Student, then Postgraduate student of the Department of Mathematics, Faculty of Physics, M.V. Lomonosov Moscow State University.

Defended his Postgraduate diploma. Theme: “On integral representations and Fourier series of differentiable functions of several variables”.

Researcher at the Moscow Research and Design Institute of Automated Control Systems in the Urban Economy.

Assistant at the Mathematics Department of the Moscow Institute of Radio Engineering, Electronics and Automation (MIREA).

Assistant professor at the Mathematics Department of MIREA.

Defended his Doctoral thesis. Theme: "Investigation of spaces of differentiable functions of several variables with generalized smoothness".

Professor at the Mathematics Department of MIREA.

Received the academic title of Professor.

Head of the Mathematics Department of MIREA.

Professor at the Department of Nonlinear Analysis and Optimization, RUDN University.

Professor of the S. Nikolsky Mathematical Institute.

Winner of the Moscow government contest.

Winner of the RUDN prize in the field of science and innovations.

Winner of the RUDN prize as the best scientific adviser of postgraduate students.

### Teaching

- Prepared a series of new training courses, the most significant are the following:
- Methodical recommendations for studying the “Functional spaces” course (Space
*L*_{p}. Hölder, Minkowski inequalities. Convergence in the*L*_{p}Classification of spaces*L*_{p}). Moscow. RUDN University. -1989. Pp. 1-49. Co-author: V. Burenkov - Methodological recommendations for the study of the "Functional spaces" course (Young's inequality. Distribution functions. Permutations. Interpolation theorems). Moscow. RUDN University -1992. S. 1-76. Co-author: V. Burenkov
- Mathematical analysis. Elements of the theory of series. Functions of complex variable (tutorial). Moscow. - MIREA-1995. S. 1-80. Co-authors: A. Vshivtsev, A. Potepalova.
- "Algebraic structures" algebra course. Moscow, RUDN University, 2007, p. 200. Co-author: E. Sivkova.
- Educational-methodical complex "Contemporary problems of mathematics" Moscow, RUDN University, 2015. P. 1-25
- Educational-methodical complex “Theory of Functional Spaces” Moscow, RUDN University, 2015. P. 1-21
- Educational-methodical complex "Fundamentals of functional analysis" Moscow, RUDN University, 2015. P. 1-21
- “Analytical geometry. Vectors: study guide. Moscow 2015. Russian Technological University (MIREA). Co-author: E. Sivkova.

- Methodical recommendations for studying the “Functional spaces” course (Space
- Conducts the following courses for bachelor degree students at RUDN University:
- "Analytical geometry" ("Mathematics" direction)
- "Theory of functional spaces" ("Mathematics" direction)

- Conducts the following courses for master’s degree students at RUDN University:
- Modern problems of mathematics and applied mathematics" ("Mathematics" direction)

- At the invitation of the L.N. Gumilyov Eurasian National University (ENU, Kazakhstan, Astana) in 2015 and 2018 conducted lectures on the theory of functional spaces for students and doctoral students.
- At the invitation of Friedrich-Schiller University (Jena, Germany) in 2011 conducted a course of lectures on the theory of functional spaces of generalized smoothness for students and doctoral students.
- At the invitation of the Vladikavkaz Scientific Center of the Russian Academy of Sciences in 2018 conducted a course on the theory of ideal shells for cones of functions with monotonicity properties at the Vladikavkaz Youth Mathematical School (VYMS-2018)
- As a foreign scientific adviser, together with professors Leyla K. Kusainova and Nurzhan A. Bokaev (ENU, Astana) supervised the training of ENU doctoral students in the field of functional space theory.

### Science

- A number of important results were obtained on optimal embeddings of spaces of differentiable functions, the theory of traces and extensions. In particular, an accurate description of the trace space for generalized Lizorkin-Triebel spaces was given. Also the space of traces was described and the absence of linear continuation operators in the limit case of the theorem on traces for generalized Besov spaces was established. The results have important applications for the correct formulation of boundary value problems for partial differential operators.
- Optimal integral properties of functions were studied for various spaces of differentiable functions, such as the generalized Sobolev, Nikol'skii-Besov and Calderon spaces, as well as the generalized Bessel and Riesz potentials and exact descriptions of rearrangement invariant envelopes were established for them. The exact characteristics of the differential properties of potentials in terms of their uniform moduli of continuity are obtained. Optimal Calderon spaces for embedding generalized Bessel and Riesz potentials were found. The results are important for constructing a theory of optimal embeddings of spaces of generalized smoothness.
- Exact descriptions of optimal normalized and quasinormed envelopes for cones of functions with monotonicity properties in terms of the theory of ideal and rearrangement invariant spaces were obtained. The estimates of integral operators in weighted Lebesgue, Lorentz and Orlicz – Lorentz spaces were investigated. Applications of spaces of generalized smoothness to the study of the conditions of convergence and summability of spectral expansions in eigenfunctions of differential operators were obtained. The results play an important role in the construction of the spectral theory of partial differential operators.

### Scientific interests

- spectral theory of differential operators
- space of functions of generalized smoothness
- optimal embeddings for spaces of functions of generalized smoothness
- integral inequalities and estimates of operators on cones of functions with monotonicity properties
- optimal embeddings for generalized Bessel and Riesz potentials rearrangement invariant envelopes of generalized Bessel and Riesz potentials
- operators in common Morrey spaces

_{k}( f ;t), and determine their continuity envelope functions. This result is then applied to estimate the approximation numbers of some embeddings.

_{1}(R

^{n}) to a general local Morrey-type space LM

_{p2}θ,w(R

^{n}) is equivalent to the boundedness of the embedding operator from Lp

_{1}(R

^{n}) to LM

_{p2}θ,w(R

^{n}) and in its turn to the boundedness of the Hardy operator from L

_{p1 /p2}(0,∞) to the weighted Lebesgue space L θ/

_{p2},v(0,∞) for a certain weight function v determined by the functional parameter w. This allows obtaining necessary and sufficient conditions on the function w ensuring the boundedness of M from Lp

_{1}(R

^{n}) to LM

_{p2}θ,w(R

^{n}) for any 0 < θ≤ ∞, 0 < p

_{2}≤ p

_{1}∞, p

_{1}> 1. These conditions with p

_{1}= p

_{2}= 1 are necessary and sufficient for the boundedness of M from L

_{1}(R

^{n}) to the weak local Morrey-type space W LM

_{1θ,w}(Rn).