Holomorphic Functions Of Several Variables Research Articles

Some results of Fekete\u2013Szeg\xf6 type for Bavrin\u2019s families of holomorphic functions in {mathbb {C}}^{n}

In the paper there is considered a generalization of the well-known Fekete–Szegö type problem onto some Bavrin’s families of complex valued holomorphic functions of several variables. The definitions of Bavrin’s families correspond to geometric properties of univalent functions of a complex variable, like as starlikeness and convexity. First of all, there are investigated such Bavrin’s families which elements satisfy also a (j, k)-symmetry condition. As application of these results there is given the solution of a Fekete–Szegö type problem for a family of normalized biholomorphic starlike mappings in {mathbb {C}}^{n}.

Vector-valued holomorphic functions in several variables

In the present paper we give some explicit proofs for folklore theorems on holomorphic functions in several variables with values in a locally complete locally convex Hausdorff space $E$ over $\mathbb{C}$. Most of the literature on vector-valued holomorphic functions is either devoted to the case of one variable or to infinitely many variables whereas the case of (finitely many) several variables is only touched or is subject to stronger restrictions on the completeness of $E$ like sequential completeness. The main tool we use is Cauchy's integral formula for derivatives for an $E$-valued holomorphic function in several variables which we derive via Pettis-integration. This allows us to generalise the known integral formula, where usually a Riemann-integral is used, from sequentially complete $E$ to locally complete $E$. Among the classical theorems for holomorphic functions in several variables with values in a locally complete space $E$ we prove are the identity theorem, Liouville's theorem, Riemann's removable singularities theorem and the density of the polynomials in the $E$-valued polydisc algebra.

Slice Holomorphic Functions in Several Variables with Bounded L-Index in Direction

In this paper, for a given direction b ∈ C n \ { 0 } we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line { z 0 + t b : t ∈ C } for any z 0 ∈ C n . Unlike to quaternionic analysis, we fix the direction b . The usage of the term slice entire function is wider than in quaternionic analysis. It does not imply joint holomorphy. For example, it allows consideration of functions which are holomorphic in variable z 1 and continuous in variable z 2 . For this class of functions there is introduced a concept of boundedness of L-index in the direction b where L : C n → R + is a positive continuous function. We present necessary and sufficient conditions of boundedness of L-index in the direction. In this paper, there are considered local behavior of directional derivatives and maximum modulus on a circle for functions from this class. Also, we show that every slice holomorphic and joint continuous function has bounded L-index in direction in any bounded domain and for any continuous function L : C n → R + .

Common reducing subspaces of several weighted shifts with operator weights

We characterize common reducing subspaces of several weighted shifts with operator weights. As applications, we study the common reducing subspaces of the multiplication operators by powers of coordinate functions on Hilbert spaces of holomorphic functions in several variables. The identification of reducing subspaces also leads to structure theorems for the commutants of von Neumann algebras generated by these multiplication operators. This general approach applies to weighted Hardy spaces, weighted Bergman spaces, Drury–Arveson spaces and Dirichlet spaces of the unit ball or polydisk uniformly.

Integration over homogeneous spaces for classical Lie groups using iterated residues at infinity

AbstractUsing the Berline-Vergne integration formula for equivariant cohomology for torus actions, we prove that integrals over Grassmannians (classical, Lagrangian or orthogonal ones) of characteristic classes of the tautological bundle can be expressed as iterated residues at infinity of some holomorphic functions of several variables. The results obtained for these cases can be expressed as special cases of one formula involving the Weyl group action on the characters of the natural representation of the torus.

Noncoherence of some rings of holomorphic functions in several variables as an easy consequence of the one-variable case

Using the facts that the disk algebra and the Wiener algebra are not coherent, we prove that the polydisc algebra, the ball algebra, and the Wiener algebra of the polydisc are not coherent.

A note on mean ergodic composition operators on spaces of holomorphic functions

We characterize those composition operators defined on spaces of holomorphic functions of several variables which are power bounded, i.e. the orbits of all the elements are bounded. This condition is equivalent to the composition operator being mean ergodic. We also describe the form of the symbol when the composition operator is mean ergodic.

On weighted spaces of holomorphic functions of several variables

We generalize the results of [11] and [12] for the unit ball \( \mathbb{B}_d \) of ℂd. In particular, we show that under the weight condition (B) the weighted H∞-space on \( \mathbb{B}_d \) is isomorphic to l∞ and thus complemented in the corresponding weighted L∞-space. We construct concrete, generalized Bergman projections accordingly. We also consider the case where the domain is the entire space ℂd. In addition, we show that for the polydisc \( \mathbb{D}^d \)d, the weighted H∞-space is never isomorphic to l∞.

On propagation of boundary continuity of holomorphic functions of several variables

We consider continuity properties of analytic functions in bounded domains in complex space, focusing on the following two questions:Question 1. Given a bounded domain G in complex affine n-space. For which subsets S of the boundary of G does the following hold? If f is a bounded function which is analytic in G and extends continuously to the union of G and S then f extends continously to the closure of G.Question 2. Given a domain G as above and a function f analytic in G and continuous in the closure of G. For which subsets S of the boundary is the rate of continuity of f on the closure of G determined already by its rate of continuity along S?Previously it was known that in some cases, the set S may be taken to be the Shilov boundary of the algebra of analytic functions on the domain which are continuous on the closure of the domain, but examples when this is not sufficient were also known.In Paper I we prove that for a bounded, smoothly bounded pseudoconvex domain G in 2-dimensional complex affine space one may take for S an open neighbourhood of the Shilov boundary (in the boundary of the domain G). This is joint work with B. Joricke.Paper II considers smoothly bounded Reinhardt domains in 2 dimensions and complete Reinhardt domains in complex affine n-space. We prove that in many cases one may let S be the Shilov boundary. Moreover, the respective continuity properties propagate to the closure of the envelope of holomorphy of the domain.Paper III considers pseudoconvex Hartogs domains with connected vertical sections in two dimensions. We prove that in some cases S may be taken to be the Shilov boundary and give examples for when this is not enough.

