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

Abstract

Suppose α > −1 and 1 ≤ p ≤ ∞. Let f = P α [F ] be an α-harmonic mapping on D with the boundary F being absolute continuous and Ḟ ∈ L p (0, 2π), where Ḟ (e iθ ) := dF (e iθ ) dθ. In this paper, we investigate the membership of f z and f z in the space H p G (D), the generalized Hardy space. We prove, if α > 0, then both f z and f z are in H p G (D). If α < 0, then f z and f z ∈ H p G (D) if and only if f is analytic. Finally, we investigate a Schwartz Lemma for α-harmonic functions.Here the boundary data f * is a distribution on T, i.e. f * ∈ D ′ (T), and the boundary condition in (1.3) is understood as f r → f * ∈ D ′ (T) as r → 1 − , where f r (e iθ ) := f (re iθ ), r ∈ [0, 1).