Inclusion relations between harmonic Bergman-Besov and weighted Bloch spaces on the unit ball

Ömer Faruk Doğan; Adem Ersin Üreyen

Displaying similar documents to “Inclusion relations between harmonic Bergman-Besov and weighted Bloch spaces on the unit ball”

Some characterizations of harmonic Bloch and Besov spaces

Xi Fu, Bowen Lu (2016)

Czechoslovak Mathematical Journal

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The relationship between weighted Lipschitz functions and analytic Bloch spaces has attracted much attention. In this paper, we define harmonic $ω$ - $α$ -Bloch space and characterize it in terms of $ω ((1 - {| x |}^{2})^{β} {(1 - | y |}^{2})^{α - β}) | \frac{f (x) - f (y)}{x - y} |$ and $ω ((1 - {| x |}^{2})^{β} {(1 - | y |}^{2})^{α - β}) | \frac{f (x) - f (y)}{| x | y - x^{'}} |$ where $ω$ is a majorant. Similar results are extended to harmonic little $ω$ - $α$ -Bloch and Besov spaces. Our results are generalizations of the corresponding ones in G. Ren, U. Kähler (2005).

On the isomorphism classes of weighted spaces of harmonic and holomorphic functions

Wolfgang Lusky (2006)

Studia Mathematica

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Let Ω be either the complex plane or the open unit disc. We completely determine the isomorphism classes of $H v = f : Ω \to ℂ h o l o m o r p h i c : s u p_{z \in Ω} | f (z) | v (z) < \infty$ and investigate some isomorphism classes of $h v = f : Ω \to ℂ h a r m o n i c : s u p_{z \in Ω} | f (z) | v (z) < \infty$ where v is a given radial weight function. Our main results show that, without any further condition on v, there are only two possibilities for Hv, namely either $H v \sim l_{\infty}$ or $H v \sim H_{\infty}$ , and at least two possibilities for hv, again $h v \sim l_{\infty}$ and $h v \sim H_{\infty}$ . We also discuss many new examples of weights.

$L^{p}$ harmonic $1$ -form on submanifold with weighted Poincaré inequality

Xiaoli Chao, Yusha Lv (2018)

Czechoslovak Mathematical Journal

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We deal with complete submanifolds with weighted Poincaré inequality. By assuming the submanifold is $δ$ -stable or has sufficiently small total curvature, we establish two vanishing theorems for $L^{p}$ harmonic $1$ -forms, which are extensions of the results of Dung-Seo and Cavalcante-Mirandola-Vitório.

Landau's theorem for p-harmonic mappings in several variables

Sh. Chen, S. Ponnusamy, X. Wang (2012)

Annales Polonici Mathematici

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A 2p-times continuously differentiable complex-valued function f = u + iv in a domain D ⊆ ℂ is p-harmonic if f satisfies the p-harmonic equation $Δ^{p} f = 0$ , where p (≥ 1) is a positive integer and Δ represents the complex Laplacian operator. If Ω ⊂ ℂⁿ is a domain, then a function $f : Ω \to ℂ^{m}$ is said to be p-harmonic in Ω if each component function $f_{i}$ (i∈ 1,...,m) of $f = (f ₁, . . ., f_{m})$ is p-harmonic with respect to each variable separately. In this paper, we prove Landau and Bloch’s theorem for a class of p-harmonic mappings...

On the dimension of $p$ -harmonic measure in space

John L. Lewis, Kaj Nyström, Andrew Vogel (2013)

Journal of the European Mathematical Society

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Let $Ω \subset ℝ^{n}, n \geq 3$ , and let $p, 1 < p < \infty, p \neq 2$ , be given. In this paper we study the dimension of $p$ -harmonic measures that arise from non-negative solutions to the $p$ -Laplace equation, vanishing on a portion of $\partial Ω$ , in the setting of $δ$ -Reifenberg flat domains. We prove, for $p \geq n$ , that there exists $\tilde{δ} = \tilde{δ} (p, n) > 0$ small such that if $Ω$ is a $δ$ -Reifenberg flat domain with $δ < \tilde{δ}$ , then $p$ -harmonic measure is concentrated on a set of $σ$ -finite $H^{n - 1}$ -measure. We prove, for $p \geq n$ , that for sufficiently flat Wolff snowflakes the Hausdorff dimension of $p$ -harmonic...

Weighted sub-Bergman Hilbert spaces

Maria Nowak, Renata Rososzczuk (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We consider Hilbert spaces which are counterparts of the de Branges-Rovnyak spaces in the context of the weighted Bergman spaces $A_{α}^{2}$ , $- 1 < α < \infty$ . These spaces have already been studied in [8], [7], [5] and [1]. We extend some results from these papers.

