Search

In this paper, we develop the theory of weighted Bergman space and obtain a general representation formula of the Bergman kernel function for the spaces on the Reinhardt domain containing the origin. As applications, we calculate the concrete forms of the Bergman kernels for some special weights on the Reinhardt domains $ \mathbb {C}^n $ C n , $ D_{n,m}:= \{(z, w)\in \mathbb {C}^n \times \mathbb {C}^m : \|w\|^2 D n , m := { (z , w) ∈ C n × C m : ‖ w ‖ 2 < e − μ 1 ‖ z ‖ μ 2 } and $ V_{\eta }:=\{\left (z, z', w\right) \in \mathbb {C}^{n} \times \mathbb {C}^{m} \times \mathbb {C} : \sum _{j=1}^{n} e^{\eta _{j}|w|^{2}}\left |z_{j}\right |^{2}+\|z'\|^{2} V η := { (z , z ′ , w) ∈ C n × C m × C : ∑ j = 1 n e η j | w | 2 | z j | 2 + ‖ z ′ ‖ 2 < 1 }. [ABSTRACT FROM AUTHOR]

We study Laplace equation and heat equation with the Gaussian white noise as the boundary/initial condition. Using the stochastic pre-orthogonal adaptive Fourier decomposition on the Bergman space, we give sparse expansions of the solutions of the boundary/initial value problems. The proposed methodology breaks away from the H - H K structure. The key point to the solution is to use the Wiener isometry property of the Gaussian white noise integral and the corresponding Green's function to represent the covariance of the solution. For the heat equation with the Gaussian white noise initial value, we define a class of parabolic Bergman space following E. Stein, prove existence of Bergman kernel, and obtain its expression, and further verify the boundary vanishing property needed for generating a sparse expansion. In order to make the heat flow satisfy a long-time decay property, we induce a subspace of the parabolic Bergman space, which corresponds to a class of homogeneous Sobolev spaces. [ABSTRACT FROM AUTHOR]

Let ω be a doubling weight and 0 < p ≤ q < ∞ . The essential norm of Riemann–Stieltjes operator T g from the weighted Bergman space A ω p to A ω q was investigated in the unit ball of ℂ n . [ABSTRACT FROM AUTHOR]

We study the problem of recovering a signal from magnitudes of its wavelet frame coefficients when the analyzing wavelet is real-valued. We show that every real-valued signal can be uniquely recovered, up to global sign, from its multiwavelet frame coefficients \[ \{\lvert \mathcal {W}_{\phi _i} f(\alpha ^{m}\beta n,\alpha ^{m}) \rvert : i\in \{1,2,3\}, m,n\in \mathbb {Z}\} \] for every \alpha >1,\beta >0 with \beta \ln (\alpha)\leq 4\pi /(1+4p), p>0, when the three wavelets \phi _i are suitable linear combinations of the Poisson wavelet P_p of order p and its Hilbert transform \mathscr {H}P_p. For complex-valued signals we find that this is not possible for any choice of the parameters \alpha >1,\beta >0, and for any window. In contrast to the existing literature on wavelet sign retrieval, our uniqueness results do not require any bandlimiting constraints or other a priori knowledge on the real-valued signals to guarantee their unique recovery from the absolute values of their wavelet coefficients. [ABSTRACT FROM AUTHOR]

Given a doubling weight ω on the unit disk D , let Aωp be the space of all the holomorphic functions f, where ∥ f ∥ A p := (∫ D ∣ f (z) ∣ p (z) d A (z)) 1 / p < ∞. We completely characterize the topological connectedness of the set of composition operators on Aωp. As an application, we construct an interesting example which reveals that two composition operators on Aαp in the same path component may fail to have a compact difference and give a negative answer to the Shapiro-Sundberg question in the (standard) weighted Bergman space. In addition, we completely describe the central compactness of any finite linear combinations of composition operators on Aωp in three terms: a Julia-Carathéodory-type function-theoretic characterization, a power-type characterization, and a Carleson-type measure-theoretic characterization. [ABSTRACT FROM AUTHOR]

Herein, the Laplace transform representations for functions of weighted holomorphic Bergman spaces on the tube domains are developed. Then a weighted version of the edge-of-the-wedge theorem is derived as a byproduct of the main results. [ABSTRACT FROM AUTHOR]

