• Title/Summary/Keyword: rough kernels

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WEIGHTED Lp-BOUNDEDNESS OF SINGULAR INTEGRALS WITH ROUGH KERNEL ASSOCIATED TO SURFACES

  • Liu, Ronghui;Wu, Huoxiong
    • Journal of the Korean Mathematical Society
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    • v.58 no.1
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    • pp.69-90
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    • 2021
  • In this paper, we prove weighted norm inequalities for rough singular integrals along surfaces with radial kernels h and sphere kernels Ω by assuming h ∈ △γ(ℝ+) and Ω ∈ ����β(Sn-1) for some γ > 1 and β > 1. Here Ω ∈ ����β(Sn-1) denotes the variant of Grafakos-Stefanov type size conditions on the unit sphere. Our results essentially improve and extend the previous weighted results for the rough singular integrals and the corresponding maximal truncated operators.

ROUGH MAXIMAL SINGULAR INTEGRAL AND MAXIMAL OPERATORS SUPPORTED BY SUBVARIETIES

  • Zhang, Daiqing
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.2
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    • pp.277-303
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    • 2021
  • Under the rough kernels Ω belonging to the block spaces B0,qr (Sn-1) or the radial Grafakos-Stefanov kernels W����(Sn-1) for some r, �� > 1 and q ≤ 0, the boundedness and continuity were proved for two classes of rough maximal singular integrals and maximal operators associated to polynomial mappings on the Triebel-Lizorkin spaces and Besov spaces, complementing some recent boundedness and continuity results in [27, 28], in which the authors established the corresponding results under the conditions that the rough kernels belong to the function class L(log L)α(Sn-1) or the Grafakos-Stefanov class ����(Sn-1) for some α ∈ [0, 1] and �� ∈ (2, ∞).

WEIGHTED ESTIMATES FOR CERTAIN ROUGH OPERATORS WITH APPLICATIONS TO VECTOR VALUED INEQUALITIES

  • Liu, Feng;Xue, Qingying
    • Journal of the Korean Mathematical Society
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    • v.58 no.4
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    • pp.1035-1058
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    • 2021
  • Under certain rather weak size conditions assumed on the kernels, some weighted norm inequalities for singular integral operators, related maximal operators, maximal truncated singular integral operators and Marcinkiewicz integral operators in nonisotropic setting will be shown. These weighted norm inequalities will enable us to obtain some vector valued inequalities for the above operators.

Adsorption Characteristics of Short Grain Rough Rice (단립종 벼의 수분흡습특성)

  • 김종순;고학균;송대빈
    • Journal of Biosystems Engineering
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    • v.23 no.5
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    • pp.465-472
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    • 1998
  • In this study short gain rough rice(Chu-cheong) with initial moisture content of around 12%(w.b.) was exposed to 3 levels of relative humidity(70, 80 and 90%) and 3 levels of temperature(20, 25 and 3$0^{\circ}C$) of the air, in order to evaluate the adsorption characteristics of rough rice and the rate of cracked kernels which will serve as the basic data when developing the quality adjusting equipment. The result showed that the moisture content of rough rice increased rapidly during the early stages of moisture adsorption like other grains, and at least 70% of the adsorption occurred within the first 24 hours of exposure to the humid environment. Adsorption rate was more related to relative humidity than the temperature of air stream, and the higher the relative humidity, the higher the adsorption rate. And the Page's equation predicted best the adsorption process of this study. Experimental results for the crack generation during the adsorption process showed that the higher the relative humidity the more the cracked kernels, and that the temperature had little effect. An empirical equation was developed to predict the crack ratio for the conditions of this study, and Nishiyama model predicted better the crack generation than Hoerl model.

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Studies on the Productivity of Individual Leaf Blade of Paddy Rice (수도의엽신별 생육효과에 관한 연구)

