On BMO-Type Characteristics for One Class of Holomorphic Functions in a Disk

By using the differential operator Dm f(z), z∈ U (Definition 1), we introduce a class of holomorphic functions, denoted by , and we obtain some inclusion relations.

Holomorphic functions differ fundamentally from real differentiable functions: they are infinitely often (real and complex) differentiable (II.3, II.7), they even admit power series expansions (II.4), their entire behaviour is determined by their values on arbitrarily small open sets (II.4, II.7), and they satisfy powerful convergence theorems and estimates (II.5). All of these properties are consequences of the Cauchy integral theorem and the integral representations that arise from it (II.1–3). Meromorphic functions extend the class of holomorphic functions (II.6); their study leads to the notion of isolated singularities and to generalizations of power series obtained by allowing negative powers (Laurent series). In addition to the phenomena that occur in the theory of functions of one complex variable, a fundamentally new phenomenon enters the picture in higher dimensions: the simultaneous holomorphic continuation of all holomorphic functions from a given domain to a larger one (II.7). Here the Cauchy integral formula (in one variable!) is again the decisive tool.

We introduce a generalized Cauchy integral formula for special classes of functions, holomorphic in several complex variables. These functions also satisfy a generalized Cachy-Riemann equation. The integral formula is then applied to demonstrate that these classes of holomorphic functions are Fréchet modules over complex Clifford algebras, and to provide maximal domains of convergence for Taylor expansions, with homogeneous terms grouped together, of real analytic solutions to Maxwell's equations in vacuo.

Recently J. Miao proved that if is a holomorphic function with Hadamard gaps on the open unit disc D then f ∈ Xp if and only if f ∈ Bp if and only if if and if only if where Xp, Bp and denote respectively the class of holomorphic functions on D which satisfy |f′(z)|p(1 − |z|2)p − 1dxdy is a finite measure, a Carleson measure and a little Carleson measure on D. In this paper we give a higher-dimensional version of Miao's result.

This article is concerned with the asymptotic behaviour of the Mahler measure (defined for the occasion) of functions of the form \(z^n + f(z)\), where f(z) is a holomorphic function on an open subset of \({\mathbb {C}}\) such that \(f(0) \ne 0\).

In this paper we introduce and examine the differential subordination related to the geometric mean of the form [p(z)]1-γp(z)+zp′(z)φp(z),zp′(z)γ≺h(z),z∈D,where D:={z∈C:|z|<1} and h is a convex univalent function with 0∈∂h(D). In particular, the above differential subordination generalizes the well-known Briot-Bouquet differential subordination. At the same time new type of ordinary differential equation has been proposed for study. The main results allow us to construct non-obvious subclasses in the class of holomorphic functions on D.

On N-Widths of Holomorphic-Functions of Several Variables

Let be a point sequence in the unit disc and the Nevanlinna characteristic of the sequence , . In terms of the Nevanlinna characteristic one finds estimates for the slowest possible growth of the characteristic as in the class of holomorphic functions in vanishing on . Let be a meromorphic function in . In terms of the Nevanlinna characteristic function of one finds estimates for the slowest possible growth of the characteristics and in the class of pairs of holomorphic functions and such that .

We establish exact-order estimates for the trigonometric widths of the Nikol’skii–Besov classes of periodic functions of many variables in the Lebesgue space with mixed norm.

The subject is traces of Sobolev spaces with mixed Lebesgue norms on Euclidean space. Specifically, restrictions to the hyperplanes given by x1 = 0 and xn = 0 are applied to functions belonging to quasi‐homogeneous, mixed norm Lizorkin–Triebel spaces ; Sobolev spaces are obtained from these as special cases. Spaces admitting traces in the distribution sense are characterised up to the borderline cases; these are also covered in case x1 = 0. For x1 the trace spaces are proved to be mixed‐norm Lizorkin–Triebel spaces with a specific sum exponent; for xn they are similarly defined Besov spaces. The treatment includes continuous right‐inverses and higher order traces. The results rely on a sequence version of Nikol'skij's inequality, Marschall's inequality for pseudodifferential operators (and Fourier multiplier assertions), as well as dyadic ball criteria. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

In this paper, we establish the sharp Fekete–Szegö inequalities of functional type over a new class of holomorphic mappings on the unit ball in a complex Banach space. This class includes some important widely studied classes. We offer simpler proofs and require less restrictive assumptions than those found in recent literature. Finally we show that some important previous results follows from ours by choosing suitable parameters.

We consider some inequalities in such classical Banach Function Spaces as Lorentz, Marcinkiewicz, and Orlicz spaces. Our aim is to explore connections between the norm of a function of two variables on the product space and the mixed norm of the same function, where mixed norm is calculated in function spaces on coordinate spaces, first in one variable, then in the other. This issue is motivated by various problems of functional analysis and theory of functions. We will currently mention just geometry of spaces of vector-valued functions and embedding theorems for Sobolev and Besov spaces generated by metrics which differ from L p. Our main results are actually counterexamples for Lorentz spaces versus the natural intuition that arises from the easier case of Orlicz spaces (Section 2). In the Appendix we give a proof for the Kolmogorov–Nagumo theorem on change of order of mixed norm calculation in its most general form. This result shows that L p is the only space where it is possible to change this order.

The class of operator-valued functions which are homogeneous of degree one, holomorphic in the open right polyhalfplane, have positive semi-definite real parts there and take selfadjoint operator values at real points, and its subclass consisting of functions representable in the form of Schur complement of a block of a linear pencil of operators with positive semidefinite operator coefficients, are investigated. The latter subclass is a generalization of the class of characteristic matrix functions of passive 2n-poles considered as functions of impedances of its elements, which was introduced by M. F. Bessmertnyi. Several equivalent characterizations of the generalized Bessmertnyi class are given, and its intimate connection with the Agler-Schur class of holomorphic contractive operator-valued functions on the unit polydisk is established.

formula omitted]-dimension in infinite dimensional hyperbolic cross approximation and application to parametric elliptic PDEs

We provide several characterizations of the bounded and the compact weighted composition operators from the Bloch space and the analytic Besov spaces (with ) into the Zygmund space . As a special case, we show that the bounded (resp., compact) composition operators from , , and to coincide. In addition, the boundedness and the compactness of the composition operator can be characterized in terms of the boundedness (resp., convergence to 0, under the boundedness assumption of the operator) of the Zygmund norm of the powers of the symbol.