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

2023/02/19 by Khalfallah, Adel, Mateljević, Miodrag
#30H20 #31A05 #Complex Variables (math.CV) #FOS: Mathematics #Primary: 30C62 #Secondary: 30H10

paper · doi:10.48550/arxiv.2302.09613

Abstract

Suppose α>-1 and 1≤ p ≤ ∞. Let f=P_α[F] be an α-harmonic mapping on \mathbbD with the boundary F being absolute continuous and F∈ L^p(0,2π), where F(e^iθ):=\fracdF(e^iθ)dθ. In this paper, we investigate the membership of f_z and f_z in the space H_G^p(\mathbbD), the generalized Hardy space. We prove, if α>0, then both f_z and f_z are in H_G^p(\mathbbD). If α<0, then f_z and f_z∈ H_G^p(\mathbbD) if and only if f is analytic. Finally, we investigate a Schwartz Lemma for α-harmonic functions.

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