Weighted Bergman Spaces Research Articles

Multiple sampling and interpolation in Bergman spaces

We study multiple sampling and interpolation problems with unbounded multiplicities in the weighted Bergman space, both in the hilbertian case and the uniform case.

Наилучшее приближение и значения поперечников некоторых классов аналитических функций в весовом пространстве Бергмана B_(2,γ)

In the paper, exact inequalities are found for the best approximation of an arbitrary analytic function f in the unit circle by algebraic complex polynomials in terms of the modulus of continuity of the m th order of the r th order derivative f^((r)) in the weighted Bergman space B_(2,γ). Also using the modulus of continuity of the m-th order of the derivative f(r), we introduce a class of functions W_m^((r) ) (h,Φ) analytic in the unit circle and defined by a given majorant Φ, h∈(0,π⁄n], n>r, monotonically increasing on the positive semiaxis. Under certain conditions on the majorant Φ, for the introduced class of functions, the exact values of some known n-widths are calculated. We use methods for solving extremal problems in normed spaces of functions analytic in a circle, as well as the method for estimating from below the n-widths of functional classes in various Banach spaces developed by V.M. Tikhomirov. The results presented in this paper are a continuation and generalization of some earlier results on the best approximations and values of widths in the weighted Bergman space B_(2,γ).

A result about the atomic decomposition of Bloch-type space in the polydisk

<abstract><p>The aim of the paper is to obtain a interesting result about the atomic decomposition of Bloch-type space in the polydisk. The existing similar results have been applied many times to the atomic decompositions of Bloch-type and weighted Bergman spaces in the unit ball.</p></abstract>

Complex symmetric Toeplitz operators on the unit polydisk

In this paper, we study the complex symmetry of Toeplitz operators on the weighted Bergman spaces over the unit polydisk. First, we completely characterize when anti-linear weighted composition operators [Formula: see text] are conjugations. We then give a sufficient and necessary condition for Toeplitz operators to be complex symmetric with respect to these conjugations. As a consequence, some interesting higher-dimensional complex symmetric phenomena appear on the unit polydisk such as the monomial Toeplitz operators [Formula: see text] with [Formula: see text] for some symmetric permutation matrix [Formula: see text]. Surprisingly, these operators [Formula: see text] are the only ones that are complex symmetric monomial Toeplitz operators on the unit bidisk.

On the solution of the operator Riccati equations and invariant subspaces in the weighted Bergman space of the unit ball

We consider the Riccati operator equations on the weighted Bergman space A2? (Bn) of the unit ball Bn in Cn and investigate the properties of their solutions. Our discussion uses the Berezin symbols method.

Generalized integration operators from weighted Bergman spaces into general function spaces

Generalized integration operators from weighted Bergman spaces into general function spaces

Helson operators and coinvariant subspaces

We give an explicit expression of the functions in the finite-dimensional coinvariant subspace of the Hardy and weighted Bergman spaces over the infinite polydisc. As an application, an alternative form of the Perfekt–Pushnitski theorem characterizing Hel

Composition operators over weighted Bergman spaces of Dirichlet series

ABSTRACT 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 ≥ 1 . But the sufficient conditions for the other case 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 . Moreover, we also obtain the compact characterizations of composition operators with c 0 ≥ 1 .

Density of polyanalytic polynomials in complex and quaternionic polyanalytic weighted Bergman spaces

We introduce the concepts of complex polyanalytic weighted Bergman spaces and of quaternionic polyanalytic weighted Bergman spaces of first and second kind. In these spaces we then prove qualitative and quantitative results in approximation by polyanalytic polynomials. The quantitative approximation results are given in terms of higher order $L^{p}$-moduli of smoothness and in terms of the best approximation quantity.

Composition operators on the polydisc

In this paper we study the boundedness of composition operators on the weighted Bergman spaces and the Hardy space over the polydisc Dn. Studying the volume of sublevel sets we show for which n the necessary conditions obtained by Bayart are sufficient. For arbitrary polydisc we prove the rank sufficiency theorem which, in particular, provides us with a simple criterion describing boundedness of composition operators on the spaces over the bidisc. Such a consistent characterization is obtained for the classical Bergman space over the tridisc.

Littlewood–Paley estimates with applications to Toeplitz and integration operators on weighted Bergman spaces

Littlewood–Paley estimates with applications to Toeplitz and integration operators on weighted Bergman spaces

Hyponormal Toeplitz operators with non-harmonic symbols on the weighted Bergman spaces

In this paper, we consider the hyponormality of Toeplitz operators acting on the weighted Bergman space A_{alpha }^2({mathbb{D}}). We establish necessary or sufficient conditions for the hyponormality of Toeplitz operators with non-harmonic symbols varphi (z) that are bivariate polynomials in z and {overline{z}} (i.e., varphi (z)=az^m{overline{z}}^n+bz^s{overline{z}}^t). In particular, we present some characteristics according to the coefficients and degree of varphi (z).

