Thin interpolating sequences in the disk

Using a kind of summability procedure we give, within this survey, a new proof of a result by Sundberg and Wolff on large perturbations of thin interpolating sequences and present a detailed proof of the statement of Dyakonov and Nicolau that asymptotic interpolation problems in H^∞ can be solved by thin Blaschke products. Dedicated to Professor Heinz K?nig on the occasion of his 80^th birthday     

Unbounded Toeplitz Operators

This partly expository article develops the basic theory of unbounded Toeplitz operators on the Hardy space H ², with emphasis on operators whose symbols are not square integrable. Unbounded truncated Toeplitz operators on coinvariant subspaces of H ² are also studied. In memory of Paul R. Halmos     

Reproducing kernels for polyharmonic polynomials

The reproducing kernel of the space of all homogeneous polynomials of degree k and polyharmonic order m is computed explicitly, solving a question of A. Fryant and M. K. Vemuri. Received: 17 May 2007, Revised: 27 March 2008     

Hankel and Toeplitz Transforms on H 1: Continuity, Compactness and Fredholm Properties

We revisit the boundedness of Hankel and Toeplitz operators acting on the Hardy space H ¹ and give a new proof of the old result stating that the Hankel operator H _a is bounded if and only if a has bounded logarithmic mean oscillation. We also establish a sufficient and necessary condition for H _a to be compact on H ¹. The Fredholm properties of Toeplitz operators on H ¹ are studied for symbols in a Banach algebra similar to C + H ^∞ under mild additional conditions caused by the differences in the boundedness of Toeplitz operators acting on H ¹ and H ². The first author was partially supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) and by the Greek Research Program “Pythagoras 2” (75% European funds and 25 National funds). The second author was fully supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) while he visited the first author at the University of Crete and later by the Academy of Finland Project 207048.     

Effect of topology on the multiplicity of solutions for some semilinear elliptic systems with critical Sobolev exponent

In this paper, we are concerned with the effect of the domain topology on the multiplicity of solutions for some semilinear elliptic systems involving critical Sobolev exponent. We show that if the interaction term is sufficiently small, then the number of solutions of the system is estimated from below by 2 catΩ. The proof of this fact requires analysis of the structure of the Nehari manifold associated with the system. This research was partially supported by the Grant-in-Aid for Young Scientists (B), The Ministry of Education, Culture, Sports, Science and Technology, Japan.     

Group gradings on diagonal algebras

We describe all group gradings on the diagonal algebra k ⁿ , where k is an arbitrary field. Received: 21 January 2008     

Universal Laurent series in finitely connected domains

Universal Laurent series where the universal approximation is valid on the boundary of the multiple connected open Ω appears for the first time in [2]. We show that it is possible to demand universal approximation only to a part of the boundary while on the remaining part the universal function can be smooth. Received: 29 May 2007     

Value distribution of the difference operator

We investigate the value distribution of difference operators for entire functions, and establish a difference analogue of the Brück conjecture for entire functions of order less than 1. This work was supported by the NNSF of China (No. 10671109).     

An Improved Cauchy Formula for Hypermonogenic Functions

A new Cauchy-type formula for hypermonogenic functions is derived. Hypermonogenic functions, introduced in [6], are a generalization of holomorphic functions to several dimensions. The power function x^m is hypermonogenic. This paper is dedicated to the memory of our friend and colleague Jarolim Bureš     

Every Biregular Function Is a Biholomorphic Map

We study Fueter-biregular functions of one quaternionic variable. We consider left-regular functions in the kernel of the Cauchy–Riemann operator

Domination in the Bergman Space and Korenblum’s Constant

Let and m be a positive integer. We obtain a condition under which . As an application we get an upper bound on Korenblum’s constant . We also give another proof of a well-known result concerning the domination problem in the Bergman space. This work was supported by NNSF of China No. 10601025 and the Natural Science Foundation of Hebei Province of China (No. 07M001).     