In the paper 'Composition operators on weighted Bergman spaces of Dirichlet series. J Math Anal Appl. 2015;426:340–363', Bailleul completely characterized the boundedness of composition operators on weighted Bergman spaces of Dirichlet series for the case of symbols with $ c_0\ge 1 $ c 0 ≥ 1. But the sufficient conditions for the other case $ c_0=0 $ c 0 = 0 were unsolved. In this paper, we follow this line and study the boundedness of composition operators on weighted Bergman spaces of Dirichlet series for the case $ c_0=0 $ c 0 = 0. Moreover, we also obtain the compact characterizations of composition operators with $ c_0\geq 1 $ c 0 ≥ 1. [ABSTRACT FROM AUTHOR]

We consider the weighted Bergman space A α 2 (픻) of analytic functions f on the unit disc 픻 = {z ∈ ℂ :|z| < 1} such that f A α 2 2 := ∫ D f z 2 d A α z < + ∞ , where d Aα(z) = (α+1) (1 −|z|2)α d A(z) and d A z = d x d y π (the normalized Lebesgue area measure). We investigate the generalized Riccati operator equation X A 1 X + A 2 X A 3 + A 4 Y A 5 + A 6 = 0 with Ai ∈ 퓑( A α 2 (픻)), i = 1, 6. Also we study the solvability of the operator equation X 1 T φ 1 + X 2 T φ 2 + ⋯ + X n T φ n = I A α 2 , in the set of Toeplitz operators on the weighted Bergman space A α 2 (픻). Moreover, we characterize compactness of the operator Tφ Tψ − Tη in terms of Berezin transforms. [ABSTRACT FROM AUTHOR]

One of the aims of this paper is to characterize 3-complex symmetric weighted composition operators induced by three types of symbols on the weighted Bergman space of the right half-plane with the conjugation J f (z) = f (z ¯) ¯ . It is well known that the complex symmetry is equivalent to 2-complex symmetry for the weighted composition operators studied in the paper. However, the interesting fact that 3-complex symmetry is not equivalent to 2-complex symmetry for such operators is found in the paper. Finally, the complex normal of such operators on the weighted Bergman space of the right half-plane with the conjugation J is characterized. [ABSTRACT FROM AUTHOR]

It is well-known that the research of linear combination of composition operators has become a topic of increasing interest. Recently, Choe, Koo and Wang proved that the compactness of combinations composition operators induced by the symbols satisfying the condition (CNC) implies that each difference is compact on the weighted Bergman space. Motivated by that work, in this paper, we discuss which difference is compact on the weighted Bergman space when the coefficients do not satisfy the condition (CNC). [ABSTRACT FROM AUTHOR]

We obtain a new boundedness criterion for the difference of two composition operators from a weighted Bergman space A α p into a Lebesgue space L q (μ) , where 0 < q < p and α > - 1 . As a consequence, we provide a direct proof that such a bounded difference operator is necessarily compact. We also characterize compact differences of composition operators from A α p into A β q explicitly for 0 < p ≤ q and α , β > - 1 . [ABSTRACT FROM AUTHOR]

This article studies the boundedness of the inclusion mapping from weighted Bergman spaces Aαp into a class of tent type space Tsp,n(μ). As an application, the boundedness, compactness and essential norm of generalized integral operators Tgn,k and Sgn,0 from Aαp to general function spaces are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, we discuss block Toeplitz operators T Φ whose symbols are in the class of normal functions Φ (z) = G ∗ (z) + F (z) , where F (z) = A m z m + A N z N and G (z) = A - m z m + A - N z N (A i ∈ M 2) , on the weighted Bergman space. Additionally, we present a necessary and sufficient condition for the hyponormality of the block Toeplitz operators T Φ . [ABSTRACT FROM AUTHOR]

Let Mu be the multiplication operator with symbol u which is a holomorphic function on the open unit ball B in Cn, Cφ the composition operator with symbol φ which is a holomorphic self-map of B, and Rm, m ∈ N,the mth iterated radial derivative operator. The metrical boundedness and metrical compactness of the operator CφRmMu from weighted Bergman spaces to weighted-type spaces on B are characterized. As an application of the proofs, the same properties of the operator ... from weighted Bergman spaces to weighted-type spaces on B are characterized. [ABSTRACT FROM AUTHOR]

