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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.

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 introduce a class of Morrey-type spaces Mp,q,Ωλ, which includes the classical Morrey spaces and discuss their properties. We prove a Marcinkiewicz-type interpolation theorem for such spaces. Thi...

The species composition and biomass of the phytoplankton, as well as the hydrophysical and hydrochemical characteristics of the water masses, were estimated for different areas of the White Sea at 24 stations for the period of August 26 to September 3, 2007. The micronutrient concentrations were lower compared to the average long-term values for the summer period. The total biomass of phytoplankton (Bt) varied from 87 to 1914 mg C/m2. For most of the studied sites, Bt was low; dinoflagellates were the most abundant species. Diatoms dominated in the areas of elevated hydrodynamic activity, i.e., in the Gorlo (vertical mixing of the water layers), in intrusions of the Gorlo water masses into the basin of the White Sea, and in the peripheral areas of the Dvina outflow current. The Bt was also the highest in the listed areas.

Results of optical studies of the Kara Sea waters are considered. The data of ship borne measurements are compared with satellite observations. The maximum values of the beam attenuation coefficient, the chlorophyll, and the yellow substance fluorescence were observed in the Ob gulf and in the coastal waters west of the Yamal Peninsula. The minimum values were observed in the central and western parts of the Kara Sea. Frontal zones with sharp changes of the parameters involved were observed. The distribution of the seawater’s optical characteristics was closely related to the hydrological structure of the Kara Sea’s waters. The data of the shipborne measurements were consistent with the satellite observations.

The distribution of the suspended particulate matter concentration in the Kara Sea is analyzed based on ship and satellite data. The statistical relationships between the suspended matter concentration and the optical characteristics were revealed. Charts of the distribution of the suspended matter concentration in the Kara Sea were plotted, which confirmed that the application of optical methods for analyzing the spatial distribution of the suspended matter is effective. The most turbid waters were observed in the Ob Gulf, Yenisei Bay, the Baidaratskaya Gulf, and the adjacent regions. The smallest concentrations of suspended matter were observed in the central and western parts of the Kara Sea.

Production parameters of surface phytoplankton were measured along three transects: La Manche-Cape Town (I); Cape Town-54°S (II); 0°-49°W (along 54°S) (III). The Canary upwelling waters were most productive along transect I, where the surface chlorophyll a (Chl 0) and the surface primary production (PP 0) were as high as 4.3 mg/m3 and 173 mg C/m3 per day, respectively. Mosaic patterns in the distribution of these parameters were recorded in the northeastern regions of the South Subtropical Anticyclonic Gyre (Chl 0 = 0.03–0.35 mg/m3; PP 0 = 1.6–12.6 mg C/m3 per day). Along transect II, the average twofold southward increase in Chl 0 (from 0.2 to 0.4 mg/m3) and the concurrent decline of the phytoplankton assimilation activity ( AN 0) resulted in deviations from typical latitudinal changes inPP 0. At most sites, PP 0 values varied between 6 and 15 mg C/m3 per day. Negligible changes in Chl 0 (0.36–0.85 mg/m3), PP 0 (8–19 mg C/m3 per day), and AN 0 (0.7–1.6 mg C/mg chl a per hour) were registered for the oceanic waters along transect III. Along all the transects, PP 0 depended on Chl 0 to a greater extent than AN 0. The values of the latter parameter were largely determined by the water temperature and showed a slight correlation with the insolation. Along transect II, the integrated primary production (PP int) and the layer-integrated chlorophyll a in the upper 200 m (Chl 0–200) generally varied from 180 to 360 mg C/m2 per day and from 30 to 70 mg/m2, respectively. In the Polar Front region, an increase in Chl 0–200, PP int, Chl 0, and PP 0 up to respective values of 190 mg/m2, 520 mg C/m2 per day, 1.2 mg/m3, and 32 mg C/m3 per day was observed. A comparison of the water column (0–100 m) stability with the vertical distribution of the primary production and chlorophyll content along transect II implies that the thick (>100 m) upper mixed layer (UML) formed in response to the strong water cooling and wind forcing was largely responsible for the limited primary production in the Subantarctic and Antarctic regions. The large UML thickness resulted in an intense removal of plant cells from the photosynthetic layer and light starvation of a significant (up to 60%) part of the phytoplankton community.

