期刊：
Rendiconti del Circolo Matematico di Palermo,2024年:1-12 ISSN：0009-725X
通讯作者：
Shaolin Chen
作者机构：
[Shaolin Chen] College of Mathematics and Statistics, Hengyang Normal University, Hengyang, People’s Republic of China;[Hidetaka Hamada] Faculty of Science and Engineering, Kyushu Sangyo University, Fukuoka, Japan
通讯机构：
[Shaolin Chen] C;College of Mathematics and Statistics, Hengyang Normal University, Hengyang, People’s Republic of China
关键词：
Harmonic function;Linear measure;Non-iterative dynamic system
摘要：
The main purpose of this paper is to determine the linear measure of non-iterative dynamic system of some classes of harmonic functions.
期刊：
Results in Mathematics,2024年79(2) ISSN：1422-6383
通讯作者：
Chen, SL
作者机构：
[Chen, Shaolin; Chen, SL] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.;[Chen, Shaolin; Chen, SL] Hunan Prov Key Lab Intelligent Informat Proc & App, Hengyang 421002, Peoples R China.;[Hamada, Hidetaka] Kyushu Sangyo Univ, Fac Sci & Engn, 3-1 Matsukadai 2 Chome,Higashi Ku, Fukuoka 8138503, Japan.
通讯机构：
[Chen, SL ] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.;Hunan Prov Key Lab Intelligent Informat Proc & App, Hengyang 421002, Peoples R China.
关键词：
Bloch type space;complex-valued harmonic function;composition operator;hardy space;pluriharmonic functions
摘要：
The main purpose of this paper is to investigate characterizations of composition operators on Bloch and Hardy type spaces. Initially, we use general doubling weights to study the composition operators from harmonic Bloch type spaces on the unit disc D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {D}}$$\end{document} to pluriharmonic Hardy spaces on the Euclidean unit ball Bn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {B}}<^>n$$\end{document}. Furthermore, we develop some new methods to study the composition operators from harmonic Bloch type spaces on D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {D}}$$\end{document} to pluriharmonic Bloch type spaces on D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {D}}$$\end{document}. Additionally, some application to new characterizations of the composition operators between pluriharmonic Lipschitz type spaces to be bounded or compact will be presented. The obtained results of this paper provide the improvements and extensions of the corresponding known results.
摘要：
The primary goal of this paper is to develop methods for investigating equivalent norms and Hardy-Littlewood-type theorems on Lipschitz-type spaces of analytic and complex-valued harmonic functions. First, we provide characterizations of equivalent norms on these spaces. Furthermore, we establish Hardy-Littlewood-type theorems for complex-valued harmonic functions. These results improve and extend the main findings of Dyakonov (1997) and Dyakonov (2005). Additionally, we apply the derived equivalent norms and Hardy-Littlewood-type theorems to the study of composition operators between Lipschitz-type spaces.
期刊：
Bulletin of the Malaysian Mathematical Sciences Society,2024年47(4):1-17 ISSN：0126-6705
通讯作者：
Hamada, H
作者机构：
[Chen, Shaolin] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.;[Hamada, Hidetaka] Kyushu Sangyo Univ, Fac Sci & Engn, 3-1 Matsukadai 2-Chome,Higashi Ku, Fukuoka 8138503, Japan.
通讯机构：
[Hamada, H ] K;Kyushu Sangyo Univ, Fac Sci & Engn, 3-1 Matsukadai 2-Chome,Higashi Ku, Fukuoka 8138503, Japan.
关键词：
Bloch space;Bounded symmetric domains;Composition operator;JB*-triple;The Kobayashi metric;Pluriharmonic function
摘要：
Let BX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {B}_X$$\end{document} be a bounded symmetric domain realized as the open unit ball of a JB*-triple X. First, we extend the definition for pluriharmonic Bloch functions to BX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {B}_X$$\end{document} by using the infinitesimal Kobayashi metric. Next, we develop some methods to investigate Bloch functions, and composition operators of pluriharmonic Bloch spaces on bounded symmetric domains. The obtained results provide the improvements and extensions of the corresponding known results.
