• Journal of the Korean Mathematical Society
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• v.54 no.2
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• pp.627-645
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• 2017

In this paper, we characterize the boundedness, compactness and essential norm of products of differentiation and composition operators from the Bloch space and weighted Dirichlet spaces to analytic Morrey type spaces.

• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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• pp.319-331
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• 2019

On the weighted Dirichlet space, by the Sobolev's embedding theorem, we characterize the compactness for operators which are finite sums of products of several Toeplitz operators. Moreover, the essential spectrum of Toeplitz operator is characterized.

• Communications of the Korean Mathematical Society
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• v.35 no.1
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• pp.217-227
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• 2020

Let ψ be an analytic function on 𝔻, the unit disc in the complex plane, and φ be an analytic self-map of 𝔻. Let 𝓑 be a Banach space of functions analytic on 𝔻. The weighted composition operator W_φ,ψ on 𝓑 is defined as W_φ,ψf = ψf ◦ φ, and the composition operator C_φ defined by C_φf = f ◦ φ for f ∈ 𝓑. Consider α > -1 and 1 ≤ p < ∞. In this paper, we prove that if φ ∈ H^∞(𝔻), then C_φ has closed range on any weighted Dirichlet space 𝒟_α if and only if φ(𝔻) satisfies the reverse Carleson condition. Also, we investigate the closed rangeness of weighted composition operators on the weighted Bergman space A^p_α.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.2
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• pp.393-402
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• 2013

For 0 < $p$ < ${\infty}$, ${\alpha}$ > -1 and 0 < $r$ < 1, we show that if $f$ is in the space of Dirichlet type $\mathfrak{D}^p_{p-1}$, then ${\int}_{1}^{0}M_{p}^{p}(r,f^{\prime})(1-r)^{p-1}rdr$ < ${\infty}$ and ${\int}_{1}^{0}M_{(2+{\alpha})p}^{(2+{\alpha})p}(r,f^{\prime})(1-r)^{(2+{\alpha})p+{\alpha}}rdr$ < ${\infty}$ where $M_p(r,f)=\[\frac{1}{2{\pi}}{\int}_{0}^{2{\pi}}{\mid}f(re^{it}){\mid}^pdt\]^{1/p}$. For 1 < $p$ < $q$ < ${\infty}$ and ${\alpha}+1$ < $p$, we show that if there exists some positive constant $c$ such that ${\parallel}f{\parallel}_{L^{q(d{\mu})}}{\leq}c{\parallel}f{\parallel}_{\mathfrak{D}^p_{\alpha}}$ for all $f{\in}\mathfrak{D}^p_{\alpha}$, then ${\parallel}f{\parallel}_{L^{q(d{\mu})}}{\leq}c{\parallel}f{\parallel}_{\mathcal{B}_p(q)}$ where $\mathcal{B}_p(q)$ is the weighted Besov space. We also find the condition of measure ${\mu}$ such that ${\sup}_{a{\in}D}{\int}_D(k_a(z)(1-{\mid}a{\mid}^2)^{(p-a-1)})^{q/p}d{\mu}(z)$ < ${\infty}$.

• Kyungpook Mathematical Journal
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• v.60 no.3
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• pp.519-534
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• 2020

Let H(𝔻) be the space of all holomorphic functions on the open unit disc 𝔻 in the complex plane ℂ. In this paper, we investigate the boundedness and compactness of the generalized integration operator $$I^{(n)}_{g,{\varphi}}(f)(z)=\normalsize\displaystyle\smashmargin{2}{\int\nolimits_0}^z\;f^{(n)}({\varphi}({\xi}))g({\xi})\;d{\xi},\;z{\in}{\mathbb{D}},$$ between Bloch-type and weighted Dirichlet-type spaces, where 𝜑 is a holomorphic self-map of 𝔻, n ∈ ℕ and g ∈ H(𝔻).