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In this paper, we give a universal description of the boundedness and compactness of Toeplitz operator T μ ω between Bergman spaces A η p and A ν q when μ is a positive Borel measure, 1 < p,q < ∞ and ω, η, ν are regular weights. By using Khinchin's inequality and Kahane's inequality, we get a new characterization of the Carleson measure for Bergman spaces induced by regular weights. [ABSTRACT FROM AUTHOR]

We completely characterize the boundedness of area operators from the Bergman spaces A α p (B n) to the Lebesgue spaces L q (S n) for all 0 < p,q < ∞. For the case n = 1, some partial results were previously obtained by Wu in [Wu, Z.: Volterra operator, area integral and Carleson measures, Sci. China Math., 54, 2487–2500 (2011)]. Especially, in the case q < p and q < s, we obtain some characterizations for the area operators to be bounded. We solve the cases left open there and extend the results to n-complex dimension. [ABSTRACT FROM AUTHOR]

Using Khinchin's inequality, Geršgorin's theorem, and the atomic decomposition of Bergman spaces, we estimate the norm and essential norm of Stević–Sharma‐type operators between weighted Bergman spaces Aωp$$ {A}_{\omega}&#x0005E;p $$ and Aμq$$ {A}_{\mu}&#x0005E;q $$ and the sum of weighted differentiation composition operators with different symbols from the weighted Bergman spaces Aωp$$ {A}_{\omega}&#x0005E;p $$ to H∞$$ {H}&#x0005E;{\infty } $$. The estimates of those between Bergman spaces remove all the restrictions of a result of Stevic, Sharma, and Bhat. As a by‐product, we also get an interpolation theorem for Bergman spaces induced by doubling weights. [ABSTRACT FROM AUTHOR]

Let μ be a self-similar measure with compact support K. The Hausdorff dimension of K is α. The Cauchy transform of μ is denoted by F (z) . For 0 < β < 1 , we define the function F [ β ] , which compares with the fractional derivative of F of order β. Let Φ (z) = F (1 / z) , | z | < 1 . In this paper, we prove that Φ [ β ] belongs to A p for 0 < p < 1 / (β + 1) , and (Φ ′) [ β ] belongs to A p for 1 ≤ p < 1 / β ≤ 1 / (2 − α) , where A p is the Bergman space. At the same time, we give a value distribution property of F, which is similar to the big Picard theorem. [ABSTRACT FROM AUTHOR]

The Friedrichs operator of a domain (in \mathbb {C}^n) is closely related to its Bergman projection and encodes crucial information (geometric, quadrature, potential theoretic etc.) about the domain. We show that the Friedrichs operator of a domain has rank one if the domain can be covered by a circular domain via a proper holomorphic map of finite multiplicity whose Jacobian is a homogeneous polynomial. As an application, we show that the Friedrichs operator is of rank one on the tetrablock, pentablock, and the symmetrized polydisc – domains of significance in the study of \mu-synthesis in control theory. [ABSTRACT FROM AUTHOR]

Let D be the unit disc in the complex plane. Given a positive finite Borel measure μ on the radius [0, 1), we denote the n-th moment of μ as μ n , that is, μ n = ∫ [ 0 , 1) t n d μ (t). The Cesàro-like operator C μ , s is defined on H (D) as follows: If f (z) = ∑ n = 0 ∞ a n z n ∈ H (D) then C μ , s (f) is defined by C μ , s (f) (z) = ∑ n = 0 ∞ μ n ∑ k = 0 n Γ (n - k + s) Γ (s) (n - k) ! a k z n , z ∈ D. In this paper, our focus is on the action of the C e s a ` r o -type operator C μ , s on spaces of analytic functions in D . We characterize the boundedness (compactness) of the C e s a ` r o -like operator C μ , s , acting between the Bloch space B and the Bergman space A p . [ABSTRACT FROM AUTHOR]

We obtain local estimates, also called propagation of smallness or Remez-type inequalities, for analytic functions in several variables. Using Carleman estimates, we obtain a three sphere-type inequality, where the outer two spheres can be any sets satisfying a boundary separation property, and the inner sphere can be any set of positive Lebesgue measure. We apply this local result to characterize the dominating sets for Bergman spaces on strongly pseudoconvex domains in terms of a density condition or a testing condition on the reproducing kernels. Our methods also yield a sufficient condition for arbitrary domains and lower-dimensional sets. [ABSTRACT FROM AUTHOR]

We study sufficient conditions on weight functions under which norm approximations by analytic polynomials are possible. The weights we study include radial, non-radial, and angular weights. [ABSTRACT FROM AUTHOR]

