arXiv subjects

Hidetaka Hamada

Publications and source records attributed to Hidetaka Hamada.

18 recordsLinked to original sources

arXiv2026

Hardy-Littlewood type phenomena and the Girela-Pel\'aez conjecture for the M\"obius invariant Laplacian operator

The purpose of this paper is twofold. First, we investigate the Hardy-Littlewood type phenomena for Dirichlet solutions to the M\"obius invariant Laplace equation on the unit ball in $\mathbb{R}^n$. Our work extends and improves several key results due to Pavlov\'c [Rev. Mat. Iberoam. 23: 831-845, 2007] and Chen et al. [J. Geom. Anal. 34: 23 pp, 2024]. In particular, we give a complete answer to a question raised by Makoto Masumoto. Second, motivated by Aikawa's work, we study the boundedness of the operator norm of $P_{\alpha}$, where $P_{\alpha}[\varphi]$ is the Dirichlet solution of such equation for the boundary data $\varphi$. By using alternative proof techniques, we obtain an equivalent characterization of the boundedness of the operator norm of $P_{\alpha}$. Finally, we show that the Girela-Pel\'aez conjecture holds positively for more general classes of functions induced by the M\"obius invariant Laplacian operator.

math.FA↗

arXiv2026

Some sharp Schwarz type estimates and their applications in Banach spaces

The primary objective of this paper is to develop methodologies for investigating Schwarz type lemmas and to present their applications in Banach spaces. First, we improve upon the main results obtained by Osserman [Proc. Am. Math. Soc. 128: 3513-3517, 2000] and Chen et al. [J. Anal. Math. 152: 181-216, 2024]. Based on these sharp estimates, we then derive several sharp boundary Schwarz type lemmas (also known as Hopf type lemmas) for holomorphic mappings in Banach spaces, as well as for solutions to certain classes of elliptic partial differential equations on the Euclidean unit ball in $\mathbb{C}^n$ or on the unit disk in $\mathbb{C}$. Furthermore, we prove some sharp Schwarz type lemmas for holomorphic mappings that send a prescribed point to another prescribed point. Finally, these lemmas are applied to establish a sharp Minda type Schwarz inequality in Banach spaces and to provide a sharp refined bound on subballs of the unit ball.

math.CV↗

arXiv2025

Remarks on "Schwarz-type lemma, Landau-type theorem, and Lipschitz-type space of solutions to inhomogeneous biharmonic equations"

Let $φ$, $ψ\in C(\mathbb{T})$, $g\in C(\overline{\mathbb{D}})$, where $\mathbb{D}$ and $\mathbb{T}$ denote the unit disk and the unit circle, respectively. Suppose that $f\in C^{4}(\mathbb{D})$ satisfies the following: (1) the inhomogeneous biharmonic equation $ Δ(Δf(z))=g(z)$ for $z\in\mathbb{D}$, (2) the Dirichlet boundary conditions $\partial_{\overline{z}}f(ζ)=φ(ζ)$ and $f(ζ)=ψ(ζ)$ for $ζ\in\mathbb{T}$. Recently, the authors in [J. Geom. Anal. 29: 2469-2491, 2019] showed that if $ω$ is a majorant with $\limsup_{t\rightarrow0^{+}}\left(ω(t)/t\right)<\infty$, $ψ=0$ and $φ_1 \in\mathscr{L}_ω(\mathbb{T})$, where $φ_1(e^{it})=φ(e^{it})e^{-it}$ for $t\in[0,2π]$, then $f\in\mathscr{L}_ω(\mathbb{D})$. The purpose of this paper is to improve and generalize this result. We not only prove that the condition "$\limsup_{t\rightarrow0^{+}}\left(ω(t)/t\right)<\infty$" is redundant, but also demonstrate that conditions "$ψ=0$" and "$φ_1\in\mathscr{L}_ω(\mathbb{T})$" can be replaced by weaker conditions.

math.CV↗

arXiv2024

Refined Bohr inequalities and a refined Bohr-Rogosinski inequality on complex Banach spaces

In this paper, we first establish refined versions of the Bohr inequalities for the class of holomorphic functions from the unit ball $B_X$ of a complex Banach space $X$ into $\mathbb{C}$. As applications, we will establish refined Bohr inequalities of functional type or of norm type for holomorphic mappings with lacunary series on the unit ball $B_X$ with values in higher dimensional spaces. Next, we obtain the Bohr-Rogosinski inequality for the class of holomorphic functions on $B_X.$ In addition, we establish an improved version of the Bohr inequality for holomorphic functions on $B_X$. All the results are proved to be sharp.

math.CV↗

arXiv2024

Bieberbach conjecture, Bohr radius, Bloch constant and Alexander's theorem in infinite dimensions

