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Vol 89, No 5 (2025)
Articles
$H^p$ spaces of separately $(\alpha, \beta)$ -harmonic functions in the unit polydisc
Abstract
We prove existence and uniqueness of a solution of the Dirichlet problem for separately $(\alpha, \beta)$ -harmonic functions on $\mathbb D^n$ with boundary data in $C(\mathbb T^n)$ using $(\alpha, \beta)$ -Poisson kernel $P_{\alpha, \beta} (z, \zeta)$ . A characterization by hypergeometric functions of separately $(\alpha, \beta)$ -harmonic functions which are also $m$ -homogeneous is given, it is used to obtain series expansion of separately $(\alpha, \beta)$ -harmonic functions. Basic $H^p$ theory of such functions is developed: integral representations by measures and $L^p$ functions on $\mathbb T^n$ , norm and weak$^\ast$ convergence at the distinguished boundary $\mathbb T^n$ . Weak $(1,1)$ -type estimate for a restricted non-tangential maximal function $M_{A, B}^{\mathrm{NT}}$ is derived. We show that slice functions $u(z_1, …, z_k, \zeta_{k+1}, …, \zeta_n)$, where some of the variables are fixed, belong in the appropriate space of separately $(\alpha', \beta')$ -harmonic functions of $k$ variables. We prove a Fatou type theorem on a.e. existence of restricted non-tangential limits for these functions and a corresponding result for unrestricted limit at a point in $\mathbb T^n$ . Our results extend earlier results for $(\alpha, \beta)$ -harmonic functions in the disc and for $n$ -harmonic functions in $\mathbb D^n$ .
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2025;89(5):3-31
3-31
On uniquely solvable Fokker–Planck–Kolmogorov equations
Abstract
In this paper we obtain broad sufficient conditions for the existence
of probability solutions to the Cauchy problem for
Fokker–Planck–Kolmogorov equations on the real line without using Lyapunov
functions. In the multidimensional case, we prove that if
the Fokker–Planck–Kolmogorov equation for an elliptic operator $L$
has a probability solution $\sigma$ , and the Cauchy problem for this
equation has a unique probability solution for every initial probability
distribution, then there exists a strongly continuous Markov operator semigroup
on the space $L^1(\sigma)$ with respect to which the measure $\sigma$
is invariant and the generator of which extends the operator $L$ .
We give an answer to the long-standing question about existence of
a sub-Markov semigroup different from the canonical semigroup with
the generator extending $L$ .
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2025;89(5):32-53
32-53
Characterization of boundedness of some commutators of fractional maximal functions in terms of $p$ -adic vector spaces
Abstract
This paper gives some characterizations of the boundedness of the maximal or non-linear commutator of the $p$ -adic fractional maximal operator $ \mathcal{M}_{\alpha}^p$ with the symbols belong to the $p$ -adic BMO spaces on (variable) Lebesgue spaces and Morrey spaces over $p$ -adic field, by which some new characterizations of BMO functions are obtained in the $p$ -adic field context. Meanwhile, some equivalent relations between the $p$ -adic BMO norm and the $p$ -adic (variable) Lebesgue or Morrey norm are given.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2025;89(5):54-79
54-79
Perverse sheaves on smooth toric varieties and stacks
Abstract
It is usually not straightforward to work with the category of
perverse sheaves on a variety using only its definition as a heart
of a $t$ -structure. In this paper, the category of perverse sheaves
on a smooth toric variety with its orbit stratification is described
explicitly as a category of finite-dimensional modules over an
algebra. An analogous result is also established for various
categories of equivariant perverse sheaves, which in particular
gives a description of perverse sheaves on toric orbifolds, and we
also compare the derived category of the category of perverse
sheaves to the derived category of constructible sheaves.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2025;89(5):80-106
80-106
Asymptotics of eigenvalues and eigenfunctions of the Dirichlet problem in thin spatial network with nodules
Abstract
We perform homogenization of a thin spatial network of quantum waveguides with small nodules (the Dirichlet problem for the Laplace operator). In contrast to a problem with the Neumann boundary
condition, the low-frequency but situated far away from the coordinate origin range of the
spectrum of the Dirichlet problem is characterized by localization of the corresponding
eigenfunctions near angular joints of ligaments or at the ligaments themselves in accordance with
distribution of eigenvalues in the discrete spectra (surely non-empty) of model problems
in junctions of semi-infinite cylindrical quantum waveguides of various shapes. The behavior of eigenvalues and eigenfunctions in the mid-frequency range of the spectrum depend crucially on
the phenomenon of threshold resonances in the above-mentioned junctions as well as the relation
between small parameters, namely, the period of distribution of the nodules and their diameter,
comparable in order but bigger than diameter of the ligaments. We consider concrete cases
of rectangular and circular cross-sections and formulate open questions.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2025;89(5):107-164
107-164
Resonances and discrete spectrum of the Laplace operator on hyperbolic surfaces
Abstract
The spectrum of the Laplace operator
on a non-compact hyperbolic Riemann surface of finite measure is studied.
A sufficient condition for the discrete spectrum to be infinite is obtained.
It is shown that this condition holds near the point
$\Gamma_0(N)/H$ , $N=p_1\cdots p_r$ , of the Teichmüller space.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2025;89(5):165-180
165-180
A prox-regular sweeping process coupled with a maximal monotone differential inclusion
Abstract
A coupled system consisting of a sweeping process and an evolution
maximal monotone inclusion is considered. The values of moving set of
the sweeping process are prox-regular sets that depend on time and state
of the system. The right-hand side of the sweeping process contains the sum
of two multivalued time- and state-dependent perturbations with
different semicontinuity properties. The perturbation in the right-hand
side of maximal monotone inclusion is a single-valued function.
A solution to the sweeping process is a right continuous function of
bounded variation (a BV-solution). A solution to the maximal monotone
inclusion is an absolutely continuous function. A theorem on existence
of a solution to this system is proved, and when the perturbations are
convex, a theorem on compactness of the solution set is established.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2025;89(5):181-232
181-232