Bloch Space Research Articles

The closure of Dirichlet spaces in the Bloch space

The closure of Dirichlet spaces in the Bloch space

Essential norm of generalized weighted composition operators from $H^\infty $ to the logarithmic Bloch space

In this paper, we give some estimates of the essential norm for generalized weighted composition operators from $H^\infty $ to the logarithmic Bloch space. Moreover, we give a new characterization for the boundedness, compactness and essential norm of the generalized weighted composition operator from $H^\infty $ to the logarithmic Bloch space.

Surjectivity of isometries between weighted spaces of holomorphic functions

We examine the surjectivity of isometries between weighted spaces of holomorphic functions. We show that for certain classical weights on the open unit disc all isometries of the weighted space of holomorphic functions, $${ {\mathcal {H}}}_{v_o}( \varDelta )$$ , are surjective. Criteria for surjectivity of isometries of $${ \mathcal H}_v(U)$$ in terms of a separation condition on points in the image of $${ {\mathcal {H}}}_{v_o}(U)$$ are also given for U a bounded open set in $${\mathbb {C}}$$ . Considering the weight $$v(z)= 1-|z|^2$$ and the isomorphism $$f\mapsto f'$$ we are able to show that all isometries of the little Bloch space are surjective.

The product-type operators from logarithmic Bloch spaces to Zygmund-type spaces

The boundedness and compactness of a product-type operator, recently introduced by S. Stevic, A. Sharma and R. Krishan, Tn?1,?2,?f(z) = ?1(z) f(n)(?(z)) + ?2(z) f(n+1)(?(z)), f ? H(D), from the logarithmic Bloch spaces to Zygmund-type spaces are characterized, where ?1, ?2 ? H(D),? is an analytic self-map of D and n a positive integer.

Equality of norms for a class of Bloch and symmetrically weighted Lipschitz spaces of vector valued functions and derivation inequalities for Pick functions

If X and Y are Banach spaces and f:BX→Y is Fréchet differentiable on the open unit ball BX of X, then for every operator monotone function φ:(−1,1)→R, which satisfies φ′′⩾0 on [a,b),(1)supa,b∈BX,a≠b⁡‖f(a)−f(b)‖φ′(‖a‖)‖a−b‖φ′(‖b‖)=supa∈BX⁡‖Df(a)‖φ′(‖a‖). This generalizes Holland–Walsh–Pavlović criterium for the membership in Bloch type spaces for functions defined in the unit ball of a Banach space and taking values in another Banach space. We also established relations of the induced Bloch and Lipschitz spaces with other spaces of vector valued functions.

SPECTRA OF LINEAR FRACTIONAL COMPOSITION OPERATORS ON THE GROWTH SPACE AND BLOCH SPACE OF THE UPPER HALF-PLANE

In this article, we provide a complete description of the spectra of linear fractional composition operators acting on the growth space and Bloch space over the upper half-plane. In addition, we also prove that the norm, essential norm, spectral radius and essential spectral radius of a composition operator acting on the growth space are all equal.

Weighted composition operators from H∞ to the Bloch space in the unit ball of Copf;n

ABSTRACTThe boundedness, compactness and essential norm of weighted composition operators from to the Bloch space in the unit ball of are investigated in this paper. In particular, some new characterizations for the boundedness, compactness and essential norm of weighted composition operators are given.

Differential-free Characterisation of Smooth Mappings with Controlled Growth

AbstractIn this paper we give some generalizations and improvements of the Pavlović result on the Holland–Walsh type characterization of the Bloch space of continuously differentiable (smooth) functions in the unit ball in Rm.

Bloch Space of a Bounded Symmetric Domain and Composition Operators

We generalize a number of finite dimensional results on Bloch functions to infinite dimensional bounded symmetric domains. In particular, we characterize the Bloch space as well as the little Bloch space of a Hilbert ball, and give one sufficient and several necessary conditions for a composition operator on a Bloch space to be an isometry. We also answer some open questions of Allen and Colonna concerning Bloch functions and composition operators.

Composition operators and closures of Dirichlet type spaces [formula omitted] in the logarithmic Bloch space

In this paper, we characterize the closures of Dirichlet type spaces Dα in the logarithmic Bloch space. Moreover, we study the boundedness and compactness of composition operators from the logarithmic Bloch space to closures of Dirichlet type spaces Dα in the logarithmic Bloch space.

Slowly growing solutions of ODEs revisited

Solutions of the differential equation $f''+Af=0$ are considered assuming that $A$ is analytic in the unit disc $\mathbb{D}$ and satisfies \begin{equation} \label{eq:dag} \sup_{z\in\mathbb{D}} \, |A(z)| (1-|z|^2)^2 \log\frac{e}{1-|z|} < \infty. \tag{$\star$} \end{equation} By recent results in the literature, such restriction has been associated to coefficient conditions which place all solutions in the Bloch space $\mathcal{B}$. In this paper it is shown that any coefficient condition implying \eqref{eq:dag} fails to detect certain cases when Bloch solutions do appear. The converse problem is also addressed: What can be said about the growth of the coefficient $A$ if all solutions of $f''+Af=0$ belong to $\mathcal{B}$? An overall revised look into slowly growing solutions is presented, emphasizing function spaces $\mathcal{B}$, $\rm{BMOA}$ and $\rm{VMOA}$.