Let \omega and \nu be radial weights on the unit disc of the complex plane such that \omega admits the doubling property \sup _{0\le r<1}\frac {\int _r^1 \omega (s)\,ds}{\int _{\frac {1+r}{2}}^1 \omega (s)\,ds}<\infty. Consider the one weight inequality \begin{equation} \|P_\omega (f)\|_{L^p_\nu }\le C\|f\|_{L^p_\nu },\quad 1

The Berezin symbol A of an operator A on the reproducing kernel Hilbert space H(Ω) over some set Ω with the reproducing kernel kλ is defined by ... We study the existence of invariant subspaces for Bergman space operators in terms of Berezin symbols. [ABSTRACT FROM AUTHOR]

We define essentially positive operators on Hilbert space as a class of self-adjoint operators whose essential spectra is contained in the non-negative real numbers and describe their basic properties. Using Toeplitz operators and the Berezin transform, we further illustrate the notion of essential positivity in the Hardy space and the Bergman space. [ABSTRACT FROM AUTHOR]

We first study the Volterra operator V acting on spaces of Dirichlet series. We prove that V is bounded on the Hardy space ℋ 0 p for any 0 < p ≤ ∞, and is compact on ℋ 0 p for 1

We study extremal problems related to the best joint polynomial approximation of functions analytic in the unit disk and belonging to the Bergman space . The problem of joint approximation of periodic functions and their derivatives by trigonometric polynomials was considered by Garkavy [1] in 1960. Then, in the same year, Timan [2] considered this problem for classes of entire functions defined on the entire axis. The problem of joint approximation of functions and their derivatives is considered in more detail in Malozemov's monograph [3], where some classical theorems of the theory of approximation of functions are presented and generalized. In the present paper, a number of exact theorems are obtained and sharp upper bounds for the best joint approximations of a function and its successive derivatives by polynomials and their respective derivatives on some classes of complex functions belonging to the Bergman space are calculated. [ABSTRACT FROM AUTHOR]

In this paper, we characterize bounded, compact, and order bounded sums of weighted composition-differentiation operators from Bergman-type spaces to weighted Banach spaces of analytic functions. [ABSTRACT FROM AUTHOR]

We provide characterizations of boundedness and compactness for composition operators between Bergman spaces of smoothly bounded convex domains in C 2 admitting the so-called geometric F-type. Such geometric type contains all domains of finite type and many cases of infinite type in the sense of Range. [ABSTRACT FROM AUTHOR]

We study a weak greedy type algorithm called the weak pre-orthogonal adaptive Fourier decomposition (WPOAFD) for the Bergman space A 2 (Ω) on a bounded pseudoconvex domain Ω with smooth boundary. We show that any function f ∈ A 2 (Ω) can be approximated by linear combinations of kernel functions in weak greedy sense. [ABSTRACT FROM AUTHOR]

Suppose α > − 1 and 1 ≤ p ≤ ∞ . Let f = P α [ F ] be an α-harmonic mapping on D with the boundary F being absolute continuous and F ˙ ∈ L p ( 0 , 2 π ) , where F ˙ ( e i θ ) := d d θ F ( e i θ ) . In this paper, we investigate the membership of f z and f z ¯ in the space H G p ( D ) , the generalized Hardy space. We prove, if α > 0 , then both f z and f z ¯ are in H G p ( D ) . If α < 0 , then f z and f z ¯ ∈ H G p ( D ) if and only if f is analytic. Finally, we investigate a Schwarz Lemma for α-harmonic functions. [ABSTRACT FROM AUTHOR]

Let Π denote the upper half-plane. In this article, we prove that every vertical operator on the Bergman space A²(Π) over the upper half-plane can be uniquely represented as an integral operator of the form (Sφ) (z) = ∫Π f (w)φ(z -- w)dµ(w), ∀ f ∈ A²(Π), z ∈ Π, where φ is an analytic function on Π given by Π (z) = ∫ℝ+ ξσ(ξ)eizξ dξ, ∀ z ∈Π for some σ ∈ L∞(ℝ+). Here dµ(w) is the Lebesgue measure on Π. Later on, with the help of above integral representation, we obtain various operator theoretic properties of the vertical operators. Also, we give integral representation of the form Sφ for all the operators in the C*-algebra generated by Toeplitz operators Ta with vertical symbols a ∈ L∞(Π). [ABSTRACT FROM AUTHOR]

We study the compact intertwining relations for the multiplication operators between Hardy spaces and Bergman spaces in the unit ball of C n. [ABSTRACT FROM AUTHOR]