A Weighted Eigenvalue Problems Driven by both $p (\cdot)$ -Harmonic and $p (\cdot)$ -Biharmonic Operators

Mohamed Laghzal, Abdelouahed El Khalil, Abdelfattah Touzani (2021)

Communications in Mathematics

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The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both $p (\cdot)$ -Harmonic and $p (\cdot)$ -biharmonic operators $\begin{matrix} Δ_{p (x)}^{2} u - Δ_{p (x)} u = λ w (x) {| u |}^{q (x) - 2} u in Ω, \\ u \in W^{2, p (\cdot)} (Ω) \cap W_{0}^{1, p (\cdot)} (Ω), \end{matrix}$ is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces $L^{p (\cdot)} (Ω)$ and $W^{m, p (\cdot)} (Ω)$ .

New characterizations for weighted composition operator from Zygmund type spaces to Bloch type spaces

Xin-Cui Guo, Ze-Hua Zhou (2015)

Czechoslovak Mathematical Journal

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Let $u$ be a holomorphic function and $ϕ$ a holomorphic self-map of the open unit disk $𝔻$ in the complex plane. We provide new characterizations for the boundedness of the weighted composition operators $u C_{ϕ}$ from Zygmund type spaces to Bloch type spaces in $𝔻$ in terms of $u$ , $ϕ$ , their derivatives, and $ϕ^{n}$ , the $n$ -th power of $ϕ$ . Moreover, we obtain some similar estimates for the essential norms of the operators $u C_{ϕ}$ , from which sufficient and necessary conditions of compactness of $u C_{ϕ}$ follows immediately. ...

Strict plurisubharmonicity of Bergman kernels on generalized annuli

Yanyan Wang (2014)

Annales Polonici Mathematici

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Let $A_{ζ} = Ω - \bar{ρ (ζ) \cdot Ω}$ be a family of generalized annuli over a domain U. We show that the logarithm of the Bergman kernel $K_{ζ} (z)$ of $A_{ζ}$ is plurisubharmonic provided ρ ∈ PSH(U). It is remarkable that $A_{ζ}$ is non-pseudoconvex when the dimension of $A_{ζ}$ is larger than one. For standard annuli in ℂ, we obtain an interesting formula for $\partial ² l o g K_{ζ} / \partial ζ \partial ζ ̅$ , as well as its boundary behavior.

Toeplitz operators on Bergman spaces and Hardy multipliers

Wolfgang Lusky, Jari Taskinen (2011)

Studia Mathematica

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We study Toeplitz operators $T_{a}$ with radial symbols in weighted Bergman spaces $A_{μ}^{p}$ , 1 < p < ∞, on the disc. Using a decomposition of $A_{μ}^{p}$ into finite-dimensional subspaces the operator $T_{a}$ can be considered as a coefficient multiplier. This leads to new results on boundedness of $T_{a}$ and also shows a connection with Hardy space multipliers. Using another method we also prove a necessary and sufficient condition for the boundedness of $T_{a}$ for a satisfying an assumption on the positivity of certain...

Complex symmetry of Toeplitz operators on the weighted Bergman spaces

Xiao-He Hu (2022)

Czechoslovak Mathematical Journal

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We give a concrete description of complex symmetric monomial Toeplitz operators $T_{z^{p} {\bar{z}}^{q}}$ on the weighted Bergman space $A^{2} (Ω)$ , where $Ω$ denotes the unit ball or the unit polydisk. We provide a necessary condition for $T_{z^{p} {\bar{z}}^{q}}$ to be complex symmetric. When $p, q \in ℕ^{2}$ , we prove that $T_{z^{p} {\bar{z}}^{q}}$ is complex symmetric on $A^{2} (Ω)$ if and only if $p_{1} = q_{2}$ and $p_{2} = q_{1}$ . Moreover, we completely characterize when monomial Toeplitz operators $T_{z^{p} {\bar{z}}^{q}}$ on $A^{2} (𝔻_{n})$ are $J_{U}$ -symmetric with the $n \times n$ symmetric unitary matrix $U$ .

Polyanalytic Besov spaces and approximation by dilatations

Ali Abkar (2024)

Czechoslovak Mathematical Journal

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Using partial derivatives $\partial f / \partial z$ and $\partial f / \partial \bar{z}$ , we introduce Besov spaces of polyanalytic functions in the open unit disk, as well as in the upper half-plane. We then prove that the dilatations of functions in certain weighted polyanalytic Besov spaces converge to the same functions in norm. When restricted to the open unit disk, we prove that each polyanalytic function of degree $q$ can be approximated in norm by polyanalytic polynomials of degree at most $q$ .

Espace de Dixmier des opérateurs de Hankel sur les espaces de Bergman à poids

Romaric Tytgat (2015)

Czechoslovak Mathematical Journal

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Nous donnons des résultats théoriques sur l’idéal de Macaev et la trace de Dixmier. Ensuite, nous caractérisons les symboles antiholomorphes $\bar{f}$ tels que l’opérateur de Hankel $H_{\bar{f}}$ sur l’espace de Bergman à poids soit dans l’idéal de Macaev et nous donnons la trace de Dixmier. Pour cela, nous regardons le comportement des normes de Schatten $𝒮^{p}$ quand $p$ tend vers $1$ et nous nous appuyons sur le résultat de Engliš et Rochberg sur l’espace de Bergman. Nous parlons aussi des puissances de tels opérateurs....