ) {L^{p}({\mathbb{R}^{n}})} bounded for 1 < p < ∞ {1<p<\infty} , but only bounded on local Hardy spaces h p ⁢ ( ℝ n ) {h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for 0 < p ≤ 1 {0<p\leq 1} . Though much work has been done on the L p ⁢ ( ℝ n 1 × ℝ n 2 ) {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for 1 < p < ∞ {1<p<\infty} and Hardy H p ⁢ ( ℝ n 1 × ℝ n 2 ) {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for 0 < p ≤ 1 {0<p\leq 1} for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of 0 < p ≤ 1 {0<p\leq 1} . The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces h p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for 0 < p ≤ 1 {0<p\leq 1} recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].

Part explanation of important recent work, and part introduction to some of the techniques of modern partial differential equations, this monograph is a self-contained exposition of the Neumann problem for the Cauchy-Riemann complex and certain of its applications. The authors prove the main existence and regularity theorems in detail, assuming only a knowledge of the basic theory of differentiable manifolds and operators on Hilbert space. They discuss applications to the theory of several complex variables, examine the associated complex on the boundary, and outline other techniques relevant to these problems. In an appendix they develop the functional analysis of differential operators in terms of Sobolev spaces, to the extent it is required for the monograph.

The Dolbeault sequence is a fundamental tool for many problems in the function theory of several complex variables. A lot of attention was paid in the last decades to its analogue in the function theory of several Clifford variables. The first operator in this resolution is the Dirac operator in several variables. The complete description is known in dimension 4 (i.e., in the case of quaternionic variables, see [1, 6, 4]). Much less is known in higher dimensions. The case of three variables was described completely (see [18]). The full description of the complex for all dimensions is not known at present. Even the case of the stable range (i.e., when the number of variables is less or equal to the half of dimension) is still not fully understood.There are two different approaches to the stable range case, one based on classical algebraic geometry (the Hilbert syzygy theory, see [8]), the other one on representation theory (differential invariants in certain parabolic geometries, see [14, 20]). Differentia...

Models in the Theory of Several Complex Variables

We wish to examine the notion of a “function of z” where z is a complex variable. To be sure, a complex variable can be viewed as nothing but a pair of real variables so that in one sense a function of z is nothing but a function of two real variables. This was the point of view we took in the last section in discussing continuous functions. But somehow this point of view is too general. There are some functions which are “direct” functions of z = x + iy and not simply functions of the separate pieces x and y.

In this paper, by using the normality criteria forK-quasimeromorphic mapping of several complex variables, we get a normality criteria for families of holomorphic functions and of meromorphic functions family.

The principal objective of this paper is to introduce the ideas of relative $\varphi $-type, relative $\varphi $-weak type of entire functions of several complex variables and study some growth properties concerning them.

Many authors, e.g., Bavrin, Jakubowski, Liczberski, Pfaltzgraff, Sitarski, Suffridge, and Stankiewicz, have discussed some families of holomorphic functions of several complex variables described by some geometrical or analytical conditions. We consider a family of holomorphic functions of several complex variables described in n-circular domain of the space C n . We investigate relations between this family and some of type of Bavrin’s families. We give estimates of G-balance of k-homogeneous polynomial, a distortion type theorem and a sufficient condition for functions belonging to this family. Furthermore, we present some examples of functions from the considered class.