Toeplitz operators in weighted Bergman spaces of polygonal domains

UDC 517.5 We study the problem of growth of the $m$th derivatives of an arbitrary algebraic polynomial in weighted Bergman spaces in unbounded domains of the complex plane, without using the recurrence relation. We consider $k$-quasidisks and quasidisks with some additional functional conditions. These conditions enable us to combine several known classes of curves into a single class and obtain estimates for the growth of $m$th derivatives in this common class. By using similar estimates obtained for bounded quasidisks, we also provide estimates valid in the whole complex plane.

Abstract A variety of norm inequalities related to Bergman and Dirichlet spaces induced by radial weights is established. Some of the results obtained can be considered as generalizations of certain known special cases while most of the estimates discovered are completely new. In particular, a Littlewood-Paley estimate recently proved by Peláez and the second author (Peláez and Rättyä Adv. Math. , 391 , 70, 2021) is improved in part. The second objective of the paper is to apply the obtained norm inequalities to relate the growth of the maximum modulus of a conformal map f , measured in terms of a weighted integrability condition, to a geometric quantity involving the area of image under f of a disc centered at the origin. Our findings in this direction yield new geometric characterizations of conformal maps in certain weighted Dirichlet and Besov spaces.

A new characterization for the essential norm of weighted composition operators and their differences on weighted Bergman spaces

Bicomplex Polyholomorphic Bergman Spaces Associated with a Bicomplex Magnetic Laplacian on the Discus

Abstract Karapetrović conjectured that the norm of the Hilbert matrix operator on the Bergman space $A^p_\alpha $ is equal to $\pi /\sin ((2+\alpha )\pi /p)$ when $-1<\alpha <p-2$ . In this article, we provide a proof of this conjecture for $0\leq \alpha \leq \frac {6p^3-29p^2+17p-2+2p\sqrt {6p^2-11p+4}}{(3p-1)^2}$ , and this range of $\alpha $ improves the best known result when $\alpha>\frac {1}{47}$ and $\alpha \not =1$ .

Abstract We study the interchange of essential norm and integration of certain families of weighted composition operators acting on the standard weighted Bergman spaces $$A^p_\alpha $$ A α p , where $$p>1$$ p > 1 and $$\alpha \ge 0$$ α ≥ 0 . To be more precise, we give a sufficient condition for $$ \left\| \int u_tC_{\phi _t}\, dt\right\| _e = \int \left\| u_tC_{\phi _t}\right\| _e \, dt $$ ∫ u t C ϕ t d t e = ∫ u t C ϕ t e d t to hold in terms of geometric properties of $$u_t$$ u t and $$\phi _t$$ ϕ t . We also provide some necessary conditions for the equality to hold and calculate the essential norm of some integral operators such as some Volterra operators.

Abstract In this short paper we will discuss recent advances on the problem of characterizing the boundedness of the composition operator acting on the Bergman spaces A β 2 ( D 2 ) ${A}_{\beta }^{2}\left({\mathbb{D}}^{2}\right)$ whenever the self map Φ of the bidisc is induced by Rational Inner Functions. The problem stated here is submitted as part of the Problem List of the Young Researchers Workshop in Complex Analysis and Operator Theory “The Bench Math Session 2025”, organized in Jagiellonian University of Kraków at 10th–11th February 2025.

In this paper, we compute the norm and spectrum of the Hilbert matrix operator on Hardy and weighted Bergman spaces of the upper half-plane.

Abstract We define the harmonic Bergman space on locally finite trees with respect to a suitable probabilistic Laplacian and a class of weighted flow measures. We characterise the corresponding Bergman projection and prove that it is bounded on $$L^p$$ for every $$p>1$$ , and of weak type (1, 1). We also prove necessary and sufficient conditions for the $$L^p$$ -boundedness of the extension of a class of Toeplitz-type operators.

In this paper, we study the behavior of the m-th(m≥0) derivatives of general algebraic polynomials in weighted Bergman spaces defined in regions of the complex plane G bounded by piecewise smooth curves L=∂G with λπ(0<λ≤2) exterior angles relative to G. Upper bounds are found for the growth of the m-th derivatives of the polynomials not only inside the unbounded region but also on the closures of this region with both exterior non-zero angles λπ(0<λ<2) and interior zero angles (i.e., exterior angles 2π). The influence of the boundary angles λπ(0<λ≤2) of the region G and the “growth rate” of the weight function on the behavior of the moduli of polynomials and their derivatives in regions of the complex plane that are “symmetric” with respect to L (bounded and unbounded) is found.

