Classical Sobolev spaces, based on Lebesgue spaces on an underlying domain with smooth boundary, are not only of considerable intrinsic interest but have for many years proved to be indispensible in the study of partial differential equations and variational problems. Of the many developments of the basic theory since its inception, two are of particular interest:
(i) the consequences of working on space domains with irregular boundaries;
(ii) the replacement of Lebesgue spaces by more general Banach function spaces.
Both of these arise in response to concrete problems, for example, with the (ubiquitous) sets with fractal boundaries.
These aspects of the theory will probably enjoy substantial further growth, but even now a connected account of those parts that have reached a degree of maturity makes a useful addition to the literature. Accordingly, the main themes of this book are Banach spaces and spaces of Sobolev type based on them; integral operators of Hardy type on intervals and on trees; and the distribution of the approximation numbers (singular numbers in the Hilbert space case) of embeddings of Sobolev spaces based on generalised ridged domains.
The significance of generalised ridged domains stems from their ability to 'unidimensionalise' the problems we study, reducing them to associated problems on trees or even on intervals.
This timely book will be of interest to all those concerned with the partial differential equations and their ramifications. A prerequisite for reading it is a good graduate course in real analysis.
From the reviews:
"This interesting monograph is the second joint book by the two authors. Whereas in their first one ? one of the main objects was the study of spectral theory of boundary value problems for elliptic differential operators, the concentration is now more on the function space side ? . surely reflects the current state of research in the above-described field. It makes a useful addition to the corresponding literature and will be of great help for those working in this field already ... ." (Dorothee D. Haroske, Mathematical Reviews, 2005g)
