A series expansion for generalized harmonic functions

Abstract

We consider a class of generalized harmonic functions in the open unit disc in the complex plane. Our main results concern a canonical series expansion for such functions. Of particular interest is a certain individual generalized harmonic function which suitably normalized plays the role of an associated Poisson kernel.

Uniqueness theorems for weighted harmonic functions in the upper half-plane

“…Condition (2.4) ensures that the series expansion (2.3) is absolutely convergent in the space C ∞ (D) of indefinitely differentiable functions in D. As a consequence we have that u ∈ C ∞ (D). We refer to Klintborg and Olofsson [16] for an updated account on these matters.…”

“…where u is as in (2.3) (see [16,Theorem 5.3]). In particular, an α-harmonic function in D is uniquely determined by its germ at the origin.…”

Hölder Continuity of Generalized Harmonic Functions in the Unit Disc

“…Let D be the open unit disc in the complex plane C, let T = ∂D be the unit circle, and denote by where α, β are real parameters. These operators have been introduced and studied in higher dimensions by Geller [10] and Ahern et al [1,2] and recently investigated in the planar case in [15]. Of particular interest are the solutions of the associated homogeneous equation in D:…”

“…Theorem B ( [1], [15]). Let α, β ∈ R. Then u is an (α, β)-harmonic function if and only if it has the form…”

“…We should point out that in the case α = 0 or β = 0, the function M α,β is constant and M α,β (z) = 1. For more properties and Hölder continuity of (α, β)-harmonic functions, see [1,5,14,15,16].…”