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An example for the holomorphic sectional curvature of the Bergman metric

Żywomir Dinew (2010)

Annales Polonici Mathematici

We study the behaviour of the holomorphic sectional curvature (or Gaussian curvature) of the Bergman metric of planar annuli. The results are then utilized to construct a domain for which the curvature is divergent at one of its boundary points and moreover the upper limit of the curvature at that point is maximal possible, equal to 2, whereas the lower limit is -∞.

Asymptotic behavior of the sectional curvature of the Bergman metric for annuli

Włodzimierz Zwonek (2010)

Annales Polonici Mathematici

We extend and simplify results of [Din 2010] where the asymptotic behavior of the holomorphic sectional curvature of the Bergman metric in annuli is studied. Similarly to [Din 2010] the description enables us to construct an infinitely connected planar domain (in our paper it is a Zalcman type domain) for which the supremum of the holomorphic sectional curvature is two, whereas its infimum is equal to -∞ .

Carleson measures and Toeplitz operators on small Bergman spaces on the ball

Van An Le (2021)

Czechoslovak Mathematical Journal

We study Carleson measures and Toeplitz operators on the class of so-called small weighted Bergman spaces, introduced recently by Seip. A characterization of Carleson measures is obtained which extends Seip’s results from the unit disk of to the unit ball of n . We use this characterization to give necessary and sufficient conditions for the boundedness and compactness of Toeplitz operators. Finally, we study the Schatten p classes membership of Toeplitz operators for 1 < p < .

Convergence of Taylor series in Fock spaces

Haiying Li (2014)

Studia Mathematica

It is well known that the Taylor series of every function in the Fock space F α p converges in norm when 1 < p < ∞. It is also known that this is no longer true when p = 1. In this note we consider the case 0 < p < 1 and show that the Taylor series of functions in F α p do not necessarily converge “in norm”.

Convexities of Gaussian integral means and weighted integral means for analytic functions

Haiying Li, Taotao Liu (2019)

Czechoslovak Mathematical Journal

We first show that the Gaussian integral means of f : (with respect to the area measure e - α | z | 2 d A ( z ) ) is a convex function of r on ( 0 , ) when α 0 . We then prove that the weighted integral means A α , β ( f , r ) and L α , β ( f , r ) of the mixed area and the mixed length of f ( r 𝔻 ) and f ( r 𝔻 ) , respectively, also have the property of convexity in the case of α 0 . Finally, we show with examples that the range α 0 is the best possible.

Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions

Sunanda Naik, Karabi Rajbangshi (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

Let f be an analytic function on the unit disk . We define a generalized Hilbert-type operator a , b by a , b ( f ) ( z ) = Γ ( a + 1 ) / Γ ( b + 1 ) 0 1 ( f ( t ) ( 1 - t ) b ) / ( ( 1 - t z ) a + 1 ) d t , where a and b are non-negative real numbers. In particular, for a = b = β, a , b becomes the generalized Hilbert operator β , and β = 0 gives the classical Hilbert operator . In this article, we find conditions on a and b such that a , b is bounded on Dirichlet-type spaces S p , 0 < p < 2, and on Bergman spaces A p , 2 < p < ∞. Also we find an upper bound for the norm of the operator a , b . These generalize...

New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space

Flavia Colonna (2013)

Open Mathematics

Let ψ and φ be analytic functions on the open unit disk 𝔻 with φ( 𝔻 ) ⊆ 𝔻 . We give new characterizations of the bounded and compact weighted composition operators W ψ,ϕ from the Hardy spaces H p, 1 ≤ p ≤ ∞, the Bloch space B, the weighted Bergman spaces A αp, α > − 1,1 ≤ p < ∞, and the Dirichlet space 𝒟 to the Bloch space in terms of boundedness (respectively, convergence to 0) of the Bloch norms of W ψ,ϕ f for suitable collections of functions f in the respective spaces. We also obtain characterizations...