期刊：
JOURNAL OF GEOMETRIC ANALYSIS,2024年34(9):1-23 ISSN：1050-6926
通讯作者：
Hamada, H
作者机构：
[Chen, Shaolin; Xie, Dou] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.;[Hamada, Hidetaka] Kyushu Sangyo Univ, Fac Sci & Engn, 3-1 Matsukadai 2-Chome,Higashi Ku, Fukuoka 8138503, Japan.
通讯机构：
[Hamada, H ] K;Kyushu Sangyo Univ, Fac Sci & Engn, 3-1 Matsukadai 2-Chome,Higashi Ku, Fukuoka 8138503, Japan.
关键词：
Hardy-Littlewood type Theorem;Hopf type lemma;Differential operator;Dirichlet solution
摘要：
The main aim of this paper is to investigate Hardy-Littlewood type Theorems and a Hopf type lemma on functions induced by a differential operator. We first prove more general Hardy-Littlewood type theorems for the Dirichlet solution of a differential operator which depends on
$$\alpha \in (-1,\infty )$$
over the unit ball
$$\mathbb {B}^n$$
of
$$\mathbb {R}^n$$
with
$$n\ge 2$$
, related to the Lipschitz type space defined by a majorant which satisfies some assumption. We find that the case
$$\alpha \in (0,\infty )$$
is completely different from the case
$$\alpha =0$$
due to Dyakonov (Adv. Math. 187 (2004), 146–172). Then a more general Hopf type lemma for the Dirichlet solution of a differential operator will also be established in the case
$$\alpha >n-2$$
.
作者机构：
[Chen, Shaolin; Chen, SL] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.
通讯机构：
[Chen, SL ] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.
关键词：
Heinz type inequality;Lipschitz continuous;Polyharmonic mapping;Quasiconformal mapping
摘要：
Suppose that f satisfies the following: (1) the polyharmonic equation triangle(m) f = triangle(triangle(m-1) f ) = phi(m) (phi(m) is an element of C((B-n) over bar, R-n)), (2) the boundary conditions triangle(0) f = phi(0), triangle(1) f = phi 1, ..., triangle(m-1) f = phi(m-1 )on Sn-1 (phi (j) is an element of C(Sn-1, R-n) for j is an element of {0, 1, ... , m - 1} and Sn-1 denotes the boundary of the unit ball B-n), and (3) f (0) = 0, where n >= 3 and m >= 1 are integers. Initially, we prove a Schwarz type lemma and use it to obtain a Heinz type inequality of mappings satisfying the polyharmonic equation with the above Dirichlet boundary value conditions. Furthermore, we establish a Bloch type theorem of mappings satisfying the above polyharmonic equation, which gives an answer to an open problem in Chen and Ponnusamy, (2019). Additionally, we show that if f is a K-quasiconformal self-mapping of Bn satisfying the above polyharmonic equation, then f is Lipschitz continuous, and the Lipschitz constant is asymptotically sharp as K -> 1(+) and parallel to phi(j) parallel to infinity -> 0(+) for j is an element of {1, ... , m}.(c) 2023 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.
期刊：
Bulletin of the Malaysian Mathematical Sciences Society,2023年46(4):1-19 ISSN：0126-6705
通讯作者：
Chen, SL
作者机构：
[Chen, Shaolin; Chen, SL] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.;[Chen, Shaolin; Chen, SL] Hunan Prov Key Lab Intelligent Informat Proc & App, Hengyang 421002, Hunan, Peoples R China.;[Hamada, Hidetaka] Kyushu Sangyo Univ, Fac Sci & Engn, 3-1 Matsukadai 2 Chome,Higashi Ku, Fukuoka 8138503, Japan.
通讯机构：
[Chen, SL ] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.;Hunan Prov Key Lab Intelligent Informat Proc & App, Hengyang 421002, Hunan, Peoples R China.
