Estimates of partial derivatives for harmonic functions on the unit disc

Abstract

Let f = P [F ] denote the Poisson integral of F in the unit disk D with F is an absolute continuous in the unit circle T and Ḟ ∈ L p (T), where Ḟ (e it ) = d dt F (e it ) and p ∈ [1, ∞]. Recently, Chen et al. [1] (J. Geom. Anal., 2021) extended Zhu's results [15] (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 (D), the Bergman spaces of D. Moreover, (ii) under additional conditions as f being harmonic quasiregular mapping in [15] or f being harmonic elliptic mapping in [1], they proved that f z and f z ∈ H p (D), the Hardy space of D, 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 (D) if and only if H( Ḟ ) ∈ L p (T ), the Hilbert transform of Ḟ and in that case, it yields zf z = P [ Ḟ +iH ( Ḟ ) 2i].The analytic Hardy space H p (D) is the set of all elements of h p (D) which are analytic. (cf. [2, 3]).Denote by L p (T) (p ∈ [1, ∞]) the space of all measurable functions F of T into C with