We find sufficient conditions which are in a sense best possible that must be satisfied by the functions of an orthonormal system in order for the Fourier coefficients of functions of bounded variation to satisfy the hypotheses of the Men’shov–Rademacher theorem. We also prove a theorem saying that every system contains a subsystem with respect to which the Fourier coefficients of functions of bounded variation satisfy those hypotheses. The results obtained complement and generalize the corresponding results in [1].
{"title":"General Fourier coefficients and convergence almost everywhere","authors":"L. Gogoladze, G. Cagareishvili","doi":"10.1070/IM8985","DOIUrl":"https://doi.org/10.1070/IM8985","url":null,"abstract":"We find sufficient conditions which are in a sense best possible that must be satisfied by the functions of an orthonormal system in order for the Fourier coefficients of functions of bounded variation to satisfy the hypotheses of the Men’shov–Rademacher theorem. We also prove a theorem saying that every system contains a subsystem with respect to which the Fourier coefficients of functions of bounded variation satisfy those hypotheses. The results obtained complement and generalize the corresponding results in [1].","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"85 1","pages":"228 - 240"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58574022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Vladimir I. Bogachev, E. Kosov, Svetlana Nikolaevna Popova
We obtain broad conditions under which distributions of homogeneous functions in Gaussian and more general random variables have bounded densities or even densities of bounded variation or densities with finite Fisher information. Analogous results are obtained for convex functions. Applications to maxima of quadratic forms are given.
{"title":"On distributions of homogeneous and convex functions in Gaussian random variables","authors":"Vladimir I. Bogachev, E. Kosov, Svetlana Nikolaevna Popova","doi":"10.1070/IM9075","DOIUrl":"https://doi.org/10.1070/IM9075","url":null,"abstract":"We obtain broad conditions under which distributions of homogeneous functions in Gaussian and more general random variables have bounded densities or even densities of bounded variation or densities with finite Fisher information. Analogous results are obtained for convex functions. Applications to maxima of quadratic forms are given.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"85 1","pages":"852 - 882"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58576034","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article we regard spherical hypersurfaces in with a fixed Reeb vector field as -dimensional Sasakian manifolds. We establish a correspondence between three different sets of parameters, namely, those arising from representing the Reeb vector field as an automorphism of the Heisenberg sphere, those used in Stanton’s description of rigid spheres, and those arising from the rigid normal forms. We also describe geometrically the moduli space for rigid spheres and provide a geometric distinction between Stanton hypersurfaces and those found in [1]. Finally, we determine the Sasakian automorphism groups of rigid spheres and detect the homogeneous Sasakian manifolds among them.
{"title":"On the classification of -dimensional spherical Sasakian manifolds","authors":"D. Sykes, G. Schmalz, V. Ezhov","doi":"10.1070/IM9046","DOIUrl":"https://doi.org/10.1070/IM9046","url":null,"abstract":"In this article we regard spherical hypersurfaces in with a fixed Reeb vector field as -dimensional Sasakian manifolds. We establish a correspondence between three different sets of parameters, namely, those arising from representing the Reeb vector field as an automorphism of the Heisenberg sphere, those used in Stanton’s description of rigid spheres, and those arising from the rigid normal forms. We also describe geometrically the moduli space for rigid spheres and provide a geometric distinction between Stanton hypersurfaces and those found in [1]. Finally, we determine the Sasakian automorphism groups of rigid spheres and detect the homogeneous Sasakian manifolds among them.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"85 1","pages":"518 - 528"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58575130","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The structure of proper holomorphic maps with multiplicity higher than one from bounded Reinhardt domains in onto two-dimensional complex manifolds is described.