关键词：
Bergman type class;Complex-valued harmonic function;Elliptic mapping;Hardy type space
摘要：
The main purpose of this paper is to discuss Hardy type spaces and Bergman type classes of complex-valued harmonic functions. We first establish a Hardy-Littlewood type theorem on complex-valued harmonic functions. Next, the relationships between the Bergman type classes and the Hardy type spaces of complex-valued harmonic functions or the relationships between the Bergman type classes and the Hardy type spaces of harmonic
$$(K,K')$$
-elliptic mappings will be discussed, where
$$K\ge 1$$
and
$$K'\ge 0$$
are constants.
期刊：
JOURNAL OF GEOMETRIC ANALYSIS,2023年33(7):1-45 ISSN：1050-6926
通讯作者：
Chen, SL
作者机构：
[Huang, Manzi; Chen, Jiaolong; Zheng, Huaqing] Hunan Normal Univ, Sch Math & Stat, MOE LCSM, Changsha 410081, Hunan, Peoples R China.;[Chen, Shaolin; Chen, SL] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.;[Chen, Shaolin; Chen, SL] Hunan Prov Key Lab Intelligent Informat Proc & App, Hengyang 421002, Peoples R China.
通讯机构：
[Chen, SL ] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.;Hunan Prov Key Lab Intelligent Informat Proc & App, Hengyang 421002, Peoples R China.
关键词：
Isoperimetric type inequality;Poisson type integral;Hardy type space;Bergman type space;\((K,K')\)-elliptic mapping
摘要：
The main purpose of this paper is to establish some isoperimetric type inequalities for mappings induced by the weighted Laplace differential operators. The obtained results of this paper provide improvements and extensions of the corresponding known results.
期刊：
JOURNAL OF GEOMETRIC ANALYSIS,2023年33(6):1-22 ISSN：1050-6926
通讯作者：
Shaolin Chen
作者机构：
[Chen, Shaolin] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.;[Hamada, Hidetaka] Kyushu Sangyo Univ, Fac Sci & Engn, 3-1 Matsukadai 2-Chome,Higashi ku, Fukuoka 8138503, Japan.
通讯机构：
[Shaolin Chen] C;College of Mathematics and Statistics, Hengyang Normal University, Hengyang, People’s Republic of China
关键词：
Lipschitz type space;Harmonic function;Composition operator;Pluriharmonic functions
摘要：
The main purpose of this paper is to develop some methods to study the composition operators between harmonic Lipschitz type spaces. Some characterizations of boundedness and w-compactness of composition operators between the harmonic Lipschitz type spaces will be given. Consequently, the obtained results improve and extend some corresponding known results.
期刊：
JOURNAL D ANALYSE MATHEMATIQUE,2023年152(1):181-216 ISSN：0021-7670
通讯作者：
Chen, SL
作者机构：
[Chen, Shaolin; Chen, SL] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.;[Chen, Shaolin; Chen, SL] Hunan Prov Key Lab Intelligent Informat Proc & App, Hengyang 421002, Hunan, Peoples R China.;[Hamada, Hidetaka] Kyushu Sangyo Univ, Fac Sci & Engn, 3-1 Matsukadai 2 Chome,Higashi ku, Fukuoka 8138503, Japan.;[Vijayakumar, Ramakrishnan; Ponnusamy, Saminathan] Indian Inst Technol Madras, Dept Math, Chennai 600036, India.;[Ponnusamy, Saminathan] Lomonosov Moscow State Univ, Moscow Ctr Fundamental & Appl Math, Moscow 119333, Russia.
通讯机构：
[Chen, SL ] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.;Hunan Prov Key Lab Intelligent Informat Proc & App, Hengyang 421002, Hunan, Peoples R China.
