Editorial

Abstract

This volume on “Topics in Function Spaces, Differential Operators, Harmonic and Fractal Analysis” is dedicated to our teacher and friend Hans Triebel on the occasion of his seventy-fifth birthday. Hans Triebel was born on February 7, 1936 in Dessau, Germany. He studied mathematics and physics at the Friedrich-Schiller-University of Jena from 1954 to 1959, graduating with a Diploma Degree in mathematics. At the beginning of his academic career he worked in classical complex analysis and received a Ph.D. in Mathematics at the Friedrich-Schiller-University in 1962. Motivated by an interest in both mathematics and physics, he studied Sobolev's famous 1950 book and learned about the theory of distributions as developed by L. Schwartz. This might have been the catalyst for his change of research topic towards partial differential operators and function spaces. As a postdoc he spent one year at the University of Leningrad (St. Petersburg), where he enjoyed the intellectual ferment of the atmosphere created by such great mathematicians as Uraltseva, Ladyzhenskaya, Birman and Solomyak, and attended lectures by Birman on functional analysis, spectral theory and quantum mechanics. Inspired by the Russian School of Mathematics he focused his research on recent developments in linear and nonlinear partial differential operators, spectral theory and functional analysis, rapidly obtaining far-reaching results with a deep impact on further research in this field. In particular, he realized the significance of function spaces and contributed to both theory and applications in a decisive way. He completed his Habilitation Thesis on function spaces and nonlinear analysis in 1966, becoming Full Professor of Analysis at the Friedrich-Schiller-University in 1970. His further studies were also strongly influenced and motivated by personal contacts with S. G. Krein, J. Peetre, as well as by new approaches to the theory of function spaces based on Fourier-analytical techniques due to S. M. Nikol'skij, E. M. Stein and C. Fefferman. The development of the modern theory of function spaces in the last 40 years and its application to various branches in both pure and applied mathematics owes much to his seminal contributions. The bare facts are impressive: he has published more than 200 papers in internationally acknowledged journals, and has written no less than 18 monographs and textbooks; the rate of production of new and interesting results shows no sign of decreasing! Perhaps he is best known by the series of books he has written which present systematic treatments of the theory of function spaces from different points of view, thus revealing its symbiotic relationship with interpolation theory, harmonic analysis, partial differential equations, nonlinear operators, entropy, spectral theory, fractal analysis, wavelet theory, and theoretical numerical analysis. In particular, his books Interpolation Theory, Differential Operators, Function Spaces (finished in 1974 and published in 1978) as well as Theory of Function Spaces (based on earlier lecture notes and published in 1983) are much-quoted standard references and have been translated into Russian. At the textbook level, his Higher Analysis and Analysis and Mathematical Physics are masterpieces, relating mathematical theory to physical applications in an extraordinarily convincing way, and made even more vivid in a series of remarkable lecture courses at Jena. Hans Triebel has supervised nearly 40 Ph.D. students, many of whom have become internationally recognised mathematicians. He is on the editorial boards of various international journals, and in particular was an editor of Mathematische Nachrichten for many years. The reputation of the analysis department at the university of Jena owes much to his pioneering scientific work and the activities of his research group on function spaces. His outstanding scientific achievements were recognised by a National Award of the German Democratic Republic for Science and Technology in 1983 and the award of a Doctor of Science honoris causa by the University of Sussex in 1990. He was elected as (Corresponding) Member of the Academy of Science of the German Democratic Republic in 1978. Since 1993 he has been a Member of the Berlin-Brandenburg Academy of Science (formerly the Prussian Academy of Science). No less remarkable than his mathematical ability are his personal qualities, coupling total integrity, resolution and great warmth with an irreverent sense of humour; stories abound of his preparedness to lecture very early in the morning, sometimes to the surprise of the students! We are glad to have this opportunity to express our deep gratitude to him for sharing with so many of his colleagues his ideas and encyclopaedic knowledge. The present collection of papers is a tribute to his distinguished work and reflects recent developments in the theory of function spaces and related fields by outstanding experts. It is a pleasure to thank all the authors for their contributions.