• Journal of the Korean Mathematical Society
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• v.59 no.2
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• pp.235-254
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• 2022

Let ℍⁿ be the Heisenberg group and Q = 2n + 2 be its homogeneous dimension. Let 𝓛 = -∆_ℍⁿ + V be the Schrödinger operator on ℍⁿ, where ∆_ℍⁿ is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class $B_{q_1}$ for q₁ ≥ Q/2. Let H^p_𝓛(ℍⁿ) be the Hardy space associated with the Schrödinger operator 𝓛 for Q/(Q+𝛿₀) < p ≤ 1, where 𝛿₀ = min{1, 2 - Q/q₁}. In this paper, we consider the Hardy type estimates for the operator T_𝛼 = V^𝛼(-∆_ℍⁿ + V )^-𝛼, and the commutator [b, T_𝛼], where 0 < 𝛼 < Q/2. We prove that T_𝛼 is bounded from H^p_𝓛(ℍⁿ) into L^p(ℍⁿ). Suppose that b ∈ BMO^𝜃_𝓛(ℍⁿ), which is larger than BMO(ℍⁿ). We show that the commutator [b, T_𝛼] is bounded from H¹_𝓛(ℍⁿ) into weak L¹(ℍⁿ).

• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1117-1127
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• 2019

Let ${\mathcal{L}}_2=(-{\Delta})^2+V^2$ be the $Schr{\ddot{o}}dinger$ type operator, where nonnegative potential V belongs to the reverse $H{\ddot{o}}lder$ class $RH_s$, s > n/2. In this paper, we consider the operator $T_{{\alpha},{\beta}}=V^{2{\alpha}}{\mathcal{L}}^{-{\beta}}_2$ and its conjugate $T^*_{{\alpha},{\beta}}$, where $0<{\alpha}{\leq}{\beta}{\leq}1$. We establish the $(L^p,\;L^q)$-boundedness of operator $T_{{\alpha},{\beta}}$ and $T^*_{{\alpha},{\beta}}$, respectively, we also show that $T_{{\alpha},{\beta}}$ is bounded from Hardy type space $H^1_{L_2}({\mathbb{R}}^n)$ into $L^{p_2}({\mathbb{R}}^n)$ and $T^*_{{\alpha},{\beta}}$ is bounded from $L^{p_1}({\mathbb{R}}^n)$ into BMO type space $BMO_{{\mathcal{L}}1}({\mathbb{R}}^n)$, where $p_1={\frac{n}{4({\beta}-{\alpha})}}$, $p_2={\frac{n}{n-4({\beta}-{\alpha})}}$.

• Bulletin of the Korean Mathematical Society
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• v.58 no.1
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• pp.235-251
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• 2021

We consider the Schrödinger type operator ��k = (-∆)k+Vk on ℝn(n ≥ 2k + 1), where k = 1, 2 and the nonnegative potential V belongs to the reverse Hölder class RHs with n/2 < s < n. In this paper, we establish the (Lp, Lq)-boundedness of the higher order Riesz transform T��,�� = V2��∇2��-��2 (0 ≤ �� ≤ 1/2 < �� ≤ 1, �� - �� ≥ 1/2) and its adjoint operator T∗��,�� respectively. We show that T��,�� is bounded from Hardy type space $H^1_{\mathcal{L}_2}({\mathbb{R}}_n)$ into Lp2 (ℝn) and T∗��,�� is bounded from ��p1 (ℝn) into BMO type space $BMO_{\mathcal{L}_1}$ (ℝn) when �� - �� > 1/2, where $p_1={\frac{n}{4({\beta}-{\alpha})-2}}$, $p_2={\frac{n}{n-4({\beta}-{\alpha})+2}}$. Moreover, we prove that T��,�� is bounded from $BMO_{\mathcal{L}_1}({\mathbb{R}}_n)$ to itself when �� - �� = 1/2.

• Kyungpook Mathematical Journal
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• v.45 no.2
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• pp.255-272
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• 2005

This paper is concerned with studying the weighted $L^P$ boundedness of a class of maximal operators related to homogeneous singular integrals with rough kernels. We obtain appropriate weighted $L^P$ bounds for such maximal operators. Our results are extensions and improvements of the main theorems in [2] and [5].

• Journal of the Korean Mathematical Society
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• v.32 no.4
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• pp.697-712
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• 1995

In the present article we consider multiparameter maximal averages and discover the crucial roles played by the number of parameters in their boundedness properties. The problem we shall deal with is initiated by Rubio de Francia [8] and will be in the spirit of an inductive extension to multiparameter cases, in which tools of our study rely on the theory of Harmonic Analysis on product spaces. Suppose that $d_\mu$ is a complex Borel measure supported on a compact subset S of $R^N$ having total mass one, $\smallint_S d_\mu = 1$.

