The trends in function spaces have always been dictated by the needs in the areas where function spaces are applied, for example partial differential equations, mathematical physics, and probability theory. The most important recent trend in function spaces is the optimality, or sharpness, of the obtained embedding results. Moreover, particular attention is paid to limiting (or critical) cases. It is very important to show that a certain result cannot be made any better unless a new category of objects is introduced. A good choice of optimal function space can often solve a difficult problem. The qualities of a function, studied in the theory of function spaces, vary in dependence on the background application where the particular problem arose.
The book under review presents a thorough investigation of two (and maybe the most) important such qualities of a function, namely its size (or growth) and its continuity (or smoothness). The author presents a new approach to the topic, based on studying the so-called growth envelopes and continuity envelopes. One of the basic ideas involved is to study how big the non-increasing rearrangement of a function can be, given that the function belongs to a certain function space. This task is perhaps similar to the question of finding the smallest Marcinkiewicz space containing the given space. The approach, built upon an impressive series of the author's results, has turned into a worthwhile general theory full of beautiful, deep results, interesting examples and plenty of applications (to name just two examples, let us mention the asymptotic behaviour of the approximation numbers and the compactness of embeddings). All this the reader will find in the text. On top of that, the book is more reader-friendly than the standard; it can be easily read as a ‘bed-time fairy tale’ (my personal experience). The book comes from one of the world’s most famous centres of function space theory, the Friedrich-Schiller University in Jena, which was created and is still led by Professor Hans Triebel. Many wonderful books have already arrived from this University and here we have another one (and certainly not the last one). Truly delightful stuff!