The boundary behavior of holomorphic functions

Holomorphic Diffusions and Boundary Behavior of Harmonic Functions

Cornell University We study a family of differential operators Lα α ≥ 0 in the unit ball D of Cn with n ≥ 2 that generalize the classical Laplacian, α = 0, and the conformal Laplacian, α = 1/2 (that is, the Laplace–Beltrami operator for Bergman metric in D). Using the diffusion processes associated with these (degenerate) differential operators, the boundary behavior of Lα-harmonic functions is...

On the Boundary Values of Holomorphic Functions in Wedge Domains

WD = {z => x + iy\ x Q R , y & D). The distinguished boundary of WD is the set {z = x+iy \ xÇzR, y = 0 } , i.e., just R. In this paper we consider the following problem. Let ƒ be a real function on R satisfying certain regularity conditions. What are the conditions for the existence of a holomorphic function F on WD such that ƒ is the limit of Re -Fas the variable approaches the distinguished b...

Boundary Values of Bounded Holomorphic Functions of Several Variables

A typical result (consequence of Theorem 1) is that if such a function tends to a limit X as x—»x° from inside an open cone with vertex at x°, then it tends "on the average" to X as x—>x° from inside any open cone with vertex at x°. In particular, if such a function tends to limits in each of two open cones with a common vertex, these limits must be equal; for n = l (when the cones are half-lin...

Large linear manifolds of noncontinuable boundary - regular holomorphic functions ✩

A special subspace of weighted spaces of holomorphic functions on the upper half plane

In this paper, we intend to define and study concepts of weight and weighted spaces of holomorphic (analytic) functions on the upper half plane. We study two special classes of these spaces of holomorphic functions on the upper half plane. Firstly, we prove these spaces of holomorphic functions on the upper half plane endowed with weighted norm supremum are Banach spaces. Then, we investigate t...