  • Dong-Sam Cho
    • KOREAN JOURNAL OF CROP SCIENCE
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    • v.18
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    • pp.1-27
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    • 1975
  • Experiment I: A field experiment was conducted in an attempt to find the effect of top-dressing at heading time in different levels of nitrogen application and of different positioned leaf blades formed by the treatment of leaf defoliation at heading time on the ripening and the yield of rice. The results obtained are as follows: 1. Average number of ears per hill and average number of grains per ear in different levels of nitrogen application were increased as the amount of nitrogen applied was increased. while the rate of ripened grains the yield of rough rice and the weight of 1, 000 kernels of brown rice were decreased respectively as the amount of nitrogen applied was increased. 2. The rate of ripened grains and the weight of 1.000 kernels of brown rice in different levels of nitrogen, top-dressing at heading time were larger than those in control and increased. The yield of rough rice although statistically significant differences were not recognized, were numerically increased. 3. The rate of ripened grains, the yield of rough rice, the weight of 1, 000 kernels of brown rice and the rate of hulling in different treatments of leaf defoliation were remarkably decreased as the degree of leaf-defoliation became larger. 4. The rate of ripened grains, the yield of rough rice, the weight of 1, 000 kernels of brown rice and the rate of hulling in different combinations of number of remained leaves positioned differently, formed the order of $L_1(flag leaf)>L_2>L_3>L_4$ when only one leaf blade was remained, and were increased as the positions of leaves were higher when two leaf blades. were, remained. 5. In case of decrease in the number of leaf blades positioned differently, by the treatment of leaf. defoliation, rate of ripened grains, the yield of rough rice, the weight of 1, 000 kernels of brown rice and the rate of hulling were increased as the area of remained leaves became larger and the nitrogen content of a leaf blade was increased. 6. There was a tendency that the increase in the amount of fertilizer application made the rate of ripened grains and the weight of 1, 000 kernels of brown rice reduced in any number of remained leaf blades, but the application of top-dressing at heading. time resulted in the reverse tendency. The yield of rough rice showed a tendency to be increased as the amount of basal dressing and top-dressing increased and for the application of top-dressing at heading time, the yield of rough rice was less at the smaller number of those. 7. The productivity effect of the rate of ripened grains and the yield of brown rice covered by leaf blades was more than 50 per cent and that of the. weight of 1, 000 kernels of brown rice was not more than 1.0 percent. As the amount of nitrogen application increased the. effect of leaf blades on the rate of ripened. grains and the weight of 1, 000 kernels of brown rice was increased. The effect of leaf blades on the weight of brown rice was increased as the amount of basal dressing-application, but the effect was decreased as the amount of top-dressing at heading time increased, 8. The productivity effects of different positioned leaf blades on the rate of ripened grains, the yield of rough rice and the weight of 1, 000 kernels of brown rice were in order of $L_1(flag leaf)>L_2>L_3>L_4$ the productivity effects of $L_1$ and $L_2$ had a tendency to be increased as the amount of nitrogen applied was increased. Experiment II: A field experiment was done in order to disclose the effect of the time of nitrogen application on yield component and the effect of different positioned leaves formed by leaf defoliation at heading time on the rate of ripened grains and the yield of rice. The results obtained are as follows: 1. Average number of ears per hill was increased in the treatment of nitrogen application from basal dressing to 22 days before heading and in the treatment of application distributed weekly. Number of grains was increased in the treatment of nitrogen application from 36 days to 15 days before heading. The rate of ripened grains was, lower in the treatment of nitrogen application from top-dressing to 15 days before heading than in that of non-application, was higher in the treatment of nitrogen application within 8 days before heading, and was the lowest in that of application 29 days before heading. The yield of rough rice was the highest in the treatment of nitrogen application from 29 days to 22 days before heading. The weight of 1, 000 kernels of brown rice was a little high in the treatment of application from 29 days to 8 days before heading. 2. The rate of ripened grains the yield of rough rice, the weight of 1, 000 kernels of brown rice and the rate of hulling in different treatments of leaf defoliation were remarkably decreased as the degree of leaf defoliation got larger and there were highly significant differences among treatments. There was also a recognized interaction between the time of nitrogen application and leaf defoliation. 3. In relation to the rate of ripened grains, the weight of 1. 000 kernels of brown rice and the rate of hulling in different numbers of remained leaves positioned differently and their combinations, the yield components were in order of $L_1(flag leaf)>L_2>L_3>L_4$ when only one leaf was remained, which indicated that the components were increased as the leaf position got higher. When two laves were remained, the rate of ripened grains, the yield of rough rice and rate of hulling were high in case of the combinations of upper positioned leaves, and the increase in the weight of 1, 000 kernels of brown rice appeared to be affected most]y by flag leaf. When three leaf blades were remained similarly the components were increased with the combination of upper positioned leaf blades. 4. In case of decreased different positioned leaf blades by treatment of leaf defoliation, there was a significant positive regression between the leaf area, the dry matter weight of leaf blades and the nitrogen contents of leaf blades, and rate of ripened grains and the yield of rough rice, but there was no constant tendency between the former components and the weight of 1. 000 kernels of brown rice. 5. The closer the time of fertilizer application to heading time, the more the rate of ripened grains and the weight of 1, 000 kernels was decreased by defoliation, and the less were the remained leaf blades, the more remarkable was the tendency. The rate of ripened grains and the weight of 1. 000 kernels was increased by the top-dressing after heading time as the number of remained leaf blades. When the number of remained leaf blades was small the yield of rough rice was increased as the time of fertilizer application was closer to heading time. 6. Discussing the productivity effects of different organs in different times of nitrogen application, the productivity effect of a leaf blade on the rate of ripened grains was higher as the time of nitrogen application got later, and in the treatment of non-fertilization the productivity effect of a leaf blade and that of culm were the same. In the productivity effect on the yield of brown rice, the effect of culm covered more than 50 percent independently on the time of nitrogen application, and the tendency was larger in the treatment of non-fertilizer. The productivity effect of culm on the weight of 1. 000 kernels of brown rice was more than 90 percent, and the productivity effect of a leaf blade was increased as the time of application got later. 7. The productivity effect of a leaf blade in different positions on the rate of ripened grains, the yield of rough rice and the weight of 1, 000 kernels of brown rice had a tendency to be increased as the time of application got later and as the position of leaf blades got higher. In the treatment of weekly application through the entire growing period, the rate of ripened grains and the yield of rough rice were affected by flag leaf and the second leaf at the same level, the but the weight of 1, 000 kernels of brown rice was affected by flag leaf with more than 60 percent of the yield of total leaves.

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Weighted LP Estimates for a Rough Maximal Operator

  • Al-Qassem, H.M.
    • 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].

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