摘要：
The main purpose of this paper is to develop some methods to investigate the Schwarz type lemmas of holomorphic mappings and pluriharmonic mappings in Banach spaces. Initially, we extend the classical Schwarz lemmas of holomorphic mappings to Banach spaces, and then we apply these extensions to establish a sharp Bloch type theorem for pluriharmonic mappings on homogeneous unit balls of DOUBLE-STRUCK CAPITAL C-n and to obtain some sharp boundary Schwarz type lemmas for holomorphic mappings in Banach spaces. Furthermore, we improve and generalize the classical Schwarz lemmas of planar harmonic mappings into the sharp forms of Banach spaces, and present some applications to sharp boundary Schwarz type lemmas for pluriharmonic mappings in Banach spaces. Additionally, using a relatively simple method of proof, we prove some sharp Schwarz-Pick type estimates of pluriharmonic mappings in JB*-triples, and the obtained results provide the improvements and generalizations of the corresponding results in [9].
作者机构：
[Chen, Shaolin; Chen, SL] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.;[Chen, Shaolin; Chen, SL] Hunan Prov Key Lab Intelligent Informat Proc & App, Hengyang 421002, Peoples R China.;[Hamada, Hidetaka] Kyushu Sangyo Univ, Fac Sci & Engn, 3-1 Matsukadai 2 Chome,Higashi Ku, Fukuoka 8138503, Japan.
通讯机构：
[Chen, SL ] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.;Hunan Prov Key Lab Intelligent Informat Proc & App, Hengyang 421002, Peoples R China.
关键词：
Riesz type inequality;Hardy–Littlewood type theorem;Smooth moduli
摘要：
The purpose of this paper is to develop some methods to study (Fejér-)Riesz type inequalities, Hardy–Littlewood type theorems and smooth moduli of holomorphic, pluriharmonic and harmonic functions in high-dimensional cases. Initially, we prove some sharp Riesz type inequalities of pluriharmonic functions on bounded symmetric domains. The obtained results extend the main results in Kalaj (Trans Am Math. Soc. 372:4031–4051, 2019). Next, some Hardy–Littlewood type theorems of holomorphic and pluriharmonic functions on John domains are established, which give analogies and extensions of a result in Hardy and Littlewood (J Reine Angew Math 167;405–423, 1931). Furthermore, we establish a Fejér–Riesz type inequality on pluriharmonic functions in the Euclidean unit ball in
$$\mathbb {C}^n$$
, which extends the main result in Melentijević and Bo
$$\breve{z}$$
in (Potential Anal 54:575–580, 2021). Additionally, we also discuss the Hardy–Littlewood type theorems and smooth moduli of holomorphic, pluriharmonic and harmonic functions. Consequently, we improve and extend the corresponding results in Dyakonov (Acta Math 178:143–167, 1997), Hardy and Littlewood (Math Z 34:403–439, 1932), Dyakonov (Adv Math 187:146–172, 2004) and Pavlović (Rev Mat Iberoam 23:831–845, 2007).
作者机构：
[Chen, Shaolin] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.;[Chen, Shaolin] Hunan Prov Key Lab Intelligent Informat Proc & Ap, Changsha 421002, Peoples R China.;[Hamada, Hidetaka] Kyushu Sangyo Univ, Fac Sci & Engn, Higashi Ku, 3-1 Matsukadai 2-Chome, Fukuoka 8138503, Japan.;[Zhu, Jian-Feng] Huaqiao Univ, Sch Math Sci, Quanzhou 362021, Peoples R China.
通讯机构：
[Shaolin Chen] C;College of Mathematics and Statistics, Hengyang Normal University, Hengyang, People’s Republic of China<&wdkj&>Hunan Provincial Key Laboratory of Intelligent Information Processing and Application, Changsha, People’s Republic of China
关键词：
Bloch type space;Complex-valued harmonic function;Composition operator;Hardy type space
摘要：
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.
期刊：
Journal of Functional Analysis,2022年282(1):109254 ISSN：0022-1236
通讯作者：
Chen, Shaolin
作者机构：
[Chen, Shaolin] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.;[Hamada, Hidetaka] Kyushu Sangyo Univ, Fac Sci & Engn, Higashi Ku, 3-1 Matsukadai 2 Chome, Fukuoka 8138503, Japan.
通讯机构：
[Chen, Shaolin] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.
