Remarks on ‘Norm Estimates of the Partial Derivatives for Harmonic Mappings and Harmonic Quasiregular Mappings’

Chen, Shaolin; Ponnusamy, Saminathan; Wang, Xiantao

doi:10.1007/s12220-021-00672-7

Cited by 13 publications

(9 citation statements)

References 8 publications

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“…In [1], the authors showed that Theorem C also holds true for harmonic elliptic mappings, which are more general than harmonic quasiregular mapping.…”

Section: Resultsmentioning

confidence: 99%

Estimates of partial derivatives for harmonic functions on the unit disc

Khalfallah¹,

Mateljević²

2023

Preprint

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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

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“…In [1], the authors showed that Theorem C also holds true for harmonic elliptic mappings, which are more general than harmonic quasiregular mapping.…”

Section: Resultsmentioning

confidence: 99%

Estimates of partial derivatives for harmonic functions on the unit disc

Khalfallah¹,

Mateljević²

2023

Preprint

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“…Let H p H (D) = H p G (D) ∩ H be the harmonic Hardy space. The classical Hardy space H p (D), that is, all the elements are analytic, is a subspace of H p H (D) (see [6,8,10,11,22,32]).…”

Section: Preliminaries and Main Resultsmentioning

confidence: 99%

Composition operators on Bloch and Hardy type spaces

Chen¹,

Hamada²,

Zhu³

2022

Preprint

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The main purpose of this paper is to discuss Hardy type spaces, Bloch type spaces and the composition operators of complex-valued harmonic functions. We first establish a sharp estimate of the Lipschitz continuity of complex-valued harmonic functions in Bloch type spaces with respect to the pseudo-hyperbolic metric, which gives an answer to an open problem. Then some classes of composition operators on Bloch and Hardy type spaces will be investigated. The obtained results improve and extend some corresponding known results.

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“…It is worth noting that a similar problem for harmonic functions was treated in [15] and improved in [1,9].…”

Section: Generalized Hardy Spacesmentioning

confidence: 99%

“…In [1], the authors showed that Theorem B also holds true for harmonic elliptic mappings, which are more general than harmonic quasiregular mapping.…”

Section: Generalized Hardy Spacesmentioning

confidence: 99%

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

Khalfallah¹,

Mateljević²

2023

Preprint

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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).

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Remarks on ‘Norm Estimates of the Partial Derivatives for Harmonic Mappings and Harmonic Quasiregular Mappings’

Cited by 13 publications

References 8 publications

Estimates of partial derivatives for harmonic functions on the unit disc

Estimates of partial derivatives for harmonic functions on the unit disc

Composition operators on Bloch and Hardy type spaces

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

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