Name: 32nd St.Petersburg Summer Meeting in Mathematical Analysis
Start: 2023-07-01T09:00:00+03:00
End: 2023-07-06T15:15:00+03:00
Location: No location set

32nd St.Petersburg Summer Meeting in Mathematical Analysis

from Saturday, 1 July 2023 (09:00) to Thursday, 6 July 2023 (15:15)

Monday, 26 June 2023
Tuesday, 27 June 2023
Wednesday, 28 June 2023
Thursday, 29 June 2023
Friday, 30 June 2023
Saturday, 1 July 2023

09:20 Registration
Registration
09:20 - 10:00
10:00 Hardy spaces of fractional order - Dmitrii Stolyarov (St. Petersburg State University)
Hardy spaces of fractional order
- Dmitrii Stolyarov (St. Petersburg State University)
10:00 - 10:45
The space of measures of bounded total variation and $L_1$ quite often lack good properties. Usually this is related to the unboundedness of the maximal function on $L_1$. The interest in these spaces is justified not only by the simplicity of their norms, but also by direct connections with the geometric measure theory, and, thus, geometry. In harmonic analysis, we sometimes replace $L_1$ with a narrower space $H_1$. Here $H_1$ is the real Hardy class, it behaves ‘better’. However, we lose the relationship with the geometric measure theory when passing to $H_1$: by ’the Riesz brothers’ theorem’ there are no analogs of the space of measures whose norm resembles that of $H_1$. We will try to suggest a scale of spaces that interpolates $L_1$ and $H_1$. These intermediate spaces do contain singular measures of fractional dimension and also posses some properties of the Hardy class; for example, some ’trace’ inequalities for fractional integration operators are valid for measures in these spaces. We do not have complete proofs yet and are also slightly unsure whether our definitions are best possible. A larger part of work is done in a martingale model that simplifies the classical setting of Euclidean spaces. Nevertheless, the story seems worth telling. For example, the aforementioned scale extends the definition of lower Hausdorff dimension to arbitrary distributions. Work in progress with Daniel Spector, Taiwan
10:45 Coffee
Coffee
10:45 - 11:10
11:10 The Nuttall decomposition of a three-sheeted torus and the asymptotics of the rational Hermite-Pade approximants - Semen Nasyrov (Kazan Federal University)
The Nuttall decomposition of a three-sheeted torus and the asymptotics of the rational Hermite-Pade approximants
- Semen Nasyrov (Kazan Federal University)
11:10 - 11:55
We study the three-sheeted Riemann surface $S$ of genus $1$,corresponding to the algebraic function $w=\sqrt[3]{(z-a_1)(z-a_2)(z-a_3)}$ with pairwise distinct complex numbers $a_j$. There exists an abelian integral $G$ on $S$ which is regular on $S$, except for three points lying over infinity where the function Re $G$ has prescribed singularities of logarithmic type. The harmonic function Re $G$ induces a partition of $S$ into three sheets; it is called the Nuttall decomposition. Such decomposition plays an important role in investigation of the asymptotics of the rational Hermite-Pade approximants. According to the Nuttall conjecture, the sheet gluing lines and their projections onto the Riemann sphere define the convergence domains for the Hermite-Pade approximants. The differential-topological structure of the Nuttall decomposition depends essentially on the mutual location of the points $a_j$. A.I.Aptekarev and and D.N.Tulyakov (2016) described the structure for the case when the triangle with vertices at the points $a_j$ is sufficiently close to the regular one. We consider the general case and completely solve the problem for isosceles triangles with the apex angle less than $\pi/3$. In the talk we also discuss other cases. The study was carried out at the expense of the grant of the Russian Science Foundation No.23-11-00066.
12:00 Uniform approximation by polynomial solutions of certain elliptic PDE and systems - Konstantin Fedorovskiy (Lomonosov Moscow State University)
Uniform approximation by polynomial solutions of certain elliptic PDE and systems
- Konstantin Fedorovskiy (Lomonosov Moscow State University)
12:00 - 12:45
We plan to discuss the problems on uniform approximation of functions on compact sets in the complex plane by elements of polynomial modules of polyanalytic type generated by antiholomorphic functions and by polynomial solutions to general second-order elliptic but not strongly elliptic systems of PDEs with constant coefficients. We will present some known and recent results in this topis and explore the special analytic characteristics of planar simply connected domains in terms of which the obtained approximation criteria are stated (the concept of a $g$-Nevanlinna domain, where $g$ is the generator of the module by elements of which the approximation is carried out, and, the concept of an $L$-special domain, where $L$ is the elliptic differential operator that determines the corresponding system). The talk is based on the results obtained in frameworks of the project 22--11--00071 by the Russian Science Foundation.
12:45 Lunch
Lunch
12:45 - 14:30
14:30 Real-valued spectral shift functions for pairs of contractions and pairs of dissipative operators - Vladimir Peller (St.Petersburg StateUniversity)
Real-valued spectral shift functions for pairs of contractions and pairs of dissipative operators
- Vladimir Peller (St.Petersburg StateUniversity)
14:30 - 15:15
The talk is based on joint results with M.M. Malamud. I am going to discuss conditions under which a pair of contractions on Hilbert space with trace class difference has a real integrable spectral shift functions. The case of pairs of dissipative operators will also be discussed. The work is supported by the Ministry of Education and Science of the Russian Federation within the framework of the state task (project number FSSF-2023-0016).