关键词：
Bohr phenomenon;Harmonic function;Pluriharmonic function;Schwarz-Pick type lemma of arbitrary order;Sharp Schwarz-Pick type estimate
摘要：
The purpose of this paper is to study the Schwarz-Pick type inequalities for harmonic or pluriharmonic functions. By analogy with the generalized Khavinson conjecture, we first give some sharp estimates of the norm of harmonic functions from the Euclidean unit ball in R-n into the unit ball of the real Minkowski space. Next, we give several sharp Schwarz-Pick type inequalities for pluriharmonic functions from the Euclidean unit ball in C-n or from the unit polydisc in C-n into the unit ball of the Minkowski space. Furthermore, we establish some sharp coefficient type Schwarz-Pick inequalities for pluriharmonic functions defined in the Minkowski space. Finally, we use the obtained Schwarz-Pick type inequalities to discuss the Lipschitz continuity, the Schwarz-Pick type lemmas of arbitrary order and the Bohr phenomenon of harmonic or pluriharmonic functions. (C) 2021 Elsevier Inc. All rights reserved.
作者机构：
[Shao Lin CHEN] College of Mathematics and Statistics,Hengyang Normal University,Hengyang 421002,P.R.China;[Shao Lin CHEN] Hunan Provincial Key Laboratory of Intelligent Information Processing and Application,Hengyang 421002,P.R.China;Department of Mathematics,Indian Institute of Technology Madras,Chennai-600 036,India;[Saminathan PONNUSAMY] 马德拉斯理工学院
通讯机构：
[Shao Lin Chen] C;College of Mathematics and Statistics, Hengyang Normal University, Hengyang, P. R. China<&wdkj&>Hunan Provincial Key Laboratory of Intelligent Information Processing and Application, Hengyang, P. R. China
关键词：
Harmonic K-quasiconformal mapping;Koebe type covering theorem;Koebe type distortion theorem;Radial John disk
摘要：
In this article, we first establish an asymptotically sharp Koebe type covering theorem for harmonic K-quasiconformal mappings. Then we use it to obtain an asymptotically Koebe type distortion theorem, a coefficients estimate, a Lipschitz characteristic and a linear measure distortion theorem of harmonic K-quasiconformal mappings. At last, we give some characterizations of the radial John disks with the help of pre-Schwarzian of harmonic mappings.
摘要：
For K >= 1, suppose that f is a K-quasiconformal self-mapping of the unit disk D, which satisfies the following: (1) the biharmonic equation Delta(Delta f) = g (g is an element of C((D) over bar)), (2) the boundary condition Delta f = phi(phi is an element of C(T) and T denotes the unit circle), (3) f (0) = 0. The purpose of this paper is to prove that f is Lipschitz continuous, and, further, it is bi-Lipschitz continuous if parallel to g parallel to(infinity) and parallel to phi parallel to(infinity) are small enough. Moreover, the estimates are asymptotically sharp as K -> 1 , parallel to g parallel to(infinity) -> 0, and parallel to phi parallel to(infinity) -> 0 and thus, such a mapping f behaves almost like a rotation for sufficiently small K, parallel to g parallel to(infinity) and parallel to phi parallel to(infinity).
摘要：
For two constants
$$K\ge 1$$
and
$$K'\ge 0$$
, suppose that f is a
$$(K,K')$$
-quasiconformal self-mapping of the unit disk
$${\mathbb {D}}$$
, which satisfies the following: (1) the inhomogeneous polyharmonic equation
$$\Delta ^{n}f=\Delta (\Delta ^{n-1} f)=\varphi _{n}$$
in
$${\mathbb {D}}$$
$$(\varphi _{n}\in {\mathcal {C}}(\overline{{\mathbb {D}}}))$$
, (2) the boundary conditions
$$\Delta ^{n-1}f=\varphi _{n-1},~\ldots ,~\Delta ^{1}f=\varphi _{1}$$
on
$${\mathbb {T}}$$
(
$$\varphi _{j}\in {\mathcal {C}}({\mathbb {T}})$$
for
$$j\in \{1,\ldots ,n-1\}$$
and
$${\mathbb {T}}$$
denotes the unit circle), and (3)
$$f(0)=0$$
, where
$$n\ge 2$$
is an integer. The main aim of this paper is to prove that f is Lipschitz continuous, and, further, it is bi-Lipschitz continuous when
$$\Vert \varphi _{j}\Vert _{\infty }$$
are small enough for
$$j\in \{1,\ldots ,n\}$$
. Moreover, the estimates are asymptotically sharp as
$$K\rightarrow 1^{+}$$
,
$$K'\rightarrow 0^{+}$$
and
$$\Vert \varphi _{j}\Vert _{\infty }\rightarrow 0^{+}$$
for
$$j\in \{1,\ldots ,n\}$$
.