Linear differential equations with slowly growing solutions

This research concerns linear differential equations in the unit disc of the complex plane. In the higher order case the separation of zeros (of maximal multiplicity) of solutions is considered, while in the second order case slowly growing solutions in H ∞ H^\infty , B M O A \mathrm {BMOA} , and the Bloch space are discussed. A counterpart of the Hardy–Stein–Spencer formula for higher derivatives is proved, and then applied to study solutions in the Hardy spaces.

Composition Operators Mapping Logarithmic Bloch Functions into Hardy Space

Characterizing the hyperbolic Hardy classes, several g-functions of hyperbolic type are introduced. Using this, necessary and sufficient conditions on the inducing self-maps are established for the boundedness of the composition operators from logarithmic Bloch spaces into Hardy spaces.

A system of conjugate functions on parabolic Bloch spaces

The parabolic Bloch space is the set of all solutions $u$ of the parabolic operator $L^{(\alpha)}$ with the finite Bloch norm $\| u \|_{\mathcal{B}_{\alpha} (\sigma)}$. In this paper, we introduce $L^{(\alpha)}$-conjugates of parabolic Bloch functions, and investigate several properties. As an application, we give an isomorphism theorem on parabolic Bloch spaces.

Möbius invariant Dirichlet type spaces

Let Dμ,p be a Dirichlet type space induced by a positive parameter p and a positive Borel measure μ on the open unit disk. Denote by M(Dμ,p) the Möbius invariant function space generated by Dμ,p. It is known that if the measure μ is finite, then M(Dμ,p) is equal to the well-known Möbius invariant space Qp. In this paper, we investigate Dμ,p and M(Dμ,p) spaces when the measures μ are not necessarily finite. We give the relation between M(Dμ,p) and the Bloch space. We characterize inner functions in M(Dμ,p) spaces. We also consider a Carleson measure problem for Dμ,p spaces.

Products of weighted composition and differentiation operators into weighted Zygmund and Bloch spaces

We characterize boundedness and compactness of products of differentiation operators and weighted composition operators between weighted Banach spaces of analytic functions and weighted Zygmund spaces or weighted Bloch spaces with general weights.

Distance Formulas on Weighted Banach Spaces of Analytic Functions

Let v be a radial weight function on the unit disc or on the complex plane. It is shown that for each analytic function $$f_0$$ in the Banach space $$H_v^\infty $$ of all analytic functions f such that v|f| is bounded, the distance of $$f_0$$ to the subspace $$H_v^0$$ of $$H_v^\infty $$ of all the functions g such that v|g| vanishes at infinity is attained at a function $$g_0 \in H_v^0$$ . Moreover a simple, direct proof of the formula of the distance of f to $$H_v^0$$ due to Perfekt is presented. As a consequence the corresponding results for weighted Bloch spaces are obtained.

Weighted Differentiation Composition Operators Between Normal Weight Zygmund Spaces and Bloch Spaces in the Unit Ball of $$\hbox {C}^{\mathrm{n}}$$ C n for $$\hbox {n}&gt;1$$ n &gt; 1

Let \(\mu \) be a normal functions on [0, 1), and H(B) be the space of all holomorphic functions on the unit ball B of \(\mathbf C^{n}\). Let \(\varphi \) be a nonconstant holomorphic self-map on B, and \(\psi \) be a holomorphic function on B. The weighted differentiation composition operator \(\psi D_{\varphi }\) is defined on the space H(B) by \(\psi D_{\varphi }(f)=\psi (Rf)\circ \varphi \), for all \(f\in H(B)\). In this paper, the authors characterize the boundedness and compactness of the weighted differentiation composition operator \(\psi D_{\varphi }\) from the normal weight Zygmund space \(Z_{\mu }(B)\) to the normal weight Bloch space \(\beta _{\mu }(B)\) for \(n>1\). As a consequence of the main results, the authors give the briefly sufficient and necessary conditions that the differentiation composition operator \( D_{\varphi }\) is compact from \(Z_{\mu }(B)\) to \(\beta _{\mu }(B)\) for \(\displaystyle {\mu (r)=(1-r)^{s}\log ^{t}\frac{e}{1-r}}\).

Superposition operators between logarithmic Bloch spaces

We characterize all entire functions $$\phi $$ that maps a logarithmic Bloch-type space $${\mathcal {B}}^\alpha _{\log ^\beta }$$ into another of the same kind by superposition. As consequences of our study, we obtain several results about the boundedness of superposition operators acting between $$\alpha $$ -Bloch spaces, Bloch–Orlicz spaces among others.

Generalized Bloch spaces, integral means of hyperbolic harmonic mappings in the unit ball

In this paper, we investigate the properties of hyperbolic harmonic mappings in the unit ball Bn in Rn(n≥2). Firstly, we establish necessary and sufficient conditions for a hyperbolic harmonic mapping to be in the Bloch space B(Bn) and the generalized Bloch space L∞,ωBα,a0(Bn), respectively. Secondly, we discuss the relationship between the integral means of hyperbolic harmonic mappings and that of their gradients. The obtained results are the generalizations of Hardy and Littlewood's related ones in the setting of hyperbolic harmonic mappings. Finally, we characterize the weak uniform boundedness property of hyperbolic harmonic mappings in terms of the quasihyperbolic metric.