Geometric Generalizations in Kresin–Maz’ya Sharp Real-Part Theorems

In the present article we present some geometric generalizations of the estimates from Chapters 5,6,7 of the monograph [7]. Received: July 24, 2007. Revised: October 27, 2007.     

On Holomorphic Perturbations of Infinite Order Differential Equations

It is proven in the present article that the solutions of infinite order differential equation with holomorphic parameter w ∈ U ? ? depend holomorphically on w in the neighborhood of characteristic points of certain directions.     

On a class of holomorphic functions representable by Carleman formulas in the interior of an equilateral cone from their values on its rigid base

Let Δ be an equilateral cone in C with vertices at the complex numbers and rigid base M (Section 1). Assume that the positive real semi-axis is the bisectrix of the angle at the origin. For the base M of the cone Δ we derive a Carleman formula representing all those holomorphic functions from their boundary values (if they exist) on M which belong to the class . The class is the class of holomorphic functions in Δ which belong to the Hardy class near the base M (Section 2). As an application of the above characterization, an important result is an extension theorem for a function fL¹(M) to a function .     

On the Bohr Radius for Two Classes of Holomorphic Functions

Using some multidimensional analogs of the inequalities of E. Landau and F. Wiener for the Taylor coefficients of special classes of holomorphic functions on Reinhardt domains we obtain some estimates for the Bohr radius.     

On asymptotic approximations of the residual currents

We use a -module approach to discuss positive examples for the existence of the unrestricted limit of the integrals involved in the approximation to the Coleff-Herrera residual currents in the complete intersection case. Our results also provide asymptotic developments for these integrals.