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. Many developments of the basic theory since
its inception arise in response to concrete problems, for example,
with the (ubiquitous) sets with fractal boundaries.
The theory will probably enjoy substantial further growth, but
even now a connected account of the mature parts of it 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.
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.
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