Estimates of partial derivatives for harmonic functions on the unit disc

2023/02/19 by Khalfallah, Adel, Mateljević, Miodrag · 1 citation
#30H20 #31A05 #Complex Variables (math.CV) #FOS: Mathematics #Primary: 30C62 #Secondary: 30H10

paper · doi:10.48550/arxiv.2302.09623

Abstract

Let f = P[F] denote the Poisson integral of F in the unit disk \mathbbD with F is an absolute continuous in the unit circle \mathbbT and F∈ L^p(\mathbbT), where F(e^it) = (d)/(dt) F(e^it) and p ∈ [1,∞]. Recently, Chen et al. (J. Geom. Anal., 2021) extended Zhu's results (J. Geom. Anal., 2020) and proved that (i) if f is a harmonic mapping and 1 ≤ p < ∞, then f_z and f_z ∈ B^p(\mathbbD), the Bergman spaces of \mathbbD. Moreover, (ii) under additional conditions as f being harmonic quasiregular mapping in \citeZhu or f being harmonic elliptic mapping in \citeCPW, they proved that f_z and f_z∈ H^p(\mathbbD), the Hardy space of \mathbbD, for 1 ≤ p ≤ ∞. The aim of this paper is to extend these results by showing that (ii) holds for p∈(1,∞) without any extra conditions and for p=1 or p=∞, f_z and f_z∈ H^p(\mathbbD) if and only if H(F)∈ L^p(\mathbbT), the Hilbert transform of F and in that case, it yields zf_z=P[\fracF+iH(F)2i].

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• Properties for (α,β)-harmonic functions

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