• Communications of the Korean Mathematical Society
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• v.34 no.4
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• pp.1135-1143
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• 2019

The matrix representation of the Toeplitz operator on the Hardy space with respect to a generalized orthonormal basis for the space of square integrable functions associated to a bounded simply connected region in the complex plane is completely computed in terms of only the Szegő kernel and the Garabedian kernels.

• Honam Mathematical Journal
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• v.44 no.2
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• pp.271-280
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• 2022

In this study, We have applied the right operator k-Riemann-Liouville is involving two orders α and β with a positive parameter p > 0, further, the left operator k-Riemann-Liouville is used with the negative parameter p < 0 to introduce a new version related to Hardy-type inequalities. These inequalities are given and reversed for the cases 0 < p < 1 and p < 0. We then improved and generalized various consequences in the framework of Hardy-type fractional integral inequalities.

• Kyungpook Mathematical Journal
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• v.60 no.1
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• pp.127-132
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• 2020

In this article, we determine sufficient conditions on the parameters of a generalized convolution operator to ensure that it belongs to the Hardy space and to the space of bounded analytic functions. We exhibit the utility of these results by deducing several interesting examples.

• Korean Journal of Mathematics
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• v.13 no.2
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• pp.193-202
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• 2005

We define a general space $H_{{\omega},p}$ of the Hardy space and improve that Aleman's results to the space $H_{{\omega},p}$. It follows that the multiplication operator on this space is cellular indecomposable and that each invariant subspace contains nontrivial bounded functions.

• Bulletin of the Korean Mathematical Society
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• v.46 no.6
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• pp.1229-1236
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• 2009

We introduce two kinds of quasi-inner functions. Since every rationally invariant subspace for a shift operator S$_K$ on a vector-valued Hardy space H$^2$(${\Omega}$, K) is generated by a quasi-inner function, we also provide relationships of quasi-inner functions by comparing rationally invariant subspaces generated by them. Furthermore, we discuss fundamental properties of quasi-inner functions and quasi-inner divisors.

• Bulletin of the Korean Mathematical Society
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• v.60 no.6
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• pp.1567-1605
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• 2023

Let d ∈ ℕ and α ∈ (0, min{2, d}). For any a ∈ [a^*, ∞), the fractional Schrödinger operator 𝓛_a is defined by 𝓛_a := (-Δ)^α/2 + a|x|^-α, where $a^*:={\frac{2^{\alpha}{\Gamma}((d+{\alpha})/4)^2}{{\Gamma}(d-{\alpha})/4)^2}}$. In this paper, we study two-weight Sobolev inequalities associated with 𝓛_a and two-weight norm estimates for several square functions associated with 𝓛_a.

• Journal of the Korean Mathematical Society
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• v.53 no.4
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• pp.847-868
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• 2016

Let M be an invariant subspace of $H^2$ over the bidisk. Associated with M, we have the fringe operator $F^M_z$ on $M{\ominus}{\omega}M$. It is studied the Fredholmness of $F^M_z$ for (generalized) zero based invariant subspaces M. Also ker $F^M_z$ and ker $(F^M_z)^*$ are described.

• Communications of the Korean Mathematical Society
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• v.39 no.1
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• pp.149-160
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• 2024

In this paper, we give sufficient conditions for the generalized Ornstein-Uhlenbeck operators perturbed by regular potentials and inverse square potentials A_Φ,G,V,c=∆-∇Φ·∇+G·∇-V+c|x|^-2 with a suitable domain generates a quasi-contractive, positive and analytic C₀-semigroup in L^p(ℝ^N , e^-Φ(x)dx), 1 < p < ∞. The proofs are based on an L^p-weighted Hardy inequality and perturbation techniques. The results extend and improve the generation theorems established by Metoui [7] and Metoui-Mourou [8].

• Kyungpook Mathematical Journal
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• v.49 no.1
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• pp.31-39
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• 2009

Let $T_{\rho}$ be the generalized fractional integral operator associated to a function ${\rho}:(0,{\infty}){\rightarrow}(0,{\infty})$, as defined in [16]. For a function W on $\mathbb{R}^n$, we shall be interested in the boundedness of the multiplication operator $f{\mapsto}W{\cdot}T_{\rho}f$ on generalized Morrey spaces. Under some assumptions on ${\rho}$, we obtain an inequality for $W{\cdot}T_{\rho}$, which can be viewed as an extension of Olsen's and Kurata-Nishigaki-Sugano's results.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.1
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• pp.35-42
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• 2012

In this paper we first introduce a Carleson inequality and study the generalized form of Carleson inequality in the context of spaces of homogeneous type. The previous inequality is known to play important roles in harmonic analysis.