We study weighted composition-differentiation operators on the Hardy and Bergman spaces in the unit disk. We first find necessary and sufficient conditions for weighted composition-differentiation operators to be Hilbert–Schmidt. We then characterize convergence of sequences of composition-differentiation operators acting on the Hardy space both in weak operator topology, and in strong operator topology. [ABSTRACT FROM AUTHOR]

We discuss reachable states for the Hermite heat equation on a segment with boundary L²-control. The Hermite heat equation corresponds to the heat equation to which the quadratic potential x² is added. We will discuss two situations: when one endpoint of the segment is the origin and when the segment is symmetric with respect to the origin. One of the main results is that reachable states extend to functions in a Bergman space on a square one diagonal of which is the segment under consideration, and that functions holomorphic in a neighborhood of this square are reachable. [ABSTRACT FROM AUTHOR]

We consider weighted Bergman spaces A μ 1 on the unit disc as well as the corresponding spaces of entire functions, defined using non-atomic Borel measures with radial symmetry. By extending the techniques from the case of reflexive Bergman spaces, we characterize the solid core of A μ 1 . Also, as a consequence of a characterization of solid A μ 1 -spaces, we show that, in the case of entire functions, there indeed exist solid A μ 1 -spaces. The second part of the article is restricted to the case of the unit disc and it contains a characterization of the solid hull of A μ 1 , when μ equals the weighted Lebesgue measure with the weight v. The results are based on the duality relation of the weighted A 1 - and H ∞ -spaces, the validity of which requires the assumption that - log v belongs to the class W 0 , studied in a number of publications; moreover, v has to satisfy the condition (b), introduced by the authors. The exponentially decreasing weight v (z) = exp (- 1 / (1 - | z |) provides an example satisfying both assumptions. [ABSTRACT FROM AUTHOR]

In this paper we develop a Sz.-Nagy-Foiaş type theory of characteristic functions and functional models on noncommutative domains and varieties in B(H)n, where B(H) is the algebra of all bounded linear operators on a Hilbert space H, associated with admissible free holomorphic functions for operator model theory. This includes, in particular, a large class of commutative domains in B(H)n. In the commutative case, the characteristic functions are multipliers of reproducing kernel Hilbert spaces on domains in Cn. We also show that the universal operator model associated with any admissible noncommutative variety has the quasiwandering property. [ABSTRACT FROM AUTHOR]

For any pair (n, p), n ∈ N and 0 < p < ∞, it has been recently proved by Peláez and Rättyä (2021) that a radial weight ! on the unit disc of the complex plane D satisfies the Littlewood-Paley equivalence... [ABSTRACT FROM AUTHOR]

In this paper we establish some sharp inequalities for holomorphic Fock spaces in C n and weighted Bergman spaces in the unit polydisk, which generalizes some results of Burbea. As an application, we prove Furdui's conjecture about the norm of composition operators between Fock spaces. [ABSTRACT FROM AUTHOR]

We show that derivatives of some classes of analytic functions, univalent or locally univalent in the unit disk, are cyclic vectors for the Bergman space B 2 (D). [ABSTRACT FROM AUTHOR]

Following the ideas of recent research in Karapetyants and Samko [Mixed norm variable exponent Bergman space on the unit disc. Complex Var Elliptic Equ. 2016;61:1090–1106], we continue the study of new weighted spaces of holomorphic functions on the unit disc with the mixed norm defined in terms of conditions on Fourier coefficients of a function. Here we present a general approach for the weighted case. We study the boundedness of the weighted Bergman projection and discuss boundedness and compactness of Toeplitz operators with radial symbols. We also provide a characterization of functions in these spaces. [ABSTRACT FROM AUTHOR]

We prove that a Toeplitz operator with a complex Borel measure symbol whose total variation is Carleson is strongly localized, thus improving the recent localization results from Sadeghi and Zorboska [J. Math. Anal. Appl. 485 (2020), p. 16]. We also show that as a corollary, we get yet another view of some of the previously known results on Toeplitz operators with BMO symbols. [ABSTRACT FROM AUTHOR]

A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and which commutes with a quasinilpotent, injective, compact operator with dense range, but unlike other examples, it acts on the Bergman space instead of the Hardy space and this operator is associated with a "hyperbolic" composition operator. [ABSTRACT FROM AUTHOR]

