We generalize a well-known sufficient condition for interpolating sequences for the Hilbert Bergman spaces to other Bergman spaces with normal weights (as defined by Shields and Williams) and obtain new results regarding the membership of the derivative of a Blashke product or a general inner function in such spaces. We also apply duality techniques to obtain further results of this type and obtain new results about interpolating Blaschke products.
We provide a characterization of the commutant of analytic Toeplitz operators $T_B$ induced by finite Blachke products $B$ acting on weighted Bergman spaces which, as a particular instance, yields the case $B(z)=z^n$ on the Bergman space solved recently by by Abkar, Cao and Zhu. Moreover, it extends previous results by Cowen and Wahl in this context and applies to other Banach spaces of analytic functions such as Hardy spaces $H^p$ for $1<p<\infty$. Finally, we apply this approach to study the reducing subspaces of $T_{B}$ in weighted Bergman spaces.
In this paper, we investigate the boundedness, compactness, essential norm and the Schatten class of weighted composition operators $uC_φ$ on Bergman type spaces $A_ω^p $ with double weight $ω$. Let $X=\{u\in H(D): uC_φ:A_ω^p\to A_ω^p \mbox{ is bounded}\}.$ For some regular weights $ω$, we obtain that $X=H^\infty$ if and only if $φ$ is a finite Blaschke product.
The relationship between the distribution of zeros of an infinite Blaschke product $B$ and the inclusion in weighted Bergman spaces $A_α^p$ of the derivative of $B$ or the derivative of functions in its model space $H^2 \ominus BH^2$ is investigated.
A finite Blaschke product, restricted to the unit circle, is a smooth covering map. The maximum and minimum values of the derivative of this map reflect the geometry of the Blaschke product. We identify two classes of extremal Blaschke products: those that maximize the difference between the maximum and minimum of the derivative, and those that minimize it. Both classes turn out to have the same algebraic structure, being the quotient of two hypergeometric polynomials.
In this note we study some basic properties of general fractional derivatives induced by weighted Bergman kernels. As an application we demonstrate a method for generating pre-images of analytic functions under weighted Bergman projections. This approach is useful for proving the surjectivity of weighted Bergman projections in cases when the target space is not a subspace of the domain space (such situations arise often when dealing with Bloch and Besov spaces). We also discuss a fractional Littlewood-Paley formula.
An equivalent norm in the weighted Bergman space $A^p_ω$, induced by an $ω$ in a certain large class of non-radial weights, is established in terms of higher order derivatives. Other Littlewood-Paley inequalities are also considered. On the way to the proofs, we characterize the $q$-Carleson measures for the weighted Bergman space $A^p_ω$ and the boundedness of a Hörmander-type maximal function. Results obtained are further applied to describe the resolvent set of the integral operators $T_g(f)(z)=\int_0^z g'(ζ)f(ζ)\,dζ$ acting on $A^p_ω$.
We establish a sharp upper bound for the absolute value of the derivative of the finite Blaschke product, provided that the critical values of this product lie in a given disk.
A finite Blaschke product is a product of finitely many automorphisms of the unit disk. This brief survey covers some of the main topics in the area, including characterizations of finite Blaschke products, approximation theorems, derivatives and residues of finite Blaschke products, geometric localization of zeros, and selected other topics.
We bound integral means of the Bergman projection of a function in terms of integral means of the original function. As an application of these results, we bound certain weighted Bergman space norms of derivatives of Bergman projections in terms of weighted $L^p$ norms of certain derivatives of the original function in the $θ$ direction. These results easily imply the well known result that the Bergman projection is bounded from the Sobolev space $W^{k,p}$ into itself for $1 < p < \infty$. We also apply our results to derive certain regularity results involving extremal problems in Bergman spa…
It is well known that a function is in a Bergman space of the unit ball if and only if it satisfies some Hardy-type inequalities. We extend this fact to Bergman-Orlicz spaces. As applications, we obtain Gustavsson-Peetre interpolation of two Bergman-Orlicz spaces and we completely characterize symbols of bounded or compact Cesàro-type operators on Bergman-Orlicz spaces, extending known results for classical weighted Bergman spaces.
