This volume is dedicated to the 70th birthday of the outstanding mathematician Victor Ivanovich Burenkov. With a broad and diverse range of interests Victor has made major contributions to many areas of both pure and applied mathematics, and has more than 150 publications in leading international journals. He is a leading authority on real analysis and the theory of function spaces, especially on Sobolev spaces and spaces with fractional order of smoothness and their applications. He has also done important work on partial differential equations and integral equations including, in particular, his contributions to the theory of hypoelliptic equations and the spectral theory of partial differential operators, and his work on ill-posed problems.
Victor's early mathematical training was in Moscow, at the Moscow Institute of Physics and Technology and the Steklov Institute of Mathematics of the Russian Academy of Sciences. His research interests were formed under the influence of his teacher, the great Russian mathematician Sergei Mikhailovich Nikol'skii.
V.I. Burenkov has developed a number of ingenious approaches and methods that led to significant breakthroughs.
His method of mollifiers with variable step and translation allowed him to obtain seminal results in the approximation of functions from general function spaces by various classes of infinitely differentiable functions and especially in the extension problem for Sobolev spaces. This method was later successfully applied by V.I. Korzyuk to some problems in partial differential equations.
His research on inequalities for intermediate derivatives with sharp constants led to active investigation of a new case of a finite interval, and he is cited as a pioneer of this new direction. He obtained sharp constants also in certain inequalities of different metrics and of Markov type for polynomials, in Hardy-type inequalities and in certain embedding theorems for Sobolev spaces.
New types of Fourier multiplier theorems were proved for the weighted Lebesgue spaces with exponential weights in which sharp conditions on a function ensuring that it is a Fourier multiplier were expressed in terms of Gevrey classes.
His paper on the composition of absolutely continuous functions was communicated by A.N. Kolmogorov, and contained the result generally referred as Burenkov's theorem.
Several significant papers were written jointly with M.L. Goldman. In particular, they successfully studied the interconnection between the norms of a wide class of local operators in general normed function spaces and the norms of their periodic analogues. These results allow the transference of many statements from the non-periodic case to the periodic one and vice versa.
The characteristic feature of Burenkov's research is his interest in non-standard effective applications of the theory of function spaces to various other areas.
One such application is the detailed study of the problem of conditional hypoellipticity (hypoellipticity depending on behaviour of solutions at infinity). He developed the method of fractional differentiation of a priori inequalities, which allowed him to obtain necessary and sufficient conditions of conditional hypoellipticity of partial differential operators with constant coefficients. This direction was further developed by the group of researchers from Yerevan headed by H.G. Ghazaryan. In their papers the aforementioned result is referred to as ‘Burenkov's theorem on conditional hypoellipticity’.
Another application belongs to the field of integral convolution-type equations (ill-posed problem). Application of the theory of spaces with low fractional smoothness allowed him to develop new flexible methods of constructing regularized approximate solutions of equations related to geophysical problems; these methods currently appear to be more effective than the traditional approaches based on using Sobolev spaces.
V.I. Burenkov worked at several universities, in particular for more than 10 years at the Moscow Institute of Radio-engineering, Electronics and Automation, at the Peoples’ Friendship University of Russia and at Cardiff University.
At Cardiff University he had fruitful collaboration with W.D. Evans which resulted in the publication of several significant papers on weighted integral inequalities and a frequently cited paper on quantum mechanics.
In the last decade V.I. Burenkov has focused on research in the following two principal directions.
The first deals with operator theory in general Morrey-type spaces. His efforts, and those of his co-researchers, were successful in determining necessary and sufficient conditions on the functional parameters characterizing these spaces, ensuring the boundedness of classical operators of real analysis (the maximal operator, the fractional maximal operator, the Riesz potential, the Hardy operator, genuine singular integrals) from one general Morrey-type space to another one. Some of these results are exhaustive and cannot be further improved, however, there are still many interesting open problems in this area. His main collaborator in this area is V.S. Guliyev.