Let ϕ be an analytic self‐map of the open unit disk D$\mathbb {D}$, and let u be an analytic function on D$\mathbb {D}$. The weighted composition operator induced by ϕ with weight u is given by (uCϕf)(z)=u(z)f(ϕ(z))$(uC_{\phi }f)(z)=u(z)f(\phi (z))$ for z in D$\mathbb {D}$ and f analytic on D$\mathbb {D}$. In this paper, we study weighted composition operators acting between two exponentially weighted Bergman spaces Aωp$A^p_{\omega }$ and Aωq$A^q_{\omega }$. We characterize the bounded, compact and Schatten class membership operators uCϕ$ uC_{\phi }$ acting from Aωp$A^p_{\omega }$ to Aωq$A^q_{\omega }$ when 0

We consider reflexive Bergman spaces A p (Ω) on polygonal domains Ω of the complex plane. With some restrictions to the angles of the boundary of Ω, we show that the boundedness of the Toeplitz operator T g : A p (Ω) → A p (Ω) with a positive symbol g is equivalent to the boundedness of the Berezin transform of g or to g times the area measure being a Carleson measure. The result is also formulated for more general simply connected domains. The main technical tool is a new weighted Forelli–Rudin-type estimate. [ABSTRACT FROM AUTHOR]

In this paper, we present necessary conditions for the hyponormality of Toeplitz operator Tϕ on the Bergman space La²(D) when the symbol ϕ is a polynomial in z and z̄. [ABSTRACT FROM AUTHOR]

For 1 ≤ p < ∞ , let A ω p be the weighted Bergman space associated with an exponential type weight ω satisfying ∫ D | K z (ξ) | ω (ξ) 1 / 2 d A (ξ) ≤ C ω (z) − 1 / 2 , z ∈ D , where K z is the reproducing kernel of A ω 2 . This condition allows us to obtain some interesting reproducing kernel estimates and more estimates on the solutions of the ∂̅-equation (Theorem 2.5) for more general weight ω ∗ . As an application, we prove the boundedness of the Bergman projection on L ω p , identify the dual space of A ω p , and establish an atomic decomposition for it. Further, we give necessary and sufficient conditions for the boundedness and compactness of some operators acting from A ω p into A ω q , 1 ≤ p , q < ∞ , such as Toeplitz and (big) Hankel operators. [ABSTRACT FROM AUTHOR]

We prove Carleson embeddings for Bergman–Orlicz spaces of the unit ball that extend the lower triangle estimates for the usual Bergman spaces. [ABSTRACT FROM AUTHOR]

We study the relationship between polynomial approximations in the Bergman space of certain simply connected domains in the complex plane and composition operators on the Dirichlet space of the unit disk. In particular, we characterize when a composition operator on the Dirichlet space has dense range, which settles a problem posed by Joseph Cima in 1976. [ABSTRACT FROM AUTHOR]

In this paper, compact double difference of composition operators acting from a space generated by the Cauchy kernel and a special measure to analytic Besov spaces is characterized. Moreover, operator norm of these operators acting from Cauchy transforms to analytic Besov spaces is obtained explicitly. [ABSTRACT FROM AUTHOR]

In this paper, we explain some sufficient conditions for unitariness of Toeplitz operators and little Hankel operators on the Bergman space. [ABSTRACT FROM AUTHOR]

We study the problem of how extremal functions for linear functionals over a Bergman space are influenced by the properties of the functions generating these functionals. For different classes of generating functions, we obtain a sufficiently exact description of qualitative properties of the corresponding extremal functions. The method developed here can be used to study similar problems in Hardy spaces. [ABSTRACT FROM AUTHOR]

In this paper, the authors characterize Carleson measures for the weighted Bergman spaces with Békollé weights on the unit ball. They apply the Carleson embedding theorem to study the properties of Toeplitz-type operators and composition operators acting on such spaces. [ABSTRACT FROM AUTHOR]

We show that several families of classical orthogonal polynomials on the real line are also orthogonal on the interior of an ellipse in the complex plane, subject to a weighted planar Lebesgue measure. In particular these include Gegenbauer polynomials C n (1 + α) (z) for α > - 1 containing the Legendre polynomials P n (z) and the subset P n (α + 1 2 , ± 1 2) (z) of the Jacobi polynomials. These polynomials provide an orthonormal basis and the corresponding weighted Bergman space forms a complete metric space. This leads to a certain family of Selberg integrals in the complex plane. We recover the known orthogonality of Chebyshev polynomials of the first up to fourth kind. The limit α → ∞ leads back to the known Hermite polynomials orthogonal in the entire complex plane. When the ellipse degenerates to a circle we obtain the weight function and monomials known from the determinantal point process of the ensemble of truncated unitary random matrices. [ABSTRACT FROM AUTHOR]