In October and November 2002, high and relatively high values of the chlorophyll a concentration at the sea surface (C chl) were observed in the English Channel (0.47 mg/m3), in the waters of the North Atlantic Current (0.25 mg/m3), in the tropical and subtropical anticyclonic gyres (0.07–0.42 mg/m3), and also in the southwestern region of the southern subtropical anticyclonic gyre (usually 0.11–0.23 mg/m3). The central regions of the southern subtropical anticyclonic gyre (SATG) and the North Atlantic tropical gyre (NATR) were characterized by lower values of C chl (0.02–0.08 mg/m3 for the SATG and 0.07–0.14 mg/m3 for the NATR). At most of the SATG stations, the values of the surface primary production (C phs) varied from 2.5 to 5.5 mg C/m3 per day and were mainly defined by the fluctuations of C chl (r = +0.78) rather than by those of the assimilation number (r = +0.54). The low assimilation activity of phytoplankton in these waters (1.3–4.6 mg chl a per hour) pointed to a lack of nutrients. An analysis of the variability of their concentration and the composition of photosynthetic pigments showed that, in the waters north of 30° N, the growth of phytoplankton was mostly restricted by the deficiency of nitrogen, while, in more southern areas, at the majority of stations (about 60%), the phosphorus concentrations were the minimum. At the low concentrations of nitrates and nitrites, ammonium represented itself as a buffer that prevented planktonic algae from extreme degrees of nitric starvation. In the tropical waters and in the waters of the SATG, the primary production throughout the water column varied from 240 to 380 mg C/m2 30° per day. This level of productivity at stations with low values of C chl (

The goal of this paper is to present in situ data for the reflectance and transparency of river plumes and to interpret the observed variability in terms of hydrodynamic processes. The data was obtained for river plumes in the Black Sea and in Arctic Ocean regions. Spectral reflectance was measured in the 400-700 nm band by a ship/helicopter-mounted spectroradiometer and was supplemented by shipboard measurements of vertical profiles of the light attenuation coefficient. Remotely-sensed data shows the transition from freshwater to seawater with both sharp boundary fronts and wide mixing zones. Internal fronts and quasi-periodic spatial variability of reflectance were also observed within the river plume. The observations are related to the relevant hydrodynamic processes (fronts, advection and settling of particulate matter, internal waves). A radiance model was developed to match the vertical profile of optical properties to the observed variation of remotely-sensed reflectance.

New interpolation theorems are formulated for the class of operators that map the cone of positive functions into the cone of monotonic decreasing functions. These theorems are based on the concept of the Kc-functional. A formula for calculating the Kc-functional for some pairs of spaces is suggested. An example of an interpolation pair of spaces is considered in which the cones obtained with the help of Peetre's K-functional are different from those obtained using the Kc-functional.

This article is concerned with extension operators for the Sobolev spaces , where is a domain in with boundary in a Lipschitz class. Two-sided bounds are obtained for the minimum norm of an extension operator that are exact with respect to the smoothness parameter .

The paper deals with convolution type integral equations of the first kind. For exact solutions it is typical that the solutions are discontinuous or unbounded. The aim of the paper is to construct approximate solutions which may also be discontinuous or unbounded. A priori information about the solution and estimates for the approximate solutions, as well as for the differences between the approximate solution and the exact one, are given in terms of Nikol'skii-Besov spaces with differentiation of fractional orders. Bibliography: 8 titles.

This article is devoted to the investigation of some properties of Sobolev spaces with mixed norm.

A study of the convergence of the differentiation formula ′(f(A))′=f′ (A)A′+fm(A)/21[A′A]+fm(A)/3l[[A′A]A]'+... where [XY]=XY−YX, and A=A(t) is a function of the real variable t with values in a Banach algebra.