S. M. Nikol'skii - Böcker

The theory of spaces of differentiable functions in several variables is treated in detail in this volume. A large number of results obtained in the past 30 years are included. The authors undertake to present the historical development of the theory of imbedding of function spaces, as well as the current state of the art in the field. The book is addressed both to researchers and to graduate students. All the definitions and formulations of the main theorems which are necessary to understand the modern literature on spaces of differentiable functions can be found in this volume.

In the Part at hand the authors undertake to give a presentation of the historical development of the theory of imbedding of function spaces, of the internal as well as the externals motives which have stimulated it, and of the current state of art in the field, in particular, what regards the methods employed today. The impossibility to cover all the enormous material connected with these questions inevitably forced on us the necessity to restrict ourselves to a limited circle of ideas which are both fundamental and of principal interest. Of course, such a choice had to some extent have a subjective character, being in the first place dictated by the personal interests of the authors. Thus, the Part does not constitute a survey of all contemporary questions in the theory of imbedding of function spaces. Therefore also the bibliographical references given do not pretend to be exhaustive; we only list works mentioned in the text, and a more complete bibliography can be found in appropriate other monographs. O.V. Besov, v.1. Burenkov, P.1. Lizorkin and V.G. Maz'ya have graciously read the Part in manuscript form. All their critical remarks, for which the authors hereby express their sincere thanks, were taken account of in the final editing of the manuscript.