In this paper, we establish sufficient conditions for the pre-compactness of sets in the global Morrey-type spaces GMpθw(·). Our main result is the compactness of the commutators of the Riesz potential b,Iα in global Morrey-type spaces from GMp1θ1w1(·) to GMp2θ2w2(·). We also present new sufficient conditions for the commutator b,Iα to be bounded from GMp1θ1w1(·) to GMp2θ2w2(·). In the proof of the theorem regarding the compactness of the commutator for the Riesz potential, we primarily utilize the boundedness condition for the commutator for the Riesz potential b,Iα in global Morrey-type spaces GMpθw(·), and the sufficient conditions derived from the theorem on pre-compactness of sets in global Morrey-type spaces GMpθw(·).

We prove analogs of Bernstein’s inequalities and inequalities of different metrics and different dimensions for entire functions of exponential type. Such inequalities are well known for Lebesgue spaces. In this paper we prove them for Morrey spaces.

In this paper, we introduce a new version of the definition of a quasi-norm (in particular, a norm) in Lebesgue spaces with variable order of summability. Using it, we prove an analogue of Holders inequality for such spaces, which is more general and more precise than those known earlier.

For all admissible values of the numerical parameters sharp sufficient conditions on the functional parameters are obtained ensuring the boundedness of the generalized Riesz potential from one general local Morrey-type space to another one, which, for a certain range of the numerical parameters, coincide with the necessary ones.

Выполнена сравнительная оценка концентрации и состава взвеси по данным экспедиций за последние 20 лет в Белое, Баренцево и Карское моря, находящихся под влиянием регионального изменения климата с 2000-х гг. Разнородное происхождение частиц взвеси и различающиеся условия седиментации изученных морей определяют существенную вариабельность оптических свойств морской воды. Необходима разработка региональных и сезонных алгоритмов, связывающих натурные и дистанционные измерения биооптических характеристик морской воды.

The phytoplankton structure in ecologically diverse areas of the sea and the role of marine invasive species in the formation of the community were studied on the meridional and latitudinal trans-Caspian sections in the autumn period of 2008, 2009, and 2012. It was established that the transformation of the phytoplankton community continues in the Caspian Sea, associated with the entry of the Black Sea diatoms Chaetoceros peruvianus, Cerataulina pelagica, and Pseudo-nitzschia seriata into their composition. During the autumn bloom of phytoplankton in November 2008, the number of invasive species reached 1.3-2.3 × 10 cells/L, C. peruvianus and P. seriata to the total weight biomass of the community reached 49-50%. The highest abundance of invasive species was recorded in the shelf areas of the Middle Caspian Sea in its eastern ( C. peruvianus , 2.3 × 10 cells/L), western ( P. seriata , 1.4 × 10 cells/L), and northern ( C. pelagica , 1.3 × 10 cells/L) parts. The north boundary of the distribution of these species in the Caspian Sea was the 5.0 isohaline, which coincided with the highly productive frontal zone (4.0-6.0) of the marginal filter of the Volga River. For the first time, it was shown that the C. peruvianus diatom was a part of the phytoplankton of the upper productive 25-30-meter water layer both in the Middle and in the Southern Caspian during the periods preceding the autumn bloom of phytoplankton (September-October). During that time the most considerable abundance of this species (1.6 × 10cells/L) was recorded in the shallow areas on the Apsheron Sill and the eastern part of the Middle Caspian in the zone of coastal wind upwelling.

We prove that block spaces defined on R-n with an arbitrary Radon measure, which are known to be the preduals of Morrey spaces, are closed under the first and the second complex interpolation method. The proof of our main theorem uses the duality theorem in the complex interpolation method, the complex interpolation of certain closed subspaces of Morrey spaces, a characterization of the preduals of block spaces, and some formulas related to the Calderon product.

В статье вводится новый вариант определения квази-нормы (в частности, нормы) в лебеговых пространствах с переменным порядком суммируемости и с его помощью доказывается аналог неравенства Гельдера для таких пространства, более общий и более точный по сравнению с известными ранее.

In the present paper, we shall give necessary and sufficient conditions for the boundedness of anisotropic fractional maximal commutator M d b,α on anisotropic local Morrey-type spaces, when b belongs to BM O spaces, by which some new characterizations for BM O spaces is obtained.As an application of this results we consider the boundedness of commutators of anisotropic fractional maximal operator [b, M d α ] on anisotropic global Morrey-type spaces.

We introduce a class of Morrey-type spaces , which includes the classical Morrey spaces and discuss their properties. We prove a Marcinkiewicz-type interpolation theorem for such spaces. This theorem is then applied to obtaining an analogue of O'Neil's inequality for convolutions and to proving the boundedness in the introduced Morrey-type spaces of the Riesz potential and singular integral operators.

In this paper, we present new interpolation theorems for nonlinear Urysohn integral operators. In particular, interpolation theorems of Marcinkiewicz-Calderon type and Stein-Weiss-Peetre type are obtained.

After 2000, interest in the Hausdorff operators grew, first in the sense of the variety of spaces on which these operators were considered. Here we give conditions ensuring the boundedness of such operators on Morrey-type spaces. The sharpness of the obtained results is studied, and classes of the Hausdorff operators are described for which the necessary and sufficient conditions coincide.

We prove estimates for the variation of the eigenvalues for a pair of self-adjoint elliptic differential operators in the case of diffeomorphic open sets.

The article describes the estimation of measurement error of seawater beam attenuation coefficient using the Monte Carlo method. Measurements of the beam attenuation coefficient and its vertical distribution are the most common type of hydro-optical investigations since it is widely used for the research of light distribution in sea medium. Moreover, the beam attenuation coefficient is closely related to the concentration of the suspended substance and may be used for its estimation. Usually, the measurements of beam attenuation coefficient are conducted with small instrument base in rather transparent waters. In this case, the method errors of the beam attenuation coefficient measurement are low. However, in practice, cases of very turbid waters (near-bottom nepheloid layers), where measurement errors of beam attenuation coefficient significantly increase due to multiple scattering within the instrument base, are often encountered. The estimation of such errors is reviewed in this work. The effect of scattering phase function on the considered errors is described to the greatest extent. Moreover, the effect of the instrument base length and angle of view of the receiving system on measurement errors of beam attenuation coefficient is studied. Corresponding estimations are provided.

Necessary and sufficient conditions on an open set Omega subset of( ) R-n are obtained ensuring that for l, m is an element of N-0, m < l the embedding (sic)(infinity)(l) (Omega) subset of W-infinity(m) (Omega) is compact, where (sic)(infinity)(m)(Omega) is the Sobolev space and (sic)(infinity)(l) (Omega) is the closure in W-infinity(l )(Omega) of the space of all infinitely continuously differentiable functions on Omega with supports compact in Omega.

In this note, the Morrey spaces and the Sobolev–Morrey spaces are considered. In particular, the K-functional with respect to these spaces is estimated from above and below. As an application, we characterize the Nikol’skii–Besov–Morrey spaces via real interpolation.

In this paper, the Kantorovich operators \(K_n, n\in \mathbb {N}\) are shown to be uniformly bounded in Morrey spaces on the closed interval [0, 1]. Also an upper estimate is obtained for the difference \(K_n(f)-f\) for functions f of regularity of order 1 measured in Morrey spaces. One of the key tools is the pointwise inequality for the Kantorovich operators and the Hardy–Littlewood maximal operator, which is of interest on its own and can be applied to other problems related to the Kantorovich operators.