期刊：
JOURNAL OF GEOMETRIC ANALYSIS,2021年31(11):11051-11060 ISSN：1050-6926
通讯作者：
Saminathan Ponnusamy
作者机构：
[Chen, Shaolin] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421002, Hunan, Peoples R China.;[Chen, Shaolin] Hunan Prov Key Lab Intelligent Informat Proc & Ap, Hengyang 421002, Peoples R China.;[Ponnusamy, Saminathan] Indian Inst Technol Madras, Dept Math, Chennai 600036, Tamil Nadu, India.;[Wang, Xiantao] Hunan Normal Univ, Key Lab High Performance Comp & Stochast Informat, Coll Math & Stat, Changsha 410081, Hunan, Peoples R China.
通讯机构：
[Saminathan Ponnusamy] D;Department of Mathematics, Indian Institute of Technology Madras, Chennai, India
关键词：
Poisson integral;Harmonic mapping;Elliptic mapping;Hardy-type space;Bergman-type space
摘要：
Let
$$f = P[F]$$
denote the Poisson integral of F in the unit disk
$${\mathbb {D}}$$
with F being absolutely continuous in the unit circle
$${\mathbb {T}}$$
and
$${\dot{F}}\in L^{p}({\mathbb {T}})$$
, where
$${\dot{F}}(e^{it})=\frac{d}{dt} F(e^{it})$$
and
$$p\ge 1$$
. Recently, the author in Zhu (J Geom Anal, 2020) proved that (1) if f is a harmonic mapping and
$$1\le p< 2$$
, then
$$f_{z}$$
and
$$\overline{f_{{\overline{z}}}}\in \mathcal {B}^{p}({\mathbb {D}}),$$
the classical Bergman spaces of
$${\mathbb {D}}$$
[12, Theorem 1.2]; (2) if f is a harmonic quasiregular mapping and
$$1\le p\le \infty $$
, then
$$f_{z},$$
$$\overline{f_{{\overline{z}}}}\in \mathcal {H}^{p}({\mathbb {D}}),$$
the classical Hardy spaces of
$${\mathbb {D}}$$
[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
$$1\le p< \infty $$
. Also, we show that [12, Theorem 1.2] is not true when
$$p=\infty $$
. 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.
期刊：
Journal of Mathematical Analysis and Applications,2020年486(2):123920 ISSN：0022-247X
通讯作者：
Ponnusamy, Saminathan
作者机构：
[Chen, Shaolin] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.;[Ponnusamy, Saminathan] Indian Inst Technol Madras, Dept Math, Chennai 600036, Tamil Nadu, India.
通讯机构：
[Ponnusamy, Saminathan] I;Indian Inst Technol Madras, Dept Math, Chennai 600036, Tamil Nadu, India.
摘要：
Let (D) over bar be the closure of the unit disk D in the complex plane C and g be a continuous function in (D) over bar. In this paper, we discuss some characterizations of elliptic mappings f satisfying the Poisson's equation Delta f = g in D, and then establish some sharp distortion theorems on elliptic mappings with the finite perimeter and the finite radial length, respectively. The obtained results are the extension of the corresponding classical results. (C) 2020 Elsevier Inc. All rights reserved.
