The Root Operator on Finite Zero Based Invariant Subspaces of the Weighted Bergman Space

AbstractBounded and compact differences of two composition operators acting from the weighted Bergman space $$A^p_\omega $$ A ω p to the Lebesgue space $$L^q_\nu $$ L ν q , where $$0<q<p<\infty $$ 0 < q < p < ∞ and $$\omega $$ ω belongs to the class "Equation missing" of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proofs a new description of q-Carleson measures for $$A^p_\omega $$ A ω p , with $$p>q$$ p > q and "Equation missing", involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q-Carleson measures for the classical weighted Bergman space $$A^p_\alpha $$ A α p with $$-1<\alpha <\infty $$ - 1 < α < ∞ to the setting of doubling weights. The case "Equation missing" is also briefly discussed and an open problem concerning this case is posed.

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The Berezin Transform of Toeplitz Operators on the Weighted Bergman Space

Potential Analysis ◽

10.1007/s11118-021-09915-2 ◽

2021 ◽

Author(s):

Cezhong Tong ◽

Junfeng Li ◽

Hicham Arroussi

Keyword(s):

Bergman Space ◽

Toeplitz Operators ◽

Reproducing Kernel ◽

Bergman Spaces ◽

Essential Norm ◽

Berezin Transform ◽

Weighted Bergman Space ◽

Weighted Bergman Spaces ◽

Kernel Estimates ◽

Berezin Transforms

AbstractIn this paper, we obtain some interesting reproducing kernel estimates and some Carleson properties that play an important role. We characterize the bounded and compact Toeplitz operators on the weighted Bergman spaces with Békollé-Bonami weights in terms of Berezin transforms. Moreover, we estimate the essential norm of them assuming that they are bounded.

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A Characterization of Bergman Spaces on the Unit Ball of ℂn. II

Canadian Mathematical Bulletin ◽

10.4153/cmb-2011-047-6 ◽

2012 ◽

Vol 55 (1) ◽

pp. 146-152 ◽

Cited By ~ 5

Author(s):

Songxiao Li ◽

Hasi Wulan ◽

Kehe Zhu

Keyword(s):

Holomorphic Function ◽

Unit Ball ◽

Bergman Space ◽

Bergman Spaces ◽

Weighted Bergman Space ◽

Double Integrals

AbstractIt has been shown that a holomorphic function f in the unit ball of ℂn belongs to the weighted Bergman space , p > n + 1 + α, if and only if the function | f(z) – f(w)|/|1 – 〈z, w〉| is in Lp( × , dvβ × dvβ), where β = (p + α – n – 1)/2 and dvβ(z) = (1 – |z|2)βdv(z). In this paper we consider the range 0 < p < n + 1 + α and show that in this case, f ∈ (i) if and only if the function | f(z) – f(w)|/|1 – hz, wi| is in Lp( × , dvα × dvα), (ii) if and only if the function | f(z)– f(w)|/|z–w| is in Lp( × , dvα × dvα). We think the revealed difference in the weights for the double integrals between the cases 0 < p < n + 1 + α and p > n + 1 + α is particularly interesting.

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