• Journal of the Korean Mathematical Society
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• v.46 no.6
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• pp.1219-1232
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• 2009

Let H(B) denote the space of all holomorphic functions on the unit ball B of $\mathbb{C}^n$. Let $\varphi$ = (${\varphi}_1,{\ldots}{\varphi}_n$) be a holomorphic self-map of B and $g{\in}2$(B) with g(0) = 0. In this paper we study the boundedness and compactness of the generalized composition operator $C_{\varphi}^gf(z)=\int_{0}^{1}{\mathfrak{R}}f(\varphi(tz))g(tz){\frac{dt}{t}}$ from generalized weighted Bergman spaces into Bloch type spaces.

• East Asian mathematical journal
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• v.33 no.1
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• pp.37-44
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• 2017

We will prove size estimates of the Bergman kernel for the generalized Fock space ${\mathcal{F}}^2_{\varphi}$, where ${\varphi}$ belongs to the class $\mathcal{W} $. The main tool for the proof is to use the estimate on the canonical solution to the ${\bar{\partial}}$-equation. We use Delin's weighted $L^2$-estimate ([3], [6]) for it.

• Journal of the Korean Mathematical Society
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• v.58 no.6
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• pp.1347-1365
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• 2021

For µ = (µ₁, …, µ_t) (µ_j > 0), ξ = (z₁, …, z_t, w) ∈ ℂ^n₁ × … × ℂ^n_t × ℂ^m, define $${\Omega}({\mu},t)=\{{\xi}{\in}\mathbb{B}_{n_1}{\times}{\cdots}{\times}\mathbb{B}_{n_t}{\times}\mathbb{C}^m:{\parallel}w{\parallel}^2 where $\mathbb{B}_{n_j}$ is the unit ball in ℂ^n_j (1 ≤ j ≤ t), C(χ, µ) is a constant only depending on χ = (n₁, …, n_t) and µ = (µ₁, …, µ_t), which is a special type of generalized Cartan-Hartogs domain. We will give some sufficient and necessary conditions for the boundedness of some type of operators on L^p(Ω(µ, t), ω) (the weighted L^p space of Ω(µ, t) with weight ω, 1 < p < ∞). This result generalizes the works from certain classes of generalized complex ellipsoids to the generalized Cartan-Hartogs domain Ω(µ, t).