Recently, several papers have considered a nonlinear analogue of Fourier series in signal analysis, referred to as either nonlinear phase unwinding or adaptive Fourier decomposition. In these processes, a signal is represented as the real component of a complex function $F: \partial\mathbb{D}\to \mathbb{C}$, and by performing an iterative method to obtain a sequence of Blaschke decompositions, the signal can be approximated using only a few terms. To better understand the convergence of these methods, the study of Blaschke decompositions on weighted Hardy spaces was studied by Coifman and Stei…
Using the approach based on sesquilinear forms, we introduce Toeplitz operator in the analytic Bergman space on the upper half-plane with strongly singular symbols, derivatives of measures. Conditions for boundedness and compactness of such operators are found. A procedure of reduction of Toeplitz operators in Bergman spaces of polyanalytic functions to operators with singular symbols in the analytic Bergman space by means of the creation-annihilation structure is elaborated, which leads to the description of the properties of the former operators
This paper is devoted to the study of reducing subspaces for multiplication operator $M_ϕ$ on the Dirichlet space with symbol of finite Blaschke product. The reducing subspaces of $M_ϕ$ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surface to study the reducing subspaces of $M_ϕ$ on the Bergman space, and we discover a new way to study the Riemann surface for $ϕ^{-1}\circϕ$. By this means, we determine the reducing subspaces of $M_ϕ$ on the Dirichlet space when the order of $ϕ$ is $5$; $6$; $7$ and answer some questions of Douglas-Putin…
In this paper, we study the complex symmetry of weighted composition-differentiation operator $D_{n, ψ, ϕ}$ on weighted Bergman spaces $\mathcal{A}^2_α$ with respect to the conjugation $C_{μ, η}$ for $μ, η\in \{z\in \mathbb{C}:|z|=1\}$. We obtain explicit conditions for which the operator $D_{n, ψ, ϕ}$ is Hermitian and normal. We also characterize the complex symmetric weighted composition-differentiation operator for derivative Hardy spaces.
Let $\mathscr J$ be the set of inner functions whose derivatives lie in Nevanlinna class. In this note, we show that the natural map $F \to \text{Inn}(F'): \mathscr J/\text{Aut}(\mathbb{D}) \to \text{Inn}/S^1$ is is injective but not surjective. More precisely, we show that that the image consists of all inner functions of the form $BS_μ$ where $B$ is a Blaschke product and $S_μ$ is the singular factor associated to a measure $μ$ whose support is contained in a countable union of Beurling-Carleson sets. Our proof is based on extending the work of D. Kraus and O. Roth on maximal Blaschke produc…
A one-component inner function $Θ$ is an inner function whose level set $$Ω_Θ(\varepsilon)=\{z\in \mathbb{D}:|Θ(z)|<\varepsilon\}$$ is connected for some $\varepsilon\in (0,1)$. We give a sufficient condition for a Blaschke product with zeros in a Stolz domain to be a one-component inner function. Moreover, a sufficient condition is obtained in the case of atomic singular inner functions. We study also derivatives of one-component inner functions in the Hardy and Bergman spaces. For instance, it is shown that, for $0<p<\infty$, the derivative of a one-component inner function $Θ$ is a member o…
We show that in a scale of weighted Dirichlet spaces $D_α$, including the Bergman space, given any finite Blaschke product $B$ there exists an equivalent norm in $D_α$ such that $B$ satisfies the wandering subspace property with respect to such norm. This extends, in some sense, previous results by Carswell, Duren and Stessin. As a particular instance, when $B(z)=z^k$ and $|α| \leq \frac{\log (2)}{\log(k+1)}$, the chosen norm is the usual one in $D_α$.
This paper is devoted to the study of the weighted Bergman space $A_ω^p $ in the unit ball $\mathbb{B}$ of $\mathbb{C}^n$ with doubling weight $ω$ satisfying $$\int_r^1ω(t)dt <C \int_{\frac{1+r}{2}}^1ω(t)dt ,\,\, 0\leq r<1.$$ The $q-$Carleson measures for $A_ω^p$ are characterized in terms of a neat geometric condition involving Carleson block. Some equivalent characterizations for $A_ω^p$ are obtained by using the radial derivative and admissible approach regions. The boundedness and compactness of Volterra integral operator $T_g:A_ω^p\to A_ω^q$ are also investigated in this paper with $0<p\l…
We consider Bergman spaces and variations of them in one or several complex variables. For some domains we show that in these spaces the generic function is totally unbounded and hence non - extendable. We also show that the generic function is not - even locally - in Bergman spaces of higher order. Finally, in certain domains we consider the space of holomorphic functions whose derivatives up to some order extend continuously to the closure of the domain endowed with its natural topology. Generically, every function in this space is proven to be non - extendable, although bounded.
We obtain estimates for integrals of derivatives of rational functions in multiply connected domains in the plane. A sharp order of the growth is found for the integral of the modulus of the derivative of a finite Blaschke product in the unit disk. We also extend the results of E.P. Dolzhenko about the integrals of the derivatives of rational functions to a wider class of domains, namely, to domains bounded by rectifiable curves without zero interior angles, and show the sharpness of the obtained results.
In this paper, we show that the Bergman functions on the Siegel upper half-space enjoy the following uniqueness property: if $f\in A_t^p(\calU)$ and $\bfL^α f\equiv 0$ for some nonnegative multi-index $α$, then $f\equiv 0$, where $\bfL^α:=(\bfL_1)^{α_1} \cdots (\bfL_n)^{α_n}$ with $\bfL_j = \frac{\partial }{\partial z_j} + 2i \bar{z}_j \frac{\partial }{\partial z_n}$ for $j=1,\ldots, n-1$ and $\bfL_n = \frac{\partial }{\partial z_n}$. As a consequence, we obtain a new integral representation for the Bergman functions on the Siegel upper half-space. In the end, as an application, we derive a re…
