Mikhail Goldman
Doctor of Physical and Mathematical Sciences
Professor,
s.M. Nikol’skii Mathematical Institute
1963 – 1972
Student, then Postgraduate student of the Department of Mathematics, Faculty of Physics, M.V. Lomonosov Moscow State University.
1972
Defended his Postgraduate diploma. Theme: “On integral representations and Fourier series of differentiable functions of several variables”.
1972-1974
Researcher at the Moscow Research and Design Institute of Automated Control Systems in the Urban Economy.
1974 – 1984
Assistant at the Mathematics Department of the Moscow Institute of Radio Engineering, Electronics and Automation (MIREA).
1984-1990
Assistant professor at the Mathematics Department of MIREA.
1989
Defended his Doctoral thesis. Theme: "Investigation of spaces of differentiable functions of several variables with generalized smoothness".
1990
Professor at the Mathematics Department of MIREA.
1991
Received the academic title of Professor.
1991-2000
Head of the Mathematics Department of MIREA.
2000-2018
Professor at the Department of Nonlinear Analysis and Optimization, RUDN University.
Since 2018
Professor of the S. Nikolsky Mathematical Institute.
2002
Winner of the Moscow government contest.
2013
Winner of the RUDN prize in the field of science and innovations.
2017
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 Lp. Hölder, Minkowski inequalities. Convergence in the Lp Classification of spaces Lp ). 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.
- 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
A characterization of the spaces dual to weighted Lorentz spaces are given by means of reverse Holder inequalities. This principle of duality is then applied to characterize weight functions for which the identity operator, the Hardy-Littlewood maximal operator and the Hilbert transform are bounded on weighted Lorentz spaces.
A criterion is obtained for the Hardy-type inequality. This criterion comes close to being necessary as well as sufficient. When an inequality in the criterion is reversed, a Poincaré-type inequality is derived in some cases.
M. L. Gol'dman, R. A. Kerman. On Optimal Embedding of Calderon Spaces and Generalized Besov Spaces. Proceedings of the Steklov Institute of Mathematics, 2003, 243, 154–184.
For isotropic Calderon-type spaces and generalized Besov spaces, criteria for embeddings into rearrangement invariant spaces are established, sharp estimates for decreasing rearrangements are found, and optimal rearrangement invariant spaces are described.
Goldman M.L. Rearrangement Invariant Envelopes of generalized Besov, Sobolev and Calderon Spaces // Contemporary Mathematics. 2007. Vol. 424. P. 53–81.
The survey is given for recent results concerning the description of the rearrangement invariant envelopes of the the generalized Besov, Sobolev and Calderon spaces. The smallest rearrangement invariant spaces are described in which these spaces are embedded. The results are based on equivalent description of the cones of decreasing rearrangements for functions in Besov, Sobolev and Calderon Spaces.
In this article, we study the spaces of potentials in n-dimensional Euclidean space. They are constructed on the basis of a rearrangement invariant space by using convolutions with some general kernels. Specifically, the treatment covers spaces of classical Bessel and Riesz potentials. We establish the equivalent characterization for the cones of decreasing rearrangements of potentials. This is the key result for description of integral properties of potentials.
In this paper we study spaces of Bessel potentials in n-dimensional Euclidean spaces. They are constructed on the basis of a rearrangement-invariant space (RIS) by using convolutions with Bessel– MacDonald kernels. Specifically, the treatment covers spaces of classical Bessel potentials. We establish two-sided estimates for the corresponding modulus of smoothness of order k ∈ N, ωk ( f ;t), and determine their continuity envelope functions. This result is then applied to estimate the approximation numbers of some embeddings.
It is proved that the boundedness of the maximal operator M from a Lebesgue space Lp1 (Rn) to a general local Morrey-type space LMp2θ,w(Rn) is equivalent to the boundedness of the embedding operator from Lp1 (Rn) to LMp2θ,w(Rn) 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 Lp1 (Rn) to LMp2θ,w(Rn) for any 0 < θ≤ ∞, 0 < p2 ≤ p1 ∞, p1 > 1. These conditions with p1 = p2 = 1 are necessary and sufficient for the boundedness of M from L1 (Rn) to the weak local Morrey-type space W LM1θ,w(Rn).
