Rubio de Francia proved a one-sided Littlewood-Paley inequality for the square function constructed from an arbitrary system of disjoint intervals. Later, Osipov proved a similar inequality for Walsh systems. We prove a similar inequality for more general Vilenkin systems. Bibliography: 11 titles.
Rubio de Francia证明了由任意不相交区间系统构造的平方函数的单侧Littlewood-Paley不等式。后来，奥西波夫证明了沃尔什系统的一个类似不等式。我们对更一般的Vilenkin系统证明了一个类似的不等式。参考书目:11篇。
{"title":"A Littlewood-Paley-Rubio de Francia inequality for bounded Vilenkin systems","authors":"A. Tselishchev","doi":"10.1070/SM9482","DOIUrl":"https://doi.org/10.1070/SM9482","url":null,"abstract":"Rubio de Francia proved a one-sided Littlewood-Paley inequality for the square function constructed from an arbitrary system of disjoint intervals. Later, Osipov proved a similar inequality for Walsh systems. We prove a similar inequality for more general Vilenkin systems. Bibliography: 11 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":"925 1","pages":"1491 - 1502"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77541600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bykovskii (2002) obtained the best current upper estimate for the minimum discrepancy of the Korobov lattice points from the uniform distribution. We show that this estimate holds for almost all -dimensional Korobov lattices of nodes, where , and is a prime number. Bibliography: 14 titles.
{"title":"A probability estimate for the discrepancy of Korobov lattice points","authors":"A. A. Illarionov","doi":"10.1070/SM9522","DOIUrl":"https://doi.org/10.1070/SM9522","url":null,"abstract":"Bykovskii (2002) obtained the best current upper estimate for the minimum discrepancy of the Korobov lattice points from the uniform distribution. We show that this estimate holds for almost all -dimensional Korobov lattices of nodes, where , and is a prime number. Bibliography: 14 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":"27 Suppl 1 1","pages":"1571 - 1587"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80309227","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A family of Cauchy-Dirichlet problems for the wave equations in unbounded domains is considered (here is a small parameter); a scattering operator is associated with each domain . For 0$?> the boundaries of the are smooth, while the boundary of the limit domain contains a conical point. The asymptotics of as is determined. Bibliography: 11 titles.
{"title":"Asymptotics of the scattering operator for the wave equation in a singularly perturbed domain","authors":"D. Korikov","doi":"10.1070/SM9462","DOIUrl":"https://doi.org/10.1070/SM9462","url":null,"abstract":"A family of Cauchy-Dirichlet problems for the wave equations in unbounded domains is considered (here is a small parameter); a scattering operator is associated with each domain . For 0$?> the boundaries of the are smooth, while the boundary of the limit domain contains a conical point. The asymptotics of as is determined. Bibliography: 11 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":"66 1","pages":"1436 - 1470"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81537971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
V. Bogachev, T. I. Krasovitskii, S. V. Shaposhnikov
The paper gives a solution to the long-standing problem of uniqueness for probability solutions to the Cauchy problem for the Fokker- Planck-Kolmogorov equation with an unbounded drift coefficient and unit diffusion coefficient. It is proved that in the one-dimensional case uniqueness holds and in all other dimensions it fails. The case of nonconstant diffusion coefficients is also investigated. Bibliography: 70 titles.
{"title":"On uniqueness of probability solutions of the Fokker-Planck-Kolmogorov equation","authors":"V. Bogachev, T. I. Krasovitskii, S. V. Shaposhnikov","doi":"10.1070/SM9427","DOIUrl":"https://doi.org/10.1070/SM9427","url":null,"abstract":"The paper gives a solution to the long-standing problem of uniqueness for probability solutions to the Cauchy problem for the Fokker- Planck-Kolmogorov equation with an unbounded drift coefficient and unit diffusion coefficient. It is proved that in the one-dimensional case uniqueness holds and in all other dimensions it fails. The case of nonconstant diffusion coefficients is also investigated. Bibliography: 70 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":"125 26 1","pages":"745 - 781"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91087910","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the homogenization of a fourth-order divergent elliptic operator with rapidly oscillating -periodic coefficients, where is a small parameter. The homogenized operator is of the same type and has constant coefficients. We apply Zhikov’s shift method to obtain an estimate in the -operator norm of order for the difference of the resolvents and . Bibliography: 25 titles.
{"title":"Approximation of resolvents in homogenization of fourth-order elliptic operators","authors":"S. Pastukhova","doi":"10.1070/SM9413","DOIUrl":"https://doi.org/10.1070/SM9413","url":null,"abstract":"We study the homogenization of a fourth-order divergent elliptic operator with rapidly oscillating -periodic coefficients, where is a small parameter. The homogenized operator is of the same type and has constant coefficients. We apply Zhikov’s shift method to obtain an estimate in the -operator norm of order for the difference of the resolvents and . Bibliography: 25 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":"21 1","pages":"111 - 134"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75039105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Finite-dimensional problems of minimizing a strongly or weakly convex function on a strongly or weakly convex set are considered. Necessary and sufficient conditions for solutions of such problems are presented, which are based on estimates for the behaviour of the objective function on the feasible set taking account of the parameters of strong or weak convexity, as well as certain local features of the set and the function. The mathematical programming problem is considered separately for strongly and weakly convex functions. In addition, necessary conditions for a global and a local solution with differentiable objective function are found, in which a ‘strong’ stationarity condition is assumed to hold. Bibliography: 33 titles.