Isometries of Privalov spaces of holomorphic functions of several variables

Complete descriptions for the sets of linear isometries of the Privalov spaces of holomorphic functions on the sphere and the polydisk are obtained. These results are generalizations, to the many-dimensional case, of those previously obtained by Iida and Mochizuki. Moreover, the general form of surjective linear isometries is established for maximal analogues of the Privalov spaces in the case of positive integer exponents. It is shown that the sets of linear isometries of the Privalov spaces with the natural and the maximal metrics are different, while their groups of surjective linear isometries coincide (in the case of positive integer exponents).

LINEAR CONTINUOUS EXTENSION OPERATORS FOR GEVREY CLASSES ON POLYSECTORS

We study Gevrey classes of holomorphic functions of several variables on a polysector, and their relation to classes of Gevrey strongly asymptotically developable functions. A new Borel-Ritt-Gevrey interpolation problem is formulated, and its solution is obtained by the construction of adequate linear continuous extension operators. Our results improve those given by Haraoka in this context, and extend to several variables the one-dimensional versions of the Borel-Ritt-Gevrey theorem given by Ramis and Thilliez, respectively. Some rigidity properties for the constructed operators are stated. 2000 Mathematics Subject Classification. 30D60, 30E05, 41A60.

Locally Convex Structure of Some Algebras of Holomorphic Functions of Several Variables

In the paper a general method is developed which allows to show that a nuclear Kothe space is a Frechet envelope of another “sequence space”. Applying this method we describe the locally convex structure of Hardy Np*(D), maximal Hardy MNp* (D), Bergman p(D), and Lumer's Hardy LNp* (D) algebras of any domain D being a product of balls in ℂ (in particular for the unit ball and the unit polydisc in ℂn).

Nonradial Hörmander algebras of several variables and convolution operators

A characterization of the closed principal ideals in nonradial Hörmander algebras of holomorphic functions of several variables in terms of the behaviour of the generator is obtained. This result is applied to study the range of convolution operators and ultradifferential operators on spaces of quasianalytic functions of Beurling type. Contrary to what is known to happen in the case of non-quasianalytic functions, an ultradistribution on a space of quasianalytic functions is constructed such that the range of the operator does not contain the real analytic functions.

Functional properties of privalov spaces of holomorphic functions in several variables

We consider Privalov classes of degreeq>1 in the unit ball and the polydisk in ℂn. They are defined, say, for the ball, as the sets of functionsf(z) such that the average of ln+q|f(z)| over a sphere centered at the origin remains bounded as the radius increases to 1. These classes, which were introduced (in the one-dimensional case) by Privalov before 1941, were often used in the foreign literature in the last 10–20 years; typically, the notation varied and Privalov was not mentioned. We discuss various equivalent definitions of these classes as well as the most general properties, such as growth estimates, properties of the natural metric, and boundedness or total boundedness of subsets.

On N-Widths of Holomorphic-Functions of Several Variables

We consider the classes of holomorphic functions whose radial derivative of order r lies in the unit ball of the Hardy space H2(Bn) or the Bergman space A2(Bn). For these classes we calculate the linear and Gel′fand N-widths in C(Sρ), where Sρ is the sphere in Cn of radius 0 < ρ < 1. Some results are obtained for analogous problems in polydiscs and for 2π-periodic functions of one variable holomorphic in a strip.

Perturbations of linear operators in spaces of holomorphic functions

The closure of the images of perturbations of convolution operators in spaces of holomorphic functions of several variables is studied. The properties of the homogeneous solutions of the corresponding perturbed equations are discovered.

On the spectrum of A(Ω) and $H^∞(Ω)$

We study some properties of the maximal ideal space of the bounded holomorphic functions in several variables. Two examples of bounded balanced domains are introduced, both having non-trivial maximal ideals.

On continuation of regular solutions of linear partial differential equations

Our research mainly concerns problems of type (2b). As a survey, however, we shall start with the general story. As is well known, the origin of this study lies in the problem of removable singularities in the theory of functions of a single complex variable (category (2a) in this classification). Already as early as 1851, Riemann [1] had pointed out that an isolated singularity of a holomorphic function is removable if the function is bounded around the singularity. Later, in 1906, Hartogs [1] proved that in the case of holomorphic functions of several variables any convex compact set is removable, irrespective of the boundedness of the function. After that, studies extending these results were made by researchers in complex analysis. Among them we mention that Hartogs’ theorem was generalized by these people to any pair K ⊂ U , where K is the part of a convex compact set lying in a half space and U is a neighborhood (although this form of the assertion also bears Hartogs’ name nowadays). See Struppa [1] for a detailed history of these. In 1952 Bochner [1] gave a new proof of Hartogs’ theorem employing the notion of an overdetermined system of differential equations. This initiated an abundant study of the problem of continuation of solutions in the theory of partial

Book Review: Introduction to holomorphic functions of several variables

Book Review: Introduction to holomorphic functions of several variables

Reconstruction of holomorphic functions of several variables from their values on a uniqueness set

Reconstruction of holomorphic functions of several variables from their values on a uniqueness set