The second research direction is dedicated to sharp spectral stability estimates for the eigenvalues of elliptic partial differential operators. It developed from joint research with E.B. Davies on spectral stability of the Neumann Laplacian. Later on V.I. Burenkov, together with P.D. Lamberti, developed the method of transition operators. This method allowed them to obtain sharp spectral stability estimates for the variation of the eigenvalues via effective geometric characteristics of the closeness of open sets for elliptic operators of arbitrary even order defined on open sets admitting arbitrarily strong degeneration for both Dirichlet and Neumann boundary conditions, thus giving a complete solution to the main aspects of this problem.
Victor is open to constructive scientific collaboration and has more than 50 co-authors. His first co-author was his student-mate A.G. Aslanyan. Jointly they published several papers on ordinary differential equations and related topics in non-commutative algebra.
V.I. Burenkov has participated in numerous research projects on analysis and partial differential equations supported by various grants, and led some of them. In particular, through 2006–2009 he was the coordinator of the large international project on function spaces and applications supported by INTAS (11 teams from 11 countries, and 65 participants altogether with more than 30 professors among them). He also led some projects on industrial mathematics. His main collaborator in this area was Yu.I. Khudak.
In addition to his research, Victor has also made a significant contribution to mathematical education through his teaching and the supervision of postgraduate students.
He is a talented and sought-after lecturer. Students of several countries have enjoyed his lectures on calculus, real, complex and functional analysis and differential equations. The distinguishing traits of Victor's character – creativity, relentless pursuit of perfection has always inspired his students. He is able to develop the student's independent thinking, jointly explore ideas and develop new insights. There are many mathematicians who have uncovered whole new fields of mathematical research after discussions with him. 25 of his research students have gained PhD degrees and are working in many countries of the world (on all continents).
V.I. Burenkov has published several text-books for students. He has developed a flexibly structured course of from 2 to 30 lectures ‘The main ideas in the theory of Sobolev spaces’ which he has delivered in many universities around the world.
His monograph ‘Sobolev spaces on domains’ became a popular text for both experts in the theory of function spaces and a wide range of mathematicians interested in applying the theory of Sobolev spaces. This monograph covers the subject at the right level of technical detail and focuses on giving a thorough exposition of the main results and ideas of the theory.
Victor is also known for his interest in languages. He has lectured in three languages and has made short introductory speeches in several more languages. Many mathematicians are thankful to him for vivid and energetic translations of their talks.
He has been a scientific editor of major books on the theory of function spaces published in Russia, and has organized and participated in translation of several books on the theory of function spaces and partial differential equations from English and Japanese into Russian.
He is a member of the editorial boards of several international journals. Recently V.I. Burenkov became one of the organizers and subsequently one of the chief editors, together with V.A. Sadovnichy and M. Otelbaev, of the new Eurasian Mathematical Journal.
Victor is active in international mathematical life, participating in and helping to organize many international conferences. He has given more than 100 invited talks at conferences in more than 30 countries. He is an honorary professor of the L.N. Gumilyov Eurasian National University and an honorary doctor of the Russian-Armenian State University.
He is an active member of the International Society for Analysis, its Application and Computation (ISAAC) and has been Vice-President of this society since 2003.
In his student years Victor played basketball for his university, but he was also keen on football, volleyball, tennis, table-tennis. He continues to enjoy sports, and is keen to encourage the participation of others. He has enjoyed organizing ‘professors versus postgraduate students’ football matches. For this reason football is now a part of the social programmes of several conferences in which he participates.
We congratulate Victor Ivanovich on his jubilee and wish him good health and further fruitful research.
Sh. Alimov, H. Begehr, O.V. Besov, D.E. Edmunds, W.D. Evans, V.M. Filippov, H.G. Ghazaryan, R. Gilbert, M.L. Goldman, V.M. Goldshtein, A.L. Gorbunov, V.S. Guliyev, V.A. Gusakov, T.Sh. Kalmenov, G.A. Kalyabin, V.M. Kokilashvili, V.I. Korzyuk, L.D. Kudryavtsev, A. Kufner, P.D. Lamberti, M. Lanza de Cristoforis, V.G. Maz’ya, S.M. Nikol'skii, M. Otelbaev, A. Pankov, S.I. Pohozhaev, A.G. Reshetnyak, H.-J. Schmeisser, A.A. Shkalikov, V.D. Stepanov, V.M. Tikhomirov, H. Triebel, M.Sh. Tuyakbaev, N.V. Vasilevski, S.K. Vodopyanov.