Weighted Bergman Spaces Associated with the Hyperbolic Group

Weighted Bergman Spaces Associated with the Hyperbolic Group

Integral means of derivatives of univalent functions in Hardy spaces

We show that the norm in the Hardy space H p H^p satisfies ( † ) ‖ f ‖ H p p ≍ ∫ 0 1 M q p ( r , f ′ ) ( 1 − r ) p ( 1 − 1 q ) d r + | f ( 0 ) | p \begin{equation} \|f\|_{H^p}^p\asymp \int _0^1M_q^p(r,f’)(1-r)^{p\left (1-\frac 1q\right )}\,dr+|f(0)|^p\tag {$\dagger $} \end{equation} for all univalent functions provided that either q ≥ 2 q\ge 2 or 2 p 2 + p > q > 2 \frac {2p}{2+p}>q>2 . This asymptotic was previously known in the cases 0 > p ≤ q > ∞ 0>p\le q>\infty and p 1 + p > q > p > 2 + 2 157 \frac {p}{1+p}>q>p>2+\frac {2}{157} by results due to Pommerenke [Math. Ann. 145 (1961/62), pp. 285–296], Baernstein, Girela and Peláez [Illinois J. Math. 48 (2004), pp. 837–859] and González and Peláez [J. Geom. Anal. 19 (2009), pp. 755–771]. It is also shown that ( † ) (\dagger ) is satisfied for all close-to-convex functions if 1 ≤ q > ∞ 1\le q>\infty . A counterpart of ( † ) (\dagger ) in the setting of weighted Bergman spaces is also briefly discussed.

Formula omitted]-harmonic Bergman projection on the real hyperbolic ball

We determine precisely when the Bergman projections Pβ are bounded from Lebesgue spaces Lαp to weighted Bergman spaces Bαp of H-harmonic functions on the real hyperbolic ball, and verify a recent conjecture of M. Stoll. We obtain upper estimates for the reproducing kernels of the H-harmonic Bergman spaces Bα2 and their partial derivatives. We also consider the projection from L∞ to the Bloch space B of H-harmonic functions.

The Generalized Volterra Integral Operator and Toeplitz Operator on Weighted Bergman Spaces

We study the boundedness and compactness of the generalized Volterra integral operator on weighted Bergman spaces with doubling weights on the unit disc. A generalized Toeplitz operator is defined and the boundedness, compactness and Schatten class membership of this operator are investigated on the Hilbert weighted Bergman space. As an application, Schatten class membership of generalized Volterra integral operators are also characterized. Finally, we also get the characterizations of Schatten class membership of generalized Toeplitz operator and generalized Volterra integral operators on the Hardy space $$H^2$$ .

Positive Toeplitz operators from a harmonic Bergman–Besov space into another

We define positive Toeplitz operators between harmonic Bergman–Besov spaces \(b^p_\alpha \) on the unit ball of \({\mathbb {R}}^n\) for the full ranges of parameters \(0<p<\infty \), \(\alpha \in {\mathbb {R}}\). We give characterizations of bounded and compact Toeplitz operators taking one harmonic Bergman–Besov space into another in terms of Carleson and vanishing Carleson measures. We also give characterizations for a positive Toeplitz operator on \(b^{2}_{\alpha }\) to be a Schatten class operator \(S_{p}\) in terms of averaging functions and Berezin transforms for \(1\le p<\infty \), \(\alpha \in {\mathbb {R}}\). Our results extend those known for harmonic weighted Bergman spaces.

Commuting Toeplitz Operators on Cartan Domains of Type IV and Moment Maps

Let us consider, for $$n \ge 3$$ , the Cartan domain $$\mathrm {D}_n^{\mathrm {IV}}$$ of type IV. On the weighted Bergman spaces $$\mathcal {A}^2_\lambda (\mathrm {D}_n^{\mathrm {IV}})$$ we study the problem of the existence of commutative $$C^*$$ -algebras generated by Toeplitz operators with special symbols. We focus on the subgroup $$\mathrm {SO}(n) \times \mathrm {SO}(2)$$ of biholomorphisms of $$\mathrm {D}_n^{\mathrm {IV}}$$ that fix the origin. The $$\mathrm {SO}(n) \times \mathrm {SO}(2)$$ -invariant symbols yield Toeplitz operators that generate commutative $$C^*$$ -algebras, but commutativity is lost when we consider symbols invariant under a maximal torus or under $$\mathrm {SO}(2)$$ . We compute the moment map $$\mu ^{\mathrm {SO}(2)}$$ for the $$\mathrm {SO}(2)$$ -action on $$\mathrm {D}_n^{\mathrm {IV}}$$ considered as a symplectic manifold for the Bergman metric. We prove that the space of symbols of the form $$a = f \circ \mu ^{\mathrm {SO}(2)}$$ , denoted by $$L^\infty (\mathrm {D}_n^{\mathrm {IV}})^{\mu ^{\mathrm {SO}(2)}}$$ , yield Toeplitz operators that generate commutative $$C^*$$ -algebras. Spectral integral formulas for these Toeplitz operators are also obtained.

Boundedness of Bergman projectors on homogeneous Siegel domains

In this paper we study the boundedness of Bergman projectors on weighted Bergman spaces on homogeneous Siegel domains of Type II. As it appeared to be a natural approach in the special case of tube domains over irreducible symmetric cones, we study such boundedness on the scale of mixed-norm weighted Lebesgue spaces. The sharp range for the boundedness of such operators is essentially known only in the case of tube domains over Lorentz cones. In this paper we prove that the boundedness of such Bergman projectors is equivalent to variuos notions of atomic decomposition, duality, and characterization of boundary values of the mixed-norm weighted Bergman spaces, extending results mostly known only in the case of tube domains over irreducible symmetric cones. Some of our results are new even in the latter simpler context. We also study the simpler, but still quite interesting, case of the “positive” Bergman projectors, the integral operator in which the Bergman kernel is replaced by its modulus. We provide a useful characterization which was previously known for tube domains.

HANKEL MEASURES FOR FOCK SPACE

AbstractInspired by Xiao’s work on Hankel measures for Hardy and Bergman spaces [‘Pseudo-Carleson measures for weighted Bergman spaces’. Michigan Math. J.47 (2000), 447–452], we introduce Hankel measures for Fock space $F^p_\alpha $ . Given $p\ge 1$ , we obtain several equivalent descriptions for such measures on $F^p_\alpha $ .