• 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.

• Journal of Chest Surgery
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• v.29 no.4
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• pp.381-385
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• 1996

From April 1984 to April 1990, seven patients with Ebstein's anomaly underwent surgical repair. Mean age at operation was 18.6 years (range, 2 to 46 years). Operations were performed using hypothermic cardiopulmonary bypass. Surgical procedures included tricupid valve replacement (n:6) and tricuspid valve reconstruction (n: 1). There were two hospital deaths. There have been no late death. All survivors are in New York Heart Association class I or II with a median follow-up of 6.2 years(range, 4 to 8.3 years).

• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1443-1455
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• 2017

Let $M_{z^N}(N{\in}{\mathbb{Z}}^d_+)$ be a bounded multiplication operator on a class of Hilbert spaces with orthogonal basis $\{z^n:n{\in}{\mathbb{Z}}^d_+\}$. In this paper, we prove that each reducing subspace of $M_{z^N}$ is the direct sum of some minimal reducing subspaces. For the case that d = 2, we find all the minimal reducing subspaces of $M_{z^N}$ ($N=(N_1,N_2)$, $N_1{\neq}N_2$) on weighted Bergman space $A^2_{\alpha}({\mathbb{B}}_2)$(${\alpha}$ > -1) and Hardy space $H^2({\mathbb{B}}_2)$, and characterize the structure of ${\mathcal{V}}^{\ast}(z^N)$, the commutant algebra of the von Neumann algebra generated by $M_{z^N}$.

• Communications of the Korean Mathematical Society
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• v.38 no.2
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• pp.401-430
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• 2023

In this paper, we prove L^p estimates of a class of singular integral operators on product domains along surfaces defined by mappings that are more general than polynomials and convex functions. We assume that the kernels are in L(log L)² (𝕊^n-1 × 𝕊^m-1). Furthermore, we prove L^p estimates of the related class of Marcinkiewicz integral operators. Our results extend as well as improve previously known results.

• Proceedings of the Korean Society of Near Infrared Spectroscopy Conference
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• 2001.06a
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• pp.1253-1253
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• 2001

The ability to accurately assess wine quality is important during the wine making process, particularly when allocating batches of wines to styles determined by consumer requirements. Grape payments are often determined by the quality category of the wine that is produced from them. Wine quality, in terms of sensory characteristics, is normally a subjective measure, performed by experienced winemakers, wine competition judges or winetasting panellists. By nature, such assessments can be biased by individual preferences and may be subject to day-to-day variation. Taste and aroma compounds are often present in concentrations below the detection limit of near infrared (NIR) spectroscopy but the more abundant organic compounds offer potential for objective quality grading by this technique. Samples were drawn from one of Australia's major wine shows and from BRL Hardy's post-vintage wine quality allocation tastings. The samples were scanned in transmission mode with a FOSS NIR Systems 6500, over the wavelength range 400-2500 ㎚. Data analysis was performed with the Vision chemometrics package. With samples from the allocation tastings, the best correlations between NIR spectra and tasting data were obtained with dry red wines. These calibrations used loadings in the wavelengths related to anthocyanins, ethanol and possibly tannins. Anthocyanins are a group of compounds responsible for colour in red wines - restricting the wavelengths to those relating to anthocyanins produced calibrations of similar accuracy to those using the full wavelength range. This was particularly marked with Merlot, a variety that tends to have relatively lower anthocyanin levels than Cabernet Sauvignon and Shiraz. For dry white wines, calibrations appeared to be more dependent on ethanol characteristics of the spectrum, implying that quality correlated with fruit maturity. The correlations between NIR spectra and sensory data obtained using the wine show samples were less significant in general. This may be related to the fact that within most classes in the show, the samples may span vintages, glowing areas and winemaking styles, even though they may be made from only one grape variety. For dry red wines, the best calibrations were obtained with a class of Pinot Noir - a variety that tends to be produced in limited areas in Australia and would represent the least matrix variation. Good correlations were obtained with a tawny port class - these wines are sweet, fortified wines, that are aged for long periods in wooden barrels. During the ageing process Maillard browning compounds are formed and the water is lost through the barrels in preference to ethanol, producing “concentrated” darkly coloured wines with high alcohol content. These calibrations indicated heaviest loadings in the water regions of the spectrum, suggesting that “concentration” of the wines was important, whilst the visible and alcohol regions of the spectrum also featured as important factors. NIR calibrations based on sensory scores will always be difficult to obtain due to variation between individual winetasters. Nevertheless, these results warrant further investigation and may provide valuable Insight into the main parameters affecting wine quality.