Hölder Continuity of Generalized Harmonic Functions in the Unit Disc

Abstract

Suppose α, β ∈ R\Z − such that α + β > −1 and 1 ≤ p ≤ ∞. Let u = P α,β [f ] be an (α, β)-harmonic mapping on D, the unit disc of C, with the boundary f being absolutely continuous and ḟ ∈ L p (0, 2π), where ḟ (e iθ ) := d dθ f (e iθ ). In this paper, we investigate the membership of the partial derivatives ∂ z u and ∂ z u in the space H p G (D), the generalized Hardy space. We prove, if α + β > 0, then both ∂ z u and ∂ z u are in H p G (D). For α + β < 0, we show if ∂ z u or ∂ z u ∈ H 1 G (D) then u = 0 or u is a polyharmonic function.

$H^p$-Norm estimates of the partial derivatives and Schwarz lemma for $α$-harmonic functions

“…In [9], we provide a refinement of the two previous theorems, we prove that for 1 < p < ∞, both f z and f z are in H p (D) without any extra conditions on f . Theorem C. [9] Suppose that F is an absolute continuous function on U and f = P [F ] is a harmonic mapping in D and Ḟ ∈ L p (T).…”

“…It is worth noting that a similar problem for harmonic functions was treated in [15] and improved in [1,9].…”

Lipschitz Continuity for Harmonic Functions and Solutions of the α¯-Poisson Equation

“…Moreover, L α u = 0 if L * α u = 0. For a, b ∈ R, which cannot be negative integers and which satisfies a + b > −1, the operator is defined in [14] as…”

“…Deeper origins of this topic can also be seen in the famous Schwartz lemma, and some newer result and history of this area can be found in [3]. For additional results, it is important to mention [13,14], where Lipschitz continuity of the solution of the hyperbolic Poisson's equation and (a, b)-harmonic functions are investigated.…”