Estimates of holomorphic functions in zero-free domains

We study functions f(z) holomorphic in having the property f(z) ≠ 0 for 0 < Im z < 1 and we obtain lower bounds for |f(z)| for 0 < Im z < 1. In our analysis we deal with scalar functions f(z) as well as with operator valued holomorphic functions I + A(z) assuming that A(z) is a trace class operator for and I + A(z) is invertible for 0 < Im z < 1 and is unitary for . A. Borichev was partially supported by the ANR project DYNOP.     

Borel–Carathéodory Type Theorem for Monogenic Functions

In this paper we give a generalization of the classical Borel–Carathéodory theorem in complex analysis to higher dimensions in the framework of Quaternionic Analysis. Submitted: November 14, 2007., Revised: February 25, 2008.     

Hyperbolic Function Theory in the Clifford Algebra {\mathcal {C}}\ell_{n+1, 0}

The aim of this paper is to give the basic principles of hyperbolic function theory on the Clifford algebra . The structure of the theory is quite similar to the case of Clifford algebras with negative generators, but the proofs are not obvious. The (real) Clifford algebra is generated by unit vectors with positive squares e²_i = + 1. The hyperbolic Dirac operator is of the form where Q₀f is represented by the composition . If is a solution of H_kf = 0, then f is called k-hypergenic in Ω, where is an open set. We introduce some basic results of hyperbolic function theory and give some representation theorems on . Received: October, 2007. Accepted: February, 2008.     

The Nielsen number for free fundamental groups and maps without remnant

For Y any space that has the homotopy type of a wedge of finitely many circles, and for g : Y → Y a map, the Nielsen number of g, N(g), is a homotopy invariant lower bound for the size of the fixed point set of any map homotopic to g. Such a map g has k-remnant if, roughly, there is limited cancellation in any product g _♯(u)g _♯(v) where g _♯ is the induced homomorphism and u, v ∈ π₁(Y) and |u| = |v| = k. We prove that such maps are (k + 1)-characteristic, meaning that in order to determine the Nielsen classes of fixed points, we need only test whether a limited, specified, set of elements z ∈ π₁(Y) are solutions to the equation z = W ⁻¹ _x f _♯(z)W _y , with x and y fixed points that are represented in the fundamental group by W _x and W _y , respectively. The number of elements to be tested is profoundly decreased by using abelianization as well. This work is a significant extension of Wagner’s results involving maps with remnant and Wagner’s algorithm. Our proofs involve new concepts and techniques. We present an algorithm for N(g) for any map g with k-remnant, and we provide examples for which no algebraic techniques previously known would work. One example shows that for any k there is a map that does not have (k − 1)-remnant but does have k-remnant. Dedicated to Edward Fadell for inspirational teaching and guidance as the thesis advisor of the first author     

Blow-up and critical exponents for parabolic equations with non-divergent operators: dual porous medium and thin film operators

Our first basic model is the fully nonlinear dual porous medium equation with source

Gap theorems for Kähler-Ricci solitons

In this paper, we prove that a gradient shrinking compact K?hler-Ricci soliton cannot have too large Ricci curvature unless it is K?hler-Einstein. Received: 23 October 2007, Revised: 28 February 2008     

Néron Models and Formal Groups

We show that formal groups can be used to simplify the construction of Néron models. Also we give a new proof of the stable reduction theorem for abelian varieties. Received: September 2007     

Irreducible <Emphasis Type="Italic">p</Emphasis>-Brauer characters of <Emphasis Type="Italic">p</Emphasis>-power degree for the symmetric and alternating groups

Starting from the question when all irreducible p-Brauer characters for a symmetric or an alternating group are of p-power degree, we classify the p-modular irreducible representations of p-power dimension in some families of representations for these groups. In particular, this then allows to confirm a conjecture by W. Willems for the alternating groups. Received: 14 June 2006     

Domains of Existence of Polymonogenic Functions

The paper addresses the Levi problem for a system of n Fueter equations in a domain in quaternionic space . This problem relates to various conditions of convexity and pseudoconvexity of the boundary of the domain. Received: October, 2007, Accepted: February, 2008.