We study composition operators between the weighted Bergman space of the polydisk and the weighted Bergman space of the unit disk. We find conditions on the symbols to guarantee the boundedness and compactness of the composition operator. We shall also find a necessary and sufficient condition for the composition operator to be Hilbert–Schmidt. [ABSTRACT FROM AUTHOR]

We study the dominating sets and reverse Carleson measures on exponentially weighted Bergman spaces A ω p under a new metric. Then, we give some applications of reverse Carleson measure and a generalization of Theorem 1 in Luecking [Sampling measures for Bergman spaces on the unit disk. Math Ann. 2000;316:659–679]. [ABSTRACT FROM AUTHOR]

We introduce a polynomial differentiation composition operator on spaces of holomorphic functions on the open unit ball in the n-dimensional complex vector space, and characterize the boundedness and compactness of the operator from the classical weighted Bergman space to the weighted-type space and the little weighted-type space of holomorphic functions on the unit ball. [ABSTRACT FROM AUTHOR]

One of the aims of this paper is to characterize 3-complex symmetric weighted composition operators induced by three types of symbols on the weighted Bergman space of the right half-plane with the conjugation Jf(z)=f(z¯)¯. It is well known that the complex symmetry is equivalent to 2-complex symmetry for the weighted composition operators studied in the paper. However, the interesting fact that 3-complex symmetry is not equivalent to 2-complex symmetry for such operators is found in the paper. Finally, the complex normal of such operators on the weighted Bergman space of the right half-plane with the conjugation J is characterized.

In this article, we mainly study the weighted composition-differentiation operator on the weighted Bergman space A α 2 and the derivative Hardy space S 1 2 , which characterize complex symmetric weighted composition-differentiation operator D u , φ . Moreover, the normality and self-adjointness of D u , φ are also discussed. [ABSTRACT FROM AUTHOR]

It is known that the norm of the Hilbert matrix operator on weighted Bergman spaces A α p was conjectured by Karapetrović to be π sin ⁡ (α + 2) π p when α > − 1 and p > α + 2. The conjecture has been confirmed by Božin and Karapetrović in the case α = 0. In this paper we prove the conjecture for the cases both α = 1 and 0 < α ≤ 1 47. Moreover, we also show that the conjecture is valid when − 1 < α < 0 and p ≥ 2 (α + 2). [ABSTRACT FROM AUTHOR]

In this paper, we consider the hyponormality of Toeplitz operators acting on the weighted Bergman space A α 2 (D). We establish necessary or sufficient conditions for the hyponormality of Toeplitz operators with non-harmonic symbols φ (z) that are bivariate polynomials in z and z ¯ (i.e., φ (z) = a z m z ¯ n + b z s z ¯ t ). In particular, we present some characteristics according to the coefficients and degree of φ (z). [ABSTRACT FROM AUTHOR]

We characterize wandering subspace property of the shift operator on zero-based invariant subspaces of reproducing kernel Hilbert spaces by using the kernels of Toeplitz operators. As applications, we prove that the wandering subspace property of the shift operator holds on the weighted Bergman space A α 2 ( α > 1) for zero-based invariant subspaces when the zeros lie in a small disk centred at the origin. [ABSTRACT FROM AUTHOR]

Motivated by the recent work of Galindo et al. (J. Funct. Anal. 265, 629–643, 2013), in this paper, we give an elegant compactness criterion for any finite linear combination of composition operators on the weighted Bergman space in terms of power type characterization. More precisely, let T be any finite linear combination of composition operators, then T is compact on A α p (D) if and only if lim n → ∞ ∥ T z n ∥ A − γ ∥ z n ∥ A − γ = 0 , which reveals that the compactness of T on the weighted Bergman space A α p (D) and Korenblum space A−γ(D) are equivalent. [ABSTRACT FROM AUTHOR]

We obtain the norm and essential norm estimates for the difference of two weighted composition operators from a weighted Bergman space A α p into a Lebesgue space L q (μ) , where 0 < p ≤ q , α > - 1 and μ is a positive Borel measure on the unit disk of C . The weights of the weighted maps in this paper are not necessarily analytic or bounded. Consequently, characterizations for the boundedness and compactness of the difference operator are established. [ABSTRACT FROM AUTHOR]