In this paper, we investigate holomorphic mappings $F$ on the unit ball $\mathbb{B}$ of a complex Banach space of the form $F(x)=f(x)x$, where $f$ is a holomorphic function on $\mathbb{B}$. First, we investigate criteria for univalence, starlikeness and quasi-convexity of type $B$ on $\mathbb{B}$. Next, we investigate a generalized Bieberbach conjecture, a covering theorem and a distortion theorem, the Fekete-Szegö inequality, lower bound for the Bloch constant, and Alexander's type theorem for such mappings.

math.CV↗

arXiv2023

Characterizations of composition operators on Bloch and Hardy type spaces

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 $\mathbb{D}$ to pluriharmonic Hardy spaces on the Euclidean unit ball $\mathbb{B}^n$. Furthermore, we develop some new methods to study the composition operators from harmonic Bloch type spaces on $\mathbb{D}$ to pluriharmonic Bloch type spaces on $\mathbb{D}$. 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.

math.CV↗

arXiv2023

Loewner PDE in infinite dimensions

In this paper, we prove the existence and uniqueness of the solution $f(z,t)$ of the Loewner PDE with normalization $Df(0,t)=e^{tA}$, where $A\in L(X,X)$ is such that $k_+(A)<2m(A)$, on the unit ball of a separable reflexive complex Banach space $X$. We also give improvements of the results obtained recently by Hamada and Kohr, but we omit their proofs for the sake of brevity. In particular, we obtain the biholomorphicity of the univalent Schwarz mappings $v(z,s,t)$ with normalization $Dv(0,s,t)=e^{-(t-s)A}$ for $t\geq s\geq 0$, where $m(A)>0$, which satisfy the semigroup property on the unit ball of a complex Banach space $X$. We further obtain the biholomorphicity of $A$-normalized univalent subordination chains under some normality condition on the unit ball of a reflexive complex Banach space $X$. We prove the existence of the biholomorphic solutions $f(z,t)$ of the Loewner PDE with normalization $Df(0,t)=e^{tA}$ on the unit ball of a separable reflexive complex Banach space $X$. The results obtained in this paper give some positive answers to the open problems and conjectures proposed by the authors in 2013.

math.CV↗

arXiv2023

A Hardy-Littlewood type Theorem and a Heinz type inequality

The main aim of this paper is to investigate the Hardy-Littlewood type Theorem and the Heinz type inequality on functions induced by a differential operator. We first prove a more general Hardy-Littlewood type theorem for the Dirichlet solution of a differential operator which depends on $α>0$ over the unit ball $\mathbb{B}^n$ of $\mathbb{R}^n$ with $n\geq 2$, related to the Lipschitz type space defined by a fast majorant. We find that the case $α>0$ is completely different from the case $α=0$. Then a more general Heinz type inequality for the Dirichlet solution of a differential operator will also be established in the case $α>n-2$.

math.FA↗

arXiv2023

Equivalent norms and Hardy-Littlewood type Theorems, and their applications

The main purpose of this paper is to develop some methods to investigate equivalent norms and Hardy-Littlewood type Theorems on Lipschitz type spaces of analytic functions and complex-valued harmonic functions. Initially, some characterizations of equivalent norms on Lipschitz type spaces of analytic functions and complex-valued harmonic functions will be given. In particular, we give an answer to an open problem posed by Dyakonov in (Math. Z.249(2005), 597--611). Furthermore, some Hardy-Littlewood type Theorems of complex-valued harmonic functions are established. The obtained results improve and extend the main results in (Acta Math.178(1997),143--167). Additionally, we apply the equivalent norms and Hardy-Littlewood type Theorems to study composition operators between Lipschitz type spaces.

math.CV↗

arXiv2022

On Riesz type inequalities, Hardy-Littlewood type theorems and smooth moduli

The purpose of this paper is to develop some methods to study 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 (\textit{Trans. Amer. Math. Soc.} {\bf 372} (2019)~ 4031--4051). Furthermore, some Hardy-Littlewood type theorems of holomorphic and pluriharmonic functions on John domains are established. Additionally, we also discuss the Hardy-Littlewood type theorems and smooth moduli of holomorphic, pluriharmonic and harmonic functions. Consequently, we improve and generalize the corresponding results in (\textit{Acta Math.} {\bf 178} (1997)~ 143--167) and (\textit{Adv. Math.} {\bf 187} (2004)~ 146--172).

math.FA↗

arXiv2022

Schwarz type lemmas and their applications in Banach spaces

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 $\C^n$ 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 \cite{CH20}.

math.CV↗

arXiv2022

Composition operators on Bloch and Hardy type spaces

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.