We study the problem of constructing a minimal Banach function space containing a given cone of nonnegative measurable functions. For the associate function norm of the norm of an optimal space, we obtain general formulas and specify them in the case of a cone defined by an integral representation. We also consider the similar problem of constructing an optimal rearrangement invariant space and compare the descriptions obtained.
The paper is devoted to generalized Bessel potentials constructed using the convolutions of generalized Bessel–McDonald kernels with functions from the basic rearrangement invariant space. If the criterion for the embedding of potentials into the space of bounded continuous functions is fulfilled, we state equivalent descriptions for the cone of moduli of continuity of potentials in the uniform norm. This makes it possible to obtain a criterion for the embedding of potentials into the Calderon space. In the case of generalized Bessel potentials constructed over the basic weighted Lorentz space, we explicitly describe the optimal Calderon space for such an embedding.
We obtain formulas for the generalized functional norm associated with the two-weight integral quasi-norm. We describe a minimal generalized Banach function space containing a given quasi-Banach space defined by the two-weight integral quasi-norm.
We study the problem of constructing a minimal quasi-Banach ideal space containing a given cone of nonnegative functions with monotonicity properties. The construction employs nondegenerate operators. We present general results on constructing optimal envelopes consistent with an order relation and obtain specifications of these constructions for various cones and various order relations. We also address the issue of order covering and order equivalence of cones.
We study the properties of generalized smoothness spaces, such as Calderon spaces, which include classical Nikolsky–Besov spaces and many of their generalizations, and describe the differential properties of generalized Bessel potentials, which include classical Bessel potentials and Sobolev spaces. The potential kernels may have non-power singularities in the neighborhood of the origin. Using order-accurate estimates of the modules of continuity of potentials, the criteria for embedding potentials in Calderon spaces are established and the optimal spaces for such embeddings are described.
In this paper we continue the study of general Morrey spaces using a general rearrangement invariant space as a basic space, and a general ideal space as an outer space. Here we consider some classes of positively homogeneous monotone operators from general rearrangement invariant spaces to general Morrey spaces and obtain the estimates for their norms. This approach covers many operators of analysis, such as the embedding and symmetrization operators, Hardy-Littlewood maximal operators, generalized Riesz potentials, Hardy-type operators.
We consider the spaces of generalized Bessel and Riesz potentials and establish criteria for the embedding of these spaces in rearrangements invariant spaces. To do this we obtain constructive equivalent descriptions for the cones of decreasing rearrangement of potentials. Covering and equivalence of cones are studied with respect to order relations which allows to weaken substantially the assumptions on the kernels of potentials.
The problems of ordinal covering and ordinal equivalence for cones of functions with monotonicity properties associated with decreasing permutations of generalized Bessel and Riesz potentials are considered.
Modular and norm inequalities are considered on the cone of all nonnegative functions as well as on the cone of all nonnegative decreasing functions in the weighted Orlicz space. Reduction theorems are proved for the norm of positively homogeneous operator on the cone. We show that it is equivalent to the norm of a certain modified operator on the cone of all nonnegative functions in this space. Analogous results are established for modular inequalities.
The criteria for the validity of modular inequalities for the Hardy operator on the cone of nonnegative decreasing functions from the weight space of Orlich with total weight are established. The result is based on the reduction theorem of modular inequalities for a positively homogeneous operator on the cone, which allows us to proceed to modular inequalities for the modified operator on the cone of all non-negative functions from the Orlich space. It is shown that for the Hardy operator, the modified operator is a generalized Hardy operator. This allowed us to establish explicit criteria for the validity of modular inequalities.
We consider a narrowing of the monotone operator P to the cone of non-negative decreasing functions from the weight space of the Orlich without additional a priori assumptions about the properties of the Orlich function and the weight function. We establish an order-accurate two-way estimate of the norm of this narrowing using a specially constructed discretization procedure. Similar estimates are also obtained for monotone operators over the corresponding Orlich-Lorentz spaces. As applications, we obtain descriptions of the associated spaces for the cone and for the Orlich–Lorentz space. The results are new and relevant in the theory of these spaces.
This paper contains the proof of general results on the calculation of the norms of monotone operators acting from one ideal space to another under matching convexity and concavity properties of the operator and the norms in ideal spaces.