Let φ and u be analytic maps on the open unit disc D such that φ (D) ⊂ D and H (D) be the spaces of analytic functions on D. For a non-negative integer n, the generalized weighted composition operator Dφ,un is defined by Dφ,un f = u. (f(n) o φ), f ∈ H (D). In this paper, we characterize the boundedness of the generalized weighted composition operator Dφ,un between the weighted Bergman spaces. We also compute the upper and lower bounds for the essential norm of this operator. [ABSTRACT FROM AUTHOR]

We give a concrete description of complex symmetric monomial Toeplitz operators T z p z ¯ q on the weighted Bergman space A2(Ω), where Ω denotes the unit ball or the unit polydisk. We provide a necessary condition for T z p z ¯ q to be complex symmetric. When p,q ∈ ℕ2, we prove that T z p z ¯ q is complex symmetric on A2(Ω) if and only if p1 = q2 and P2 = q1. Moreover, we completely characterize when monomial Toeplitz operators T z p z ¯ q on A 2 (D n) are JU-symmetric with the n × n symmetric unitary matrix U. [ABSTRACT FROM AUTHOR]

We shall give a description of the commutant of a particular multiplication operator in the weighted Bergman spaces on n-dimensional complex polydisk../files/site1/files/71/1.pdf

Holomorphic functions on the maximal Grauert tube of a hyperbolic compact Riemann surface are studied. It is shown that their weighted Bergman spaces are infinite dimensional for arbitrary weight order greater than - 1 in spite of the fact that they do not admit any non-constant bounded holomorphic functions. The key ingredient of the proof is a computation of weighted norms of analytic continuations of eigenfunctions of the Laplacian in terms of the hypergeometric function. This result complements our previous work (Adachi in Trans Am Math Soc 374(10):7499–7524, 2021) where it was shown that the space of geodesic segments on the Riemann surface has exactly the same property. [ABSTRACT FROM AUTHOR]

In this paper, we give a characterization of a complex symmetric Toeplitz operator T φ on the weighted Bergman space A α 2 (D). We first give properties of complex symmetric Toeplitz operators T φ on A α 2 (D). Next, we prove that if T φ is complex symmetric with finite symbol, then T φ is hyponormal on A α 2 (D) if and only if it is hyponormal on the Hardy space H 2 (T). Finally, we consider the complex symmetric Toeplitz operator T φ on A α 2 (D) when the conjugation is a special case. [ABSTRACT FROM AUTHOR]

In Koo and Wang (Joint Carleson measure and the difference of composition operators on A α p (B n). J Math Anal Appl. 2014;419:1119–1142), the authors introduced a concept of joint Carleson measure and used it to characterize when the difference of two composition operators on weighted Bergman space over the unit ball is bounded or compact. In this paper, we extend the concept of joint Carleson measure to the polydisk setting and obtain analogue characterizations of the boundedness (compactness, resp.) of the difference of two composition operators on the weighted Bergman spaces over the unit polydisk, which may provide a unified approach for various ad hoc studies on the boundedness or the compactness of the difference of composition operators on polydisk. Moreover, we construct a concrete example to show that both the boundedness and the compactness depend on the index p when the dimension n ≥ 2 , which is in sharp contrast with the one-variable case where the boundedness and the compactness of the difference of two composition operators are independent of p>0. Due to the complexity of the Carleson measure on the unit polydisk, some new techniques are required in the polydisk setting. [ABSTRACT FROM AUTHOR]

We obtain conditions that imply reverse Carleson inequalities for weighted Bergman spaces with Békollé weights. Our main theorem generalizes the results in the unweighted case by Luecking. [ABSTRACT FROM AUTHOR]

The spectrum of a weighted composition operator C ψ , φ that is induced by an automorphism has been investigated for over 50 years. However, many results are got only under the condition that the weight function ψ is continuous up to the boundary. In this paper we study the spectra and essential spectra of C ψ , φ on weighted Bergman spaces when φ is an elliptic automorphism, without the assumption that ψ is continuous up to the boundary. [ABSTRACT FROM AUTHOR]

