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Remarks on ‘Norm Estimates of the Partial Derivatives for Harmonic Mappings and Harmonic Quasiregular Mappings’
, Chen S., Wang X.
Published in Springer Nature
Let f= P[F] denote the Poisson integral of F in the unit disk D with F being absolutely continuous in the unit circle T and F˙ ∈ Lp(T) , where F˙(eit)=ddtF(eit) and p≥ 1. Recently, the author in Zhu (J Geom Anal, 2020) proved that (1) if f is a harmonic mapping and 1 ≤ p< 2 , then fz and fz¯¯ ∈ Bp(D) , the classical Bergman spaces of D [12, Theorem 1.2]; (2) if f is a harmonic quasiregular mapping and 1 ≤ p≤ ∞, then fz, fz¯¯ ∈ Hp(D) , the classical Hardy spaces of D [12, Theorem 1.3]. These are the main results in Zhu (J Geom Anal, 2020). The purpose of this paper is to generalize these two results. First, we prove that, under the same assumptions, [12, Theorem 1.2] is true when 1 ≤ p< ∞. Also, we show that [12, Theorem 1.2] is not true when p= ∞. Second, we demonstrate that [12, Theorem 1.3] still holds true when the assumption f being a harmonic quasiregular mapping is replaced by the weaker one f being a harmonic elliptic mapping. © 2021, Mathematica Josephina, Inc.
About the journal
JournalData powered by TypesetJournal of Geometric Analysis
PublisherData powered by TypesetSpringer Nature
Open AccessNo