Reduction theorems for operators on the cones of monotone functions

For a quasilinear operator on the semiaxis a reduction theorem is proved on the cones of monotone functions in Lp - Lq setting for 0 < q < ∞, 1<= p < ∞. The case 0 < p < 1 is also studied for operators with additional properties. In particular, we obtain critera for three-weight inequalities for the Hardy-type operators with Oinarov' kernel on monotone functions in the case 0 < q < p <= 1

On a supremum operator

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Весовые неравенства для квазилинейных интегральных операторов на полуоси и приложения к пространствам Лоренца

A precise characterization of inequalities in weighted Lebesgue spaces with positive quasilinear integral operators of iterative type on the half-axis is given. All cases of positive integration parameters are treated, including the case of supremum. Applications to the solution of the well-known problem of the boundedness of the Hardy-Littlewood maximal operator in weighted Lorentz Γ-spaces are given.Дается точная характеризация неравенств в весовых пространствах Лебега на полуоси с положительными квазилинейными интегральными операторами итерационного типа. Рассмотрены все варианты положительных параметров суммирования, включая супремальный случай. Дано приложение к решению известной задачи об ограниченности максимального оператора Харди–Литтлвуда в весовых Γ-пространствах Лоренца

Reduction theorems for operators on the cones of monotone functions

For a quasilinear operator on the semiaxis a reduction theorem is proved on the cones of monotone functions in Lp - Lq setting for 0 &lt; q &lt; ∞, 1&lt;= p &lt; ∞. The case 0 &lt; p &lt; 1 is also studied for operators with additional properties. In particular, we obtain critera for three-weight inequalities for the Hardy-type operators with Oinarov' kernel on monotone functions in the case 0 &lt; q &lt; p &lt;= 1

Reduction theorems for operators on the cones of monotone functions

For a quasilinear operator on the semiaxis a reduction theorem is proved on the cones of monotone functions in Lp - Lq setting for 0 < q < ∞, 1<= p < ∞. The case 0 < p < 1 is also studied for operators with additional properties. In particular, we obtain critera for three-weight inequalities for the Hardy-type operators with Oinarov' kernel on monotone functions in the case 0 < q < p <= 1

Evaluation of the Modern State of Water Ecosystems and the Issues with Protecting Biological Resources During Development of the Kruzenshternskoye Gas Condensate Field

In the article, the results of the studies of the present state of freshwater ecosystems and their biotic components in the western part of the Yamal Peninsula are presented. Based on the evaluation of the structure of the communities of phytoplankton, zooplankton, benthos and whitefishes, the range of the problems related to the protection of biological resources at the development of the Kruzenshternskoye gas field is defined. Data on species composition and quantitative indicators of hydrobionts of different types of waterbodies and watercourses in the lower reaches of the Mordyyakha and Naduyyakha rivers basins are the basis for environmental monitoring of water objects at development and exploitation of the Kruzenshternskoye gas field. According to the monitoring program, evaluation of the fish fauna state and their food base on the territory of the Kruzenshternskoye gas condensate field (GCF), is present. The zones of rivers deltas are the most important areas of the salmonid and whitefishes valuable fish species feeding at the territory of Kruzenshternskoye GCF. In the cases where complete demolish of waterbodies and watercourses for construction of facilities for GCF does not occur, changes of quantitative and qualitative characteristics of communities of hydrobionts after cease of works are reversible. River ecosystems are restored within a more short period of time in comparison to lacustrine ones. On the basis of conducted comprehensive studies, the proposals for the protection of fisheries resources and monitoring of aquatic ecosystems are reported. Recommendations for reducing the anthropogenic impact on aquatic ecosystems in the development period are presented. The results of the investigation were used in the designing the environmental protection part of the Kruzenshternskoye deposit project. At present, the disturbances in the territory of Kruzenshternskoye deposit of gas does not impact the aquatic ecosystem