描述了从有界Reinhardt域到二维复流形的多重度大于1的真全纯映射的结构。
{"title":"Proper holomorphic maps of bounded two-dimensional Reinhardt domains. I","authors":"N. Kruzhilin","doi":"10.1070/IM9066","DOIUrl":"https://doi.org/10.1070/IM9066","url":null,"abstract":"The structure of proper holomorphic maps with multiplicity higher than one from bounded Reinhardt domains in onto two-dimensional complex manifolds is described.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"85 1","pages":"388 - 406"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58576000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the Cauchy problem for a model partial differential equation of third order with non-linearity of the form , where for and . We construct a fundamental solution for the linear part of the equation and use it to obtain analogues of Green’s third formula for elliptic operators, first in a bounded domain and then in unbounded domains. We derive an integral equation for classical solutions of the Cauchy problem. A separate study of this equation yields that it has a unique inextensible-in-time solution in weighted spaces of bounded and continuous functions. We prove that every solution of the integral equation is a local-in-time weak solution of the Cauchy problem provided that 3$?> . When , we use Pokhozhaev’s non-linear capacity method to show that the Cauchy problem has no local-in-time weak solutions for a large class of initial functions. When , this method enables us to prove that the Cauchy problem has no global-in-time weak solutions for a large class of initial functions.
{"title":"On critical exponents for weak solutions of the Cauchy problem for a non-linear equation of composite type","authors":"M. O. Korpusov, A. K. Matveeva","doi":"10.1070/IM8954","DOIUrl":"https://doi.org/10.1070/IM8954","url":null,"abstract":"We consider the Cauchy problem for a model partial differential equation of third order with non-linearity of the form , where for and . We construct a fundamental solution for the linear part of the equation and use it to obtain analogues of Green’s third formula for elliptic operators, first in a bounded domain and then in unbounded domains. We derive an integral equation for classical solutions of the Cauchy problem. A separate study of this equation yields that it has a unique inextensible-in-time solution in weighted spaces of bounded and continuous functions. We prove that every solution of the integral equation is a local-in-time weak solution of the Cauchy problem provided that 3$?> . When , we use Pokhozhaev’s non-linear capacity method to show that the Cauchy problem has no local-in-time weak solutions for a large class of initial functions. When , this method enables us to prove that the Cauchy problem has no global-in-time weak solutions for a large class of initial functions.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"85 1","pages":"705 - 744"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58573534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider Klein–Gordon equations in , , with constant or variable coefficients and study the Cauchy problem with random initial data. We investigate the distribution of a random solution at moments of time . We prove the convergence of correlation functions of the measure to a limit as . The explicit formulae for the limiting correlation functions and the energy current density (in mean) are obtained in terms of the initial covariance. Furthermore, we prove the weak convergence of to a limiting measure as . We apply these results to the case when the initial random function has the Gibbs distribution with different temperatures in some infinite “parts” of the space. In this case, we find states in which the limiting energy current density does not vanish. Thus, for the model being studied, we construct a new class of stationary non-equilibrium states.
{"title":"Convergence to stationary non-equilibrium states for Klein–Gordon equations","authors":"T. V. Dudnikova","doi":"10.1070/IM9044","DOIUrl":"https://doi.org/10.1070/IM9044","url":null,"abstract":"We consider Klein–Gordon equations in , , with constant or variable coefficients and study the Cauchy problem with random initial data. We investigate the distribution of a random solution at moments of time . We prove the convergence of correlation functions of the measure to a limit as . The explicit formulae for the limiting correlation functions and the energy current density (in mean) are obtained in terms of the initial covariance. Furthermore, we prove the weak convergence of to a limiting measure as . We apply these results to the case when the initial random function has the Gibbs distribution with different temperatures in some infinite “parts” of the space. In this case, we find states in which the limiting energy current density does not vanish. Thus, for the model being studied, we construct a new class of stationary non-equilibrium states.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"85 1","pages":"932 - 952"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58575027","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct an integrable function whose Fourier series possesses the following property. After an appropriate choice of signs of the coefficients of this series, the partial sums of the resulting series are dense in , .