15:20 B. Ya. Levin function for some sets of segments - Nikolai Shirokov (St.-Petersburg State University and National Research University High School of Economics in SPb)
B. Ya. Levin function for some sets of segments
- Nikolai Shirokov (St.-Petersburg State University and National Research University High School of Economics in SPb)
15:20 - 15:50
Let $\{I_k\}_{k\in \mathbb{Z}}$ be a set of disjoint of segments of real axis, $E=\bigcup_{k\in \mathbb{Z}}{I_k}$. We call the B.Ya.Levin function $f_{E,\sigma}(z)$, $\sigma >0$, a function satisfying the following conditions: 1. $f_{E,\sigma}(z)$ is subharmonic on the complex plane $\mathbb{C}$ and harmonic on $\mathbb{C}\setminus E$; 2. $f_{E,\sigma}(z)=0$, $x\in E$; $f_{E,\sigma}(z)>0$, $z\in\mathbb{C} \setminus E$; 3. $\limsup_{z\rightarrow \infty}\frac{f_{E,\sigma}(z)}{|z|}=\sigma$, $f_{E,\sigma}(\bar{z})=f_{E,\sigma}(z)$; 4. If $g$ is subharmonic on $\mathbb{C}$, $g(x)\leq 0$, $x\in E$, $\limsup_{z\rightarrow \infty}\frac{g(z)}{|z|}\leq\sigma$, then $g(z)\leq f_{E,\sigma}(z)$. We construct the B.Ya.Levin function for some sets of segments such that $|I_k|_{|k|\rightarrow \infty}\rightarrow 0$.
15:50 Coffee
Coffee
15:50 - 16:15
16:15 Refinement of powered Bohr inequality - Diana Khammatova (Kazan Federal University)
Refinement of powered Bohr inequality
- Diana Khammatova (Kazan Federal University)
16:15 - 16:45
Let B denote the class of all analytic functions mapping the unit disk into itself. We consider the powered Bohr sum of a function from the class B with an additional term including the area of the image of the disk of radius r. In this setting, we prove a counterpart of the Bohr theorem for two different cases.
16:50 The Bohr radius of a pair of operators - Ramis Khasyanov (SPbU)
The Bohr radius of a pair of operators
- Ramis Khasyanov (SPbU)
16:50 - 17:20
In 1914, H. Bohr, studying Dirichlet series, discovered the following interesting fact in complex analysis, which is now called the Bohr phenomenon: Theorem. Let $f(z)=\sum_{n\ge 0}a_nz^n$ and $\|f\|_{\infty}:=\sup_{z\in \mathbb{D}}|f(z) |$ in the unit disc $\mathbb{D}=\{|z|<1\}$. Then $M_rf:=\sum_{n\ge 0}^{} |a_n| r^n\le \|f\|_{\infty}, \:\: 0\le r\le 1/3.$ Moreover, the constant $1/3$ is the best possible. We introduce the concept of the Bohr radius of a pair of operators, in terms of which many well-known results related to the Bohr inequality can be formulated. The report will discuss questions about the Bohr radius for the differential operators, Hadamard convolution operators, and other operators. Using the concept of the Bohr radius of a pair of operators, we generalize the theorem of B.Bhowmik and N.Das on the comparison of majorant series of subordinate functions.
17:25 Nontriviality of Siddiqi class and nonspanning systems of exponentials on curves - Rashit Gaisin (Institute of Mathematics with Computing Centre - Subdivision of the Ufa Federal Research Centre of Russian Academy of Science)
Nontriviality of Siddiqi class and nonspanning systems of exponentials on curves
- Rashit Gaisin (Institute of Mathematics with Computing Centre - Subdivision of the Ufa Federal Research Centre of Russian Academy of Science)
17:25 - 17:55
We study problem of noncompleteness of the system of exponentials $\{e^{\pm \lambda_n z}\}$ $(\lambda_n>0)$ in the space of continuous functions on the arc of bounded slope. In the terms of minorant sequence we have obtained new equivalent condition, which is sufficient for noncompleteness of the given system of exponentials. We also have proved that Siddiqi class is nontrivial if sequence $\{M_n^c\}$ satisfyes the A-condition.
Sunday, 2 July 2023
10:00 On Gelfand-Shilov type spaces - Ildar Musin (Institute of Mathematics with Computing Centre of RAS)
On Gelfand-Shilov type spaces
- Ildar Musin (Institute of Mathematics with Computing Centre of RAS)
10:00 - 10:45
Following the scheme of constructing of Gelfand-Shilov spaces of type $S$ three new spaces of rapidly decreasing infinitely differentiable functions in ${\mathbb R}^n$ will be defined with a help of families of separately radial convex weight functions in ${\mathbb R}^n$. Under some mild restrictions on weight functions these spaces will be characterized via Fourier transforms. Some interesting particular cases of such spaces corresponding to special families of weight functions will be considered. This is a joint work with R.S. Yulmukhametov and A.V. Lutsenko.