In this paper, we solve a separation of singularities problem in the Bergman space. More precisely, we show that if P ⊂ C is a convex polygon which is the intersection of n half planes, then the Bergman space on P decomposes into the sum of the Bergman spaces on these half planes. The result applies to the characterization of the reachable space of the one-dimensional heat equation on a finite interval with boundary controls. We prove that this space is a Bergman space of the square which has the given interval as a diagonal. This gives an affirmative answer to a conjecture raised in [17]. [ABSTRACT FROM AUTHOR]

In this paper, we present some necessary and sufficient conditions for the hyponormality of Toeplitz operator T φ on the Bergman space A 2 (D) with non-harmonic symbols under certain assumptions. [ABSTRACT FROM AUTHOR]

Following the idea of the paper by Contreras and Hernández-Díaz, we first give the characterization of the boundedness of the weighted composition operator W φ , ϕ on B p . Then, we investigate the boundedness and the compactness of composition operators C φ on B p and study the spectrum of the multiplication operator M ϕ on B p , as well. Finally, motivated by the paper by Čučković and Paudyal, we describe the relationships between the invariant subspaces of M z on B 0 p and T on A p , where T is the sum of multiplication operator and Volterra operator. Moreover, we provide some Beurling-type invariant subspaces of M z on B p and B 0 p , respectively. [ABSTRACT FROM AUTHOR]

Two central objects in constructive approximation, the Christoffel–Darboux kernel and the Christoffel function, encode ample information about the associated moment data and ultimately about the possible generating measures. We develop a multivariate theory of the Christoffel–Darboux kernel in C d , with emphasis on the perturbation of Christoffel functions and their level sets with respect to perturbations of small norm or low rank. The statistical notion of leverage score provides a quantitative criterion for the detection of outliers in large data. Using the refined theory of Bergman orthogonal polynomials, we illustrate the main results, including some numerical simulations, in the case of finite atomic perturbations of area measure of a 2D region. Methods of function theory of a complex variable and (pluri) potential theory are widely used in the derivation of our perturbation formulas. [ABSTRACT FROM AUTHOR]

In this paper we introduce a class A ⊂ L∞(D) such that if ø ∈ A and satisfies certain positive-definite condition, then there exists a ψ ∈ A such that ø(z) ≤ αeψ(z), for some constant α > 0. Further, if ø(z) = hAkz, kzi, for some bounded positive, invertible operator A from the Bergman space L ² a (D) into itself then ψ(z) = h(log A)kz, kzi. Here kz, z ∈ D are the normalized reproducing kernel of L ² a (D). Applications of these results are also discussed. [ABSTRACT FROM AUTHOR]

We generalize the Riesz conjugate functions theorem for planar harmonic K -quasiregular mappings (when 1 < p ≤ 2) and harmonic K -quasiconformal mappings (when 2 < p < ∞) in the unit disk. Moreover, if K = 1 , then our constant coincides with the classical analytic case. For the n dimensional case (n > 2), we also obtain the Riesz conjugate functions theorem for invariant harmonic K -quasiregular mappings when 1 < p ≤ 2. [ABSTRACT FROM AUTHOR]

The boundedness of the small Hankel operator h f ν (g) = P ν (f g ‾) , induced by an analytic symbol f and the Bergman projection P ν associated to ν , acting from the weighted Bergman space A ω p to A ν q is characterized on the full range 0 < p , q < ∞ of parameters when ω and ν belong to the class D of radial weights admitting certain two-sided doubling conditions. Moreover, an asymptotic formula for the operator norm of h f ν is established in terms of a suitable norm of f (n) depending upon the inducing weights and parameters. Certain results obtained are equivalent to the boundedness of bilinear Hankel forms, which are in turn used to establish the weak factorization A η q = A ω p 1 ⊙ A ν p 2 , where 1 < q , p 1 , p 2 < ∞ such that q − 1 = p 1 − 1 + p 2 − 1 and η ˜ 1 q ≍ ω ˜ 1 p 1 ν ˜ 1 p 2 . Here τ ˜ (r) = ∫ r 1 τ (t) d t / (1 − r) for all 0 ≤ r < 1. [ABSTRACT FROM AUTHOR]

In this paper, we establish necessary conditions and sufficient conditions for the boundedness and compactness of the small Hankel product Γ φ Γ ψ ⁎ on the Bergman space, where φ and ψ are square integrable on the open unit disk. [ABSTRACT FROM AUTHOR]