{"title":"Functions universal with respect to the trigonometric system","authors":"M. Grigoryan, L. Galoyan","doi":"10.1070/IM8964","DOIUrl":"https://doi.org/10.1070/IM8964","url":null,"abstract":"We construct an integrable function whose Fourier series possesses the following property. After an appropriate choice of signs of the coefficients of this series, the partial sums of the resulting series are dense in , .","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"85 1","pages":"241 - 261"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58573600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the manifold of non-singular points of a stable real caustic germ of type and the manifolds of points of transversal intersection of its smooth branches consist only of contractible connected components. We also calculate the number of these components.
{"title":"On a real caustic of type","authors":"V. Sedykh","doi":"10.1070/IM9015","DOIUrl":"https://doi.org/10.1070/IM9015","url":null,"abstract":"We prove that the manifold of non-singular points of a stable real caustic germ of type and the manifolds of points of transversal intersection of its smooth branches consist only of contractible connected components. We also calculate the number of these components.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"24 1","pages":"279 - 305"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58574282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define a two-index scale , , of homeomorphisms of spatial domains in , the geometric description of which is determined by the control of the behaviour of the -capacity of condensers in the target space in terms of the weighted -capacity of condensers in the source space. We obtain an equivalent functional and analytic description of based on the properties of the composition operator (from weighted Sobolev spaces to non-weighted ones) induced by the inverses of the mappings in . When , the class of mappings coincides with the set of so-called -homeomorphisms which have been studied extensively in the last 25 years.
{"title":"Functional and analytic properties of a class of mappings in quasi-conformal analysis","authors":"S. Vodopyanov, A. Tomilov","doi":"10.1070/IM9082","DOIUrl":"https://doi.org/10.1070/IM9082","url":null,"abstract":"We define a two-index scale , , of homeomorphisms of spatial domains in , the geometric description of which is determined by the control of the behaviour of the -capacity of condensers in the target space in terms of the weighted -capacity of condensers in the source space. We obtain an equivalent functional and analytic description of based on the properties of the composition operator (from weighted Sobolev spaces to non-weighted ones) induced by the inverses of the mappings in . When , the class of mappings coincides with the set of so-called -homeomorphisms which have been studied extensively in the last 25 years.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"85 1","pages":"883 - 931"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58576419","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the problem of identifying domains of univalence on classes of holomorphic maps of the unit disc into itself. In 1926 E. Landau found the exact value of the radius of the disc of univalence on the class of such maps with a given value of the derivative at an interior fixed point. In 2017 V. V. Goryainov discovered the existence of univalence domains on classes of holomorphic maps of the unit disc into itself with an interior and a boundary fixed points, with a restriction on the value of the angular derivative at the boundary fixed point. However, the question of finding unimprovable domains of univalence remained open. In this paper, this extremal problem is solved completely: we find an exact univalence domain on the indicated class of holomorphic maps of the disc into itself. This result is a strengthening of Landau’s theorem for functions of the corresponding class.
{"title":"The exact domain of univalence on the class of holomorphic maps of a disc into itself with an interior and a boundary fixed points","authors":"A. Solodov","doi":"10.1070/IM9053","DOIUrl":"https://doi.org/10.1070/IM9053","url":null,"abstract":"We consider the problem of identifying domains of univalence on classes of holomorphic maps of the unit disc into itself. In 1926 E. Landau found the exact value of the radius of the disc of univalence on the class of such maps with a given value of the derivative at an interior fixed point. In 2017 V. V. Goryainov discovered the existence of univalence domains on classes of holomorphic maps of the unit disc into itself with an interior and a boundary fixed points, with a restriction on the value of the angular derivative at the boundary fixed point. However, the question of finding unimprovable domains of univalence remained open. In this paper, this extremal problem is solved completely: we find an exact univalence domain on the indicated class of holomorphic maps of the disc into itself. This result is a strengthening of Landau’s theorem for functions of the corresponding class.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"85 1","pages":"1008 - 1035"},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58575457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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