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Vol 85, No 5 (2021)
Articles
Vladimir Leonidovich Popov (congratulation)
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(5):3-3
3-3
The Calderon construction for a couple of global Morrey spaces
Abstract
We employ a new approach to show that the Calderon construction for a couple of global Morrey spaces coincides with the Morrey space with appropriate parameters only under rather strong assumptions on the couples of ideal spaces that parameterize the original Morrey spaces. We show that, in the case of classical examples of global Morrey spaces, these assumptions are necessary and sufficient. Applying a well-known reduction, we use the Calderon construction for a couple of global Morrey spaces to describe the spaces given by the complex interpolation method and also to prove new interpolation theorems for global Morrey spaces.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(5):5-24
5-24
On distributions of homogeneous and convex functions in Gaussian random variables
Abstract
We obtain broad conditions under which distributions of homogeneousfunctions in Gaussian and more general random variables have bounded densities or evendensities of bounded variation or densities with finite Fisher information.Analogous results are obtained for convex functions.Applications to maxima of quadratic forms are given.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(5):25-57
25-57
Functional and analytic properties of a class of mappings in quasi-conformal analysis
Abstract
We define a two-index scale $\mathcal Q_{q,p}$, $n-1< q\leq p<\infty$, of homeomorphisms of spatial domains in $\mathbb R^n$, the geometric description of which is determined by the control of the behaviour of the $q$-capacity of condensers in the target space in terms of the weighted $p$-capacity of condensers in the source space. We obtain an equivalent functional and analytic description of $\mathcal Q_{q,p}$ based on the properties of the composition operator (from weighted Sobolev spaces to non-weighted ones) induced by the inverses of the mappings in $\mathcal Q_{q,p}$.
When $q=p=n$, the class of mappings $\mathcal Q_{n,n}$ coincides with the set of so-called $Q$-homeomorphisms which have been studied extensively in the last 25 years.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(5):58-109
58-109
Convergence to stationary non-equilibrium states for Klein–Gordon equations
Abstract
We consider Klein–Gordon equations in $\mathbb{R}^d$, $d\ge2$, with constant or variable coefficients and study the Cauchy problem with random initial data. We investigate the distribution $\mu_t$ of a random solution at moments of time $t\in\mathbb{R}$. We prove the convergence of correlation functions of the measure $\mu_t$ to a limit as $t\to\infty$. 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 $\mu_t$ to a limiting measure as $t\to\infty$. 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.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(5):110-131
110-131
Arithmetic of certain $\ell$-extensions ramified at three places. II
Abstract
Let $\ell$ be a regular odd prime, $k$ the $\ell$ th cyclotomic field and $K=k(\sqrt[\ell]{a})$, where $a$ is a positive integer. Under the assumption that there are exactly three places not over $\ell$that ramify in $K_\infty/k_\infty$, we continue to study the structure of the Tate module (Iwasawa module) $T_\ell(K_\infty)$ as a Galois module. In the case $\ell=3$, we prove that for finite $T_\ell(K_\infty)$ we have $|T_\ell(K_\infty)| {=} \ell^r$ for some odd positive integer $r$. Under the same assumptions, if $\overline T_\ell(K_\infty)$ is the Galois group of the maximal unramified Abelian $\ell$-extension of $K_\infty$, then the kernel of the natural epimorphism $\overline T_\ell(K_\infty)\to T_\ell (K_\infty)$ is of order $9$. Some other results are obtained.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(5):132-151
132-151
On the classification problem for polynomials $f$ with a periodic continued fraction expansion of $\sqrt{f}$ in hyperelliptic fields
Abstract
The classical problem of the periodicity of continued fractions for elements of hyperelliptic fieldshas a long and deep history. This problem has up to now been far from completely solved.A surprising result was obtained in [1] for quadratic extensions defined by cubic polynomialswith coefficients in the field $\mathbb{Q}$ of rational numbers: except for trivial cases there areonly three (up to equivalence) cubic polynomials over $\mathbb{Q}$ whose square root has a periodic continued fraction expansion in the field $\mathbb{Q}((x))$ of formal power series.In view of the results in [1], we completely solve the classification problem for polynomials$f$ with periodic continued fraction expansion of $\sqrt{f}$ in elliptic fields with the field ofrational numbers as the field of constants.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(5):152-189
152-189
The exact domain of univalence on the class of holomorphic maps of a disc into itself with an interior and a boundary fixed points
Abstract
We consider the problem of identifying domains of univalence on classes ofholomorphic maps of the unit disc into itself. In 1926 E. Landau found the exactvalue of the radius of the disc of univalence on the class of such maps with a givenvalue of the derivative at an interior fixed point. In 2017 V. V. Goryainovdiscovered the existence of univalence domains on classes of holomorphic maps of theunit 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. Inthis paper, this extremal problem is solved completely: we find an exact univalencedomain 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 correspondingclass.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(5):190-218
190-218