We completely describe the boundedness and compactness of Hankel operators with general symbols acting on Bergman spaces with exponential type weights. [ABSTRACT FROM AUTHOR]

In this work, we introduce a family of weighted Bergman spaces { A α , n } n ∈ N . This family satisfies the continuous inclusions A α , n ⊂ ⋯ ⊂ A α , 2 ⊂ A α , 1 ⊂ A α , 0 = A α , where A α is the classical weighted Bergman space. Next, we define and study the derivative operator ∇ = d d z and its adjoint operator L α = z 2 d d z + (α + 2) z on the weighted Bergman space A α , and we establish an uncertainty inequality of Heisenberg-type for this space. A more general uncertainty inequality for the space A α , n is also given when we considered the operators ∇ n = ∇ n and L α , n : = (L α) n . Afterward, we give Heisenberg-type and Laeng-Morpurgo-type uncertainty inequalities for the Bargmann transform B α , which is an isometric isomorphism between the space A α and the Lebesgue space L 2 (R + , d μ α) , where d μ α is an appropriate measure. [ABSTRACT FROM AUTHOR]

We study the dominating sets and reverse Carleson measures on weighted Bergman spaces A ω p with invariant weights in the spirit of Lueckings results from 1981 and 1985. Then, we characterize integrating derivatives of A ω p functions, and combine this with reverse Carleson measures to characterize the reverse Carleson inequality for the derivatives of A ω p functions. [ABSTRACT FROM AUTHOR]

The aim of this study is to understand to what extent a 1-convex domain with Levi-flat boundary is capable of holomorphic functions with slow growth. This paper discusses a typical example of such domain, the space of all the geodesic segments on a hyperbolic compact Riemann surface. Our main finding is an integral formula that produces holomorphic functions on the domain from holomorphic differentials on the Riemann surface. This construction can be seen as a non-trivial example of L2 jet extension of holomorphic functions with optimal constant. As its corollary, it is shown that the weighted Bergman spaces of the domain is infinite dimensional for any weight order greater than -1 in spite of the fact that the domain does not admit any non-constant bounded holomorphic functions. [ABSTRACT FROM AUTHOR]

We characterise bounded and compact generalised weighted composition operators acting from the weighted Bergman space Apω, where 0 and ω belongs to the class D of radial weights satisfying a two-sided doubling condition, to a Lebesgue space Lqν. On the way, we establish a new embedding theorem on weighted Bergman spaces Apω which generalises the well-known characterisation of the boundedness of the differentiation operator Dnn(ƒ) = ƒ(n) from the classical weighted Bergman space Apα to the Lebesgue space Lqμ, induced by a positive Borel measure μ, to the setting of doubling weights. [ABSTRACT FROM AUTHOR]

Abstract Closures of weighted Bergman spaces in Bloch type spaces are investigated in the paper. Moreover, the boundedness and compactness of the product of composition and differentiation operators from Bloch type spaces to closures of weighted Bergman spaces spaces in Bloch type spaces are characterized.

In this article, we prove that the two-parameter family of operators Ab,c is bounded on the weighted Bergman spaces B p α+c−1 if α + 2 < p and unbounded if α + 2 = p.

For an analytic self-map φ of the unit disk D in the complex plane C, a nonnegative integer n, and u analytic function on D, weighted differentiation composition operator is defined by (Dφ,unƒ)(z) = u(z)ƒ(n)(φ(z)), where f is an analytic function on D and z ∊ D. In this paper, we study the boundedness and compactness of Dφ,un, from weighted Bergman spaces with admissible weights to Bloch-type spaces. [ABSTRACT FROM AUTHOR]

We consider operators acting on a Hilbert space that can be written as the sum of a shift and a diagonal operator and determine when the operator is hyponormal. The condition is presented in terms of the norm of an explicit block Jacobi matrix whose entries are explicitly computable in many applications. We apply this result to the Toeplitz operator with symbol z n + c | z | s acting on certain weighted Bergman spaces and determine for what values of the constant c this operator is hyponormal. Our result strengthens an earlier result by the second author, which in turn addressed a question posed by Fleeman and Liaw in 2019. [ABSTRACT FROM AUTHOR]