10:45 Coffee
Coffee
10:45 - 11:10
11:10 Gaussian multiplicative chaos and excess one for the sine-process - Alexander Bufetov (Steklov Mathematical Institute of Russian Academy of Sciences)
Gaussian multiplicative chaos and excess one for the sine-process
- Alexander Bufetov (Steklov Mathematical Institute of Russian Academy of Sciences)
11:10 - 11:55
The main result of the talk is that the random entire function associated to the sine-process converges to the Gaussian multiplicative chaos under suitable normalization. As a corollary of the developed formalism, we obtain that almost every realization of the sine-process, after removing one particle, becomes a complete minimal set for the Paley-Wiener space.
12:00 Some asymptotic formulas connected with Bessel analysis - Sergey Platonov (Petrozavovsk State University)
Some asymptotic formulas connected with Bessel analysis
- Sergey Platonov (Petrozavovsk State University)
12:00 - 12:45
In various sections of the classical Fourier Analysis, the problem of the convergence of the integrals $$ \int\limits_0^\infty f(\lambda t)\, g(t)\, dt $$ as $\lambda\to+0$ and $\lambda\to +\infty$ under various assumptions on the functions $f$ and $g$ are considered. In the talk we study some analogues of such problems for weight integrals of the form $$ \int\limits_0^\infty f(\lambda t)\, g(t)\,t^{2\alpha+1} \, dt , \quad \alpha>-1/2, $$ for functions $f$ and $g$ from some weighted functional classes that are connected with Fourier -- Bessel harmonic analysis.
12:45 Lunch
Lunch
12:45 - 14:30
14:30 Regularized sums for eigenvalues of bounded perturbations of model second-order partial derivative operators - Ziganur Fazullin (Ufa University of Science and Technology)
Regularized sums for eigenvalues of bounded perturbations of model second-order partial derivative operators
- Ziganur Fazullin (Ufa University of Science and Technology)
14:30 - 15:15
We are going to discuss problems and analyze calculation of regularized trace formulas for perturbed model second-order partial derivative operators if the perturbation is performed by the operator of multiplication by function. In particular, we will formulate certain results on the question for Laplace-Beltrami operator on $S^2$, for harmonic oscillator acting in the whole plane $\mathbb R^2$ or in a strip, for Schr\”odinger operator in homogeneous magnetic field. We will also discuss certain approaches and results for Dirichlet problem on a square
15:20 Lp-Hardy-type inequalities for special weight functions - Ramil Nasibullin (Kazan Federal University)
Lp-Hardy-type inequalities for special weight functions
- Ramil Nasibullin (Kazan Federal University)
15:20 - 15:50
We prove one-dimensional $𝐿_𝑝$-Hardy inequalities with additional terms and use them for justifying spatial analogues in convex domains with finite volumes. We consider spatial inequalities in arbitrary domains, and then we simplify them for convex domains. The constants in the additional terms in these spatial inequalities depend on the volume or on the diameter of the domain.
15:50 Coffee
Coffee
15:50 - 16:15
16:15 Exponential series with a given growth majorant near the boundary - Galiya Gaisina (Ufa University of Science and Technology)
Exponential series with a given growth majorant near the boundary
- Galiya Gaisina (Ufa University of Science and Technology)
16:15 - 16:45
We will talk about the behavior of series of exponents near the boundary in cases where the domain of regularity of the sum of the series is the entire plane, a half-plane, a bounded convex domain. Next, we will touch the representation of analytic in the right half-plane functions by series of exponents, taking into account a given convex majorant of growth.
16:50 Blashke $C^*$-algebras - Alla Kuznetsova (Kazan Federal University)
Blashke $C^*$-algebras
- Alla Kuznetsova (Kazan Federal University)
16:50 - 17:20
The idea of the talk is to bring together the well known Blaschke product and operator algebras. We suggest the notion of the Blaschke $C^*$-algebra, which is the universal $C^*$-algebra generated by some relations. The relations are defined by a family of finite Blaschke products. In this way we extend the Coburn’s Theorem for a family of non-unitary isometries connected by a family of finite Blaschke products. We show that the Blaschke $C^*$-algebras is isomorphic to the inductive limit of Toeplitz algebras. The talk is based on the joint work with Tamara Grigoryan.
17:25 Hypercubic generalizations and refinements of Cauchy-Bunyakovsky-Schwarz inequality - Armenak Gasparyan (Ailamazyan Program Systems Institute of Russian Academy of Sciences)
Hypercubic generalizations and refinements of Cauchy-Bunyakovsky-Schwarz inequality
- Armenak Gasparyan (Ailamazyan Program Systems Institute of Russian Academy of Sciences)
17:25 - 17:55
The classical Cauchy-Bunyakovsky-Schwarz inequality is one of fundamental inequalities of mathematics that have been generalized and applied by many mathematicians in different directions. In the talk a family of new type generalizations are presented for CBS inequality. These results are applied to give infinite sequence of refinements both of same CBS inequality and its generalizations.