Abstract In this note, the author recalls the Calderon–Zygmund theory on the unit ball and derives the weak (1,1) boundedness of the projection for $\mathcal {H}$ -harmonic Bergman space.

We investigate the behavior of continuous frames in the weighted Bergman space Aα2 over the unit disc under the action of weighted composition operators. Motivated by developments in the discrete frame setting, we provide a comprehensive characterization of those weighted composition operators that preserve continuous frames, including tight and Parseval frames. Additionally, we examine the structure of dual frames in this context and establish necessary and sufficient conditions under which dual frame pairs are preserved by such operators. Explicit constructions of dual pairs induced by weighted composition operators are also presented. The study concludes with an analysis of the scalability of continuous frames and explores its invariance under the action of weighted composition operators.

A major open problem in the theory of Toeplitz operators on the analytic Bergman space over the unit disk is the characterization of the commutant of a given Toeplitz operator, that is, the set of all bounded Toeplitz operators that commute with it. In this paper, we provide a complete description of bounded Toeplitz operators T f , where the symbol f has a truncated polar decomposition, that commute with a Toeplitz operator, whose symbol is the sum of a quasihomogeneous function and a bounded analytic function.

Approximation in holomorphic Bergman spaces

In this study, we examine the problem of maximal simultaneous approximation in Bergman spaces defined over unbounded continuum of the complex plane, using classical Faber series.We derive upper bounds for the approximation error that explicitly depend on the best polynomial approximation numbers, as well as on structural parameters of the canonical domain under consideration.These results provide quantitative insights into how the geometry of the domain influences the convergence behavior of the Faber series.

Abstract A new family of Volterra‐type operators based on bona fide fractional calculus is introduced in [12] by constructing analytic paraproducts acting on and their boundedness between Hardy spaces is characterized for certain parameter ranges there. This paper is a natural companion to [12] in the sense that it characterizes those ’s such that is bounded from weighted Bergman spaces to Hardy spaces for the range The case extends earlier results of Wu [32] and Miihkinen et al. [22]. Besides standard techniques in this area, the proof relies on certain recent results on fractional integration operators obtained in [14, 35].

Dual H-Toeplitz Operators on the Harmonic Bergman Space

Abstract A classical result of Hardy and Littlewood says that if $$f=u+iv$$ f = u + i v is analytic in the unit disk $${\mathbb {D}}$$ D and u is in the harmonic Bergman space $$a^p$$ a p ( $$0<p<\infty $$ 0 < p < ∞ ), then v is also in $$a^p$$ a p . This complements a celebrated result of M. Riesz on Hardy spaces, which only holds for $$1<p<\infty $$ 1 < p < ∞ . These results do not extend directly to complex-valued harmonic functions. We prove that the Hardy-Littlewood theorem holds for a harmonic function $$f=u+iv$$ f = u + i v if we place the assumption that f is quasiregular in $${\mathbb {D}}$$ D . This makes further progress on the recent Riesz type theorems for harmonic quasiregular mappings by several authors. Then we consider univalent harmonic mappings in $${\mathbb {D}}$$ D and study their membership in Bergman spaces. In particular, we produce a non-trivial range of $$p>0$$ p > 0 such that every univalent harmonic function f (and the partial derivatives $$f_\theta ,\, rf_r$$ f θ , r f r ) is of class $$a^p$$ a p . This result extends nicely to harmonic quasiconformal mappings in $${\mathbb {D}}$$ D .

We examine Toeplitz operators on weighted Bergman spaces A V p A_\mathcal {V}^p , 1 > p > ∞ 1 > p > \infty , on the unit disk of C \mathbb {C} with symbols satisfying the simple geometric condition that the values are contained in an angle with vertex in the origin and magnitude less than π \pi . The condition is used to relax the conventional positivity assumption of the symbol, yet it is possible to give characterizations of the boundedness and compactness of such Toeplitz operators. The radial weight V \mathcal {V} defining the space A V p A_\mathcal {V}^p may be doubling or exponentially decreasing, but the geometric condition depends only on the symbol and not on V \mathcal {V} . It is known that there are significant differences between doubling and exponential weights for example as regards to the boundedness of Bergman projections. Nevertheless, we give a unified approach which includes both weight classes.