• East Asian mathematical journal
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• v.34 no.1
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• pp.17-20
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• 2018

Hardy-Bennett inequality is an inequality of Hardy-type having logarithmic weight. It appeared in 1973, and several generalizations followed. We, in this note, present another generalization with a simple proof.

• East Asian mathematical journal
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• v.34 no.5
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• pp.549-570
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• 2018

We establish existence and bifurcation of global positive solutions for parametrized nonhomogeneous elliptic equations involving critical Sobolev-Hardy exponents and Hardy terms. The main approach to the problem is the variational method.

• Kyungpook Mathematical Journal
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• v.46 no.3
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• pp.315-328
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• 2006

In this paper, the Boundedness for the multilinear Littlewood-Paley operator on certain Hardy and Herz-Hardy spaces are obtained.

• Journal of the Korean Mathematical Society
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• v.40 no.1
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• pp.41-60
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• 2003

In this paper, the continuity for the commutators of Littlewood-Paley operators on certain Hardy and Herz-Hardy spaces are obtained.

• Bulletin of the Korean Mathematical Society
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• v.60 no.5
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• pp.1409-1425
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• 2023

In this paper, we are devoted to studying the mixed radial-angular integrabilities for Hardy type operators. As an application, the upper and lower bounds are obtained for the fractional Hardy operator. In addition, we also establish the sharp weak-type estimate for the fractional Hardy operator.

• Bulletin of the Korean Mathematical Society
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• v.59 no.5
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• pp.1215-1235
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• 2022

In this paper we study Szegö kernel projections for Hardy spaces in quaternionic Clifford analysis. At first we introduce the matrix Szegö projection operator for the Hardy space of quaternionic Hermitean monogenic functions by the characterization of the matrix Hilbert transform in the quaternionic Clifford analysis. Then we establish the Kerzman-Stein formula which closely connects the matrix Szegö projection operator with the Hardy projection operator onto the Hardy space, and we get the matrix Szegö projection operator in terms of the Hardy projection operator and its adjoint. At last, we construct the explicit matrix Szegö kernel function for the Hardy space on the sphere as an example, and get the solution to a Diriclet boundary value problem for matrix functions.

• Journal of the Korean Mathematical Society
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• v.59 no.5
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• pp.927-944
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• 2022

Let $\vec{p}{\in}(0,\;1]^n$ be an n-dimensional vector and A a dilation. Let $H^{\vec{p}}_A(\mathbb{R}^n)$ denote the anisotropic mixed-norm Hardy space defined via the radial maximal function. Using the known atomic characterization of $H^{\vec{p}}_A(\mathbb{R}^n)$ and establishing a uniform estimate for corresponding atoms, the authors prove that the Fourier transform of $f{\in}H^{\vec{p}}_A(\mathbb{R}^n)$ coincides with a continuous function F on ℝⁿ in the sense of tempered distributions. Moreover, the function F can be controlled pointwisely by the product of the Hardy space norm of f and a step function with respect to the transpose matrix of A. As applications, the authors obtain a higher order of convergence for the function F at the origin, and an analogue of Hardy-Littlewood inequalities in the present setting of $H^{\vec{p}}_A(\mathbb{R}^n)$.

• Journal of the Korean Mathematical Society
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• v.57 no.3
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• pp.603-616
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• 2020

In the paper, we establish the validity of the weighted discrete and integral Hardy inequalities with periodic weights and find the best possible constants in these inequalities. In addition, by applying the established discrete Hardy inequality to a certain second-order difference equation, we discuss some oscillation and nonoscillation results.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.1
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• pp.43-46
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• 2018

It is concerned with the continuity of the Hardy-Little wood maximal function between the classical Lebesgue spaces or the Orlicz spaces. A new approach to the continuity of the Hardy-Littlewood maximal function is presented through the observation that the continuity is closely related to the existence of solutions for a certain type of first order ordinary differential equations. It is applied to verify the continuity of the Hardy-Littlewood maximal function from $L^p({\mathbb{R}}^n)$ to $L^q({\mathbb{R}}^n)$ for 1 ${\leq}$ q < p < ${\infty}$.

• Journal of Microbiology and Biotechnology
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• v.30 no.6
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• pp.912-919
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• 2020

Hardy kiwifruits (Actinidia arguta Planch.) have high amounts of antioxidants, including ascorbic acid (vitamin C) and phenolics. The anti-cholinesterase activity and neuroprotective effects of three different cultivars of hardy kiwifruits, cv. Mansu (A. arguta × A. deliciosa), cv. Haeyeon (A. arguta), and cv. Chiak (A. arguta), on PC-12 and SH-SY5Y cells were evaluated. Extraction of phenolics and vitamin C was carried out using 80% (v/v) aqueous ethanol and metaphosphoric acid assisted with homogenization, respectively. Hardy kiwifruit of cv. Mansu showed higher total phenolic, total flavonoid, and vitamin C contents and antioxidant capacity compared to the other two cultivars of hardy kiwifruits, cv. Haeyeon and cv. Chiak. Analysis of high-performance liquid chromatography results revealed the presence of procyanidin B2, (-)-epicatechin, neochlorogenic acid, cryptochlorogenic acid, rutin, hyperoside, isoquercitrin, and astragalin in hardy kiwifruits. The three cultivars of hardy kiwifruits had a wide range of vitamin C content of 55.2-130.0 mg/100 g fresh weight. All three cultivars of hardy kiwifruits had protective effects on neuronal PC-12 and SH-SY5Y cells exposed to hydrogen peroxide by increasing cell viability and reducing intracellular oxidative stress. Furthermore, the hardy kiwifruits inhibited acetylcholinesterase and butyrylcholinesterase. Collectively, these results suggest that hardy kiwifruits rich in antioxidants like phenolics and vitamin C have good potential as functional materials in neuroprotective applications.