On products of some Toeplitz operators on polyanalytic Fock spaces

Irène Casseli (2020)

Czechoslovak Mathematical Journal

The purpose of this paper is to study the Sarason’s problem on Fock spaces of polyanalytic functions. Namely, given two polyanalytic symbols f and g , we establish a necessary and sufficient condition for the boundedness of some Toeplitz products T f T g ¯ subjected to certain restriction on f and g . We also characterize this property in terms of the Berezin transform.

On some spaces of holomorphic functions of exponential growth on a half-plane

Marco M. Peloso, Maura Salvatori (2016)

Concrete Operators

In this paper we study spaces of holomorphic functions on the right half-plane R, that we denote by Mpω, whose growth conditions are given in terms of a translation invariant measure ω on the closed half-plane R. Such a measure has the form ω = ν ⊗ m, where m is the Lebesgue measure on R and ν is a regular Borel measure on [0, +∞). We call these spaces generalized Hardy–Bergman spaces on the half-plane R. We study in particular the case of ν purely atomic, with point masses on an arithmetic progression...

On the powers of quasihomogeneous Toeplitz operators

Aissa Bouhali, Zohra Bendaoud, Issam Louhichi (2021)

Czechoslovak Mathematical Journal

We present sufficient conditions for the existence of p th powers of a quasihomogeneous Toeplitz operator T e i s θ ψ , where ψ is a radial polynomial function and p , s are natural numbers. A large class of examples is provided to illustrate our results. To our best knowledge those examples are not covered by the current literature. The main tools in the proof of our results are the Mellin transform and some classical theorems of complex analysis.

On the weighted estimate of the Bergman projection

Benoît Florent Sehba (2018)

Czechoslovak Mathematical Journal

We present a proof of the weighted estimate of the Bergman projection that does not use extrapolation results. This estimate is extended to product domains using an adapted definition of Békollé-Bonami weights in this setting. An application to bounded Toeplitz products is also given.

On Volterra composition operators from Bergman-type space to Bloch-type space

Zhi Jie Jiang (2011)

Czechoslovak Mathematical Journal

Let ϕ be an analytic self-mapping of 𝔻 and g an analytic function on 𝔻 . In this paper we characterize the bounded and compact Volterra composition operators from the Bergman-type space to the Bloch-type space. We also obtain an asymptotical expression of the essential norm of these operators in terms of the symbols g and ϕ .

On weighted composition operators acting between weighted Bergman spaces of infinite order and weighted Bloch type spaces

Elke Wolf (2011)

Annales Polonici Mathematici

Let ϕ: → and ψ: → ℂ be analytic maps. They induce a weighted composition operator ψ C ϕ acting between weighted Bergman spaces of infinite order and weighted Bloch type spaces. Under some assumptions on the weights we give a characterization for such an operator to be bounded in terms of the weights involved as well as the functions ψ and ϕ

Polyanalytic Besov spaces and approximation by dilatations

Ali Abkar (2024)

Czechoslovak Mathematical Journal

Using partial derivatives f / z and f / z ¯ , we introduce Besov spaces of polyanalytic functions in the open unit disk, as well as in the upper half-plane. We then prove that the dilatations of functions in certain weighted polyanalytic Besov spaces converge to the same functions in norm. When restricted to the open unit disk, we prove that each polyanalytic function of degree q can be approximated in norm by polyanalytic polynomials of degree at most q .

Some Banach spaces of Dirichlet series

Maxime Bailleul, Pascal Lefèvre (2015)

Studia Mathematica

The Hardy spaces of Dirichlet series, denoted by p (p ≥ 1), have been studied by Hedenmalm et al. (1997) when p = 2 and by Bayart (2002) in the general case. In this paper we study some L p -generalizations of spaces of Dirichlet series, particularly two families of Bergman spaces, denoted p and p . Each could appear as a “natural” way to generalize the classical case of the unit disk. We recover classical properties of spaces of analytic functions: boundedness of point evaluation, embeddings between...

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