We prove that continuity properties of bounded analytic functions in bounded smoothly bounded pseudoconvex domains in two-dimensional affine space are determined by their behaviour near the Shilov boundary. Namely, if the function has continuous extension to an open subset of the boundary containing the Shilov boundary it extends continuously to the whole boundary. If it is e.g. Hölder continuous on such a boundary set, it is Hölder continuous on the closure of the domain. The statements may fail...

We study the properties of the group Aut(D) of all biholomorphic transformations of a bounded circular domain D in ${ℂ}^n$ containing the origin. We characterize the set of all possible roots for the Lie algebra of Aut(D). There exists an n-element set P such that any root is of the form α or -α or α-β for suitable α,β ∈ P.

We prove that the self-commutator of a Toeplitz operator with unbounded analytic rational symbol has a dense domain in both the Bergman space and the Hardy space of the unit disc. This is a basic step towards establishing whether the self-commutator has a compact or trace-class extension.

We present a description of the diagonal of several spaces in the polydisk. We also generalize some previously known contentions and obtain some new assertions on the diagonal map using maximal functions and vector valued embedding theorems, and integral representations based on finite Blaschke products. All our results were previously known in the unit disk.

MSC 2010: 33-00, 33C45, 33C52, 30C15, 30D20, 32A17, 32H02, 44A05The 6th International Conference "Transform Methods and Special Functions' 2011", 20 - 23 October 2011 was dedicated to the 80th anniversary of Professor Peter Rusev, as one of the founders of this series of international meetings in Bulgaria, since 1994. It is a pleasure to congratulate the Jubiliar on behalf of the Local Organizing Committee and International Steering Committee, and to present shortly some of his life achievements...

Following the line of Ouyang et al. (1998) to study the ${}_p$ spaces of holomorphic functions in the unit ball of ℂⁿ, we present in this paper several results and relations among ${}_p\left(ₙ\right)$, the α-Bloch, the Dirichlet ${}_p$ and the little ${}_{p,0}$ spaces.

We show that the holomorphic functions on polysectors whose derivatives remain bounded on proper subpolysectors are precisely those strongly asymptotically developable in the sense of Majima. This fact allows us to solve two Borel-Ritt type interpolation problems from a functional-analytic viewpoint.

For a domain $\Omega \subset {ℂ}^n$ let $H\left(\Omega \right)$ be the holomorphic functions on $\Omega$ and for any $k\in \mathbb{N}$ let $A^k\left(\Omega \right)=H\left(\Omega \right)\cap C^k\left(\overline{\Omega }\right)$. Denote by ${𝒜}_D^k\left(\Omega \right)$ the set of functions $f\Omega \to [0,\infty )$ with the property that there exists a sequence of functions $f_j\in A^k\left(\Omega \right)$ such that $\left\{\right|f_j\left|\right\}$ is a nonincreasing sequence and such that $f\left(z\right)={lim}_{j\to \infty }\left|f_j\left(z\right)\right|$. By ${𝒜}_I^k\left(\Omega \right)$ denote the set of functions $f\Omega \to (0,\infty )$ with the property that there exists a sequence of functions $f_j\in A^k\left(\Omega \right)$ such that $\left\{\right|f_j\left|\right\}$ is a nondecreasing sequence and such that $f\left(z\right)={lim}_{j\to \infty }\left|f_j\left(z\right)\right|$. Let $k\in \mathbb{N}$ and let ${\Omega }_1$ and ${\Omega }_2$ be bounded $A^k$-domains of holomorphy in ${ℂ}^{m_1}$ and ${ℂ}^{m_2}$ respectively. Let $g_1\in {𝒜}_D^k\left({\Omega }_1\right)$, $g_2\in {𝒜}_I^k\left({\Omega }_1\right)$ and $h\in {𝒜}_D^k\left({\Omega }_2\right)\cap {𝒜}_I^k\left({\Omega }_2\right)$. We prove that the...