{"title":"Characterization of solutions of strong-weak convex programming problems","authors":"S. Dudov, M. Osiptsev","doi":"10.1070/SM9431","DOIUrl":"https://doi.org/10.1070/SM9431","url":null,"abstract":"Finite-dimensional problems of minimizing a strongly or weakly convex function on a strongly or weakly convex set are considered. Necessary and sufficient conditions for solutions of such problems are presented, which are based on estimates for the behaviour of the objective function on the feasible set taking account of the parameters of strong or weak convexity, as well as certain local features of the set and the function. The mathematical programming problem is considered separately for strongly and weakly convex functions. In addition, necessary conditions for a global and a local solution with differentiable objective function are found, in which a ‘strong’ stationarity condition is assumed to hold. Bibliography: 33 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":"97 4 1","pages":"782 - 809"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83602437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The class of Smale regular homeomorphisms of closed topological manifolds, with nonwandering set consisting of a finite number of periodic orbits of hyperbolic type, is considered. This class contains the Morse-Smale diffeomorphisms of smooth closed manifolds. For two Smale regular homomorphisms necessary and sufficient conditions for being conjugate are presented. Bibliography: 26 titles.
{"title":"Necessary and sufficient conditions for the conjugacy of Smale regular homeomorphisms","authors":"E. Zhuzhoma, V. Medvedev","doi":"10.1070/SM9244","DOIUrl":"https://doi.org/10.1070/SM9244","url":null,"abstract":"The class of Smale regular homeomorphisms of closed topological manifolds, with nonwandering set consisting of a finite number of periodic orbits of hyperbolic type, is considered. This class contains the Morse-Smale diffeomorphisms of smooth closed manifolds. For two Smale regular homomorphisms necessary and sufficient conditions for being conjugate are presented. Bibliography: 26 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":"62 1","pages":"57 - 69"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88863853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider continuous mappings between two Banach spaces that depend on a parameter with values in a topological space. These mappings are assumed to be continuously differentiable for each value of the parameter. Under normality (regularity) assumptions of the mappings under consideration, we obtain sufficient conditions for the existence of global and semilocal implicit functions. A priori estimates for solutions are given. As an application of these results, we obtain, in particular, a theorem on extending an implicit function from a given closed set to the whole parameter space and a theorem on coincidence points of mappings. Bibliography: 32 titles.
{"title":"Global and semilocal theorems on implicit and inverse functions in Banach spaces","authors":"A. Arutyunov, S. Zhukovskiy","doi":"10.1070/SM9483","DOIUrl":"https://doi.org/10.1070/SM9483","url":null,"abstract":"We consider continuous mappings between two Banach spaces that depend on a parameter with values in a topological space. These mappings are assumed to be continuously differentiable for each value of the parameter. Under normality (regularity) assumptions of the mappings under consideration, we obtain sufficient conditions for the existence of global and semilocal implicit functions. A priori estimates for solutions are given. As an application of these results, we obtain, in particular, a theorem on extending an implicit function from a given closed set to the whole parameter space and a theorem on coincidence points of mappings. Bibliography: 32 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":"41 1","pages":"1 - 41"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79366916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For unbounded simply connected domains in the complex plane, bounded by several simple curves with regular asymptotic behaviour at infinity, we obtain necessary conditions and sufficient conditions for simple partial fractions (logarithmic derivatives of polynomials) with poles on the boundary of to be dense in the space of holomorphic functions in (with the topology of uniform convergence on compact subsets of ). In the case of a strip bounded by two parallel lines, we give estimates for the convergence rate to zero in the interior of of simple partial fractions with poles on the boundary of and with one fixed pole. Bibliography: 16 titles.
{"title":"Approximation by simple partial fractions in unbounded domains","authors":"P. Borodin, K. Shklyaev","doi":"10.1070/SM9298","DOIUrl":"https://doi.org/10.1070/SM9298","url":null,"abstract":"For unbounded simply connected domains in the complex plane, bounded by several simple curves with regular asymptotic behaviour at infinity, we obtain necessary conditions and sufficient conditions for simple partial fractions (logarithmic derivatives of polynomials) with poles on the boundary of to be dense in the space of holomorphic functions in (with the topology of uniform convergence on compact subsets of ). In the case of a strip bounded by two parallel lines, we give estimates for the convergence rate to zero in the interior of of simple partial fractions with poles on the boundary of and with one fixed pole. Bibliography: 16 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":"20 2","pages":"449 - 474"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72481708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For simple trigonometric series it is shown, in particular, that if the trigonometric series is Riemann summable in measure to an integrable function and if the Riemann majorant is finite everywhere except possibly on a countable set, then this series is the Fourier series of the function . Uniqueness theorems for multiple trigonometric series are obtained on the basis of this result. Bibliography: 14 titles.
{"title":"Uniqueness theorems for simple trigonometric series with application to multiple series","authors":"G. G. Gevorkyan","doi":"10.1070/SM9525","DOIUrl":"https://doi.org/10.1070/SM9525","url":null,"abstract":"For simple trigonometric series it is shown, in particular, that if the trigonometric series is Riemann summable in measure to an integrable function and if the Riemann majorant is finite everywhere except possibly on a countable set, then this series is the Fourier series of the function . Uniqueness theorems for multiple trigonometric series are obtained on the basis of this result. Bibliography: 14 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":"25 1","pages":"1675 - 1693"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89815484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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