math.CV↗

arXiv2020

Some sharp Schwarz-Pick type estimates and their applications of harmonic and pluriharmonic functions

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 $\mathbb{R}^n$ 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 $\mathbb{C}^n$ or from the unit polydisc in $\mathbb{C}^n$ 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.

math.CV↗

arXiv2020

Support points for families of univalent mappings on bounded symmetric domains

In this paper we study some extremal problems for the family $S_g^0(\mathbb{B}_X)$ of normalized univalent mappings with $g$-parametric representation on the unit ball $\mathbb{B}_X$ of an $n$-dimensional JB$^*$-triple $X$ with $r\geq 2$, where $r$ is the rank of $X$ and $g$ is a convex (univalent) function on the unit disc $\mathbb{U}$, which satisfies some natural assumptions. We obtain sharp coefficient bounds for the family $S_g^0(\mathbb{B}_X)$, and examples of bounded support points for various subsets of $S_g^0(\mathbb{B}_X)$. Our results are generalizations to bounded symmetric domains of known recent results related to support points for families of univalent mappings on the Euclidean unit ball $\mathbb{B}^n$ and the unit polydisc $\mathbb{U}^n$ in $\mathbb{C}^n$. Certain questions will be also mentioned. Finally, we point out sharp coefficient bounds and bounded support points for the family $S_g^0(\mathbb{B}^n)$ and for special compact subsets of $S_g^0(\mathbb{B}^n)$, in the case $n\geq 2$.

math.CV↗

arXiv2018

Bloch-type spaces and extended Cesàro operators in the unit ball of a complex Banach space

Let $\mathbb{B}$ be the unit ball of a complex Banach space $X$. In this paper, we will generalize the Bloch-type spaces and the little Bloch-type spaces to the open unit ball $\mathbb{B}$ by using the radial derivative. Next, we define an extended Cesàro operator $T_φ$ with holomorphic symbol $φ$ and characterize those $φ$ for which $T_φ$ is bounded between the Bloch-type spaces and the little Bloch-type spaces. We also characterize those $φ$ for which $T_φ$ is compact between the Bloch-type spaces and the little Bloch-type spaces under some additional assumption on the symbol $φ$. When $\mathbb{B}$ is the open unit ball of a finite dimensional complex Banach space $X$, this additional assumption is automatically satisfied.

math.CV↗

arXiv2017

Approximation properties of univalent mappings on the unit ball in $\mathbb{C}^n$

Let $n\geq 2$. In this paper, we obtain approximation properties of various families of normalized univalent mappings $f$ on the Euclidean unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$ by automorphisms of $\mathbb{C}^n$ whose restrictions to $\mathbb{B}^n$ have the same geometric property of $f$. First, we obtain approximation properties of spirallike, convex and $g$-starlike mappings $f$ on $\mathbb{B}^n$ by automorphisms of $\mathbb{C}^n$ whose restrictions to $\mathbb{B}^n$ have the same geometric property of $f$, respectively. Next, for a nonresonant operator $A$ with $m(A)>0$, we obtain an approximation property ofmappings which have $A$-parametric representation by automorphisms of $\mathbb{C}^n$ whose restrictions to $\mathbb{B}^n$ have $A$-parametric representation. Certain questions will be also mentioned. Finally, we obtain an approximation property by automorphisms of $\mathbb{C}^n$ for a subset of $S_{I_n}^0(\mathbb{B}^n)$ consisting of mappings $f$ which satisfy the condition $\|Df(z)-I_n\|<1$, $z\in\mathbb{B}^n$. Related results will be also obtained.

math.CV↗

arXiv2015

Variation of Loewner chains, extreme and support points in the class $S^0$ in higher dimensions

We introduce a family of natural normalized Loewner chains in the unit ball, which we call "geräumig"---spacious---which allow to construct, by means of suitable variations, other normalized Loewner chains which coincide with the given ones from a certain time on. We apply our construction to the study of support points, extreme points and time-$\log M$-reachable functions in the class $S^0$ of mappings admitting parametric representation.

math.CV↗

arXiv2011

An abstract approach to Loewner chains

We present a new geometric construction of Loewner chains in one and several complex variables which holds on a complete hyperbolic complex manifold M and prove that there is essentially a one-to-one correspondence between evolution families of order d and Loewner chains of the same order. As a consequence we obtain a solution for any Loewner-Kufarev PDE, given by univalent mappings (f_t) from M to a complex manifold N. The problem of finding solutions given by univalent mappings with range in C^n is reduced to investigating whether the union of the images f_t(M) is biholomorphic to a domain in C^n. We apply such results to the study of univalent mappings from the unit ball B^n to C^n.

math.CV↗