The first edition of this book was published in 1980 in the LMS Lecture Note Series, and a Russian translation by V.V. Peller and A.G. Tumarkin, made under the direction of V.P. Havin, the editor, appeared in 1984.

Both versions of the book are now out of print, and for the past couple of years people have been asking me how they might procure a copy of it. The Cambridge University Press has therefore decided to put out a second edition, and I am grateful to Dr. David Tranah, the Press' senior mathematics editor, for his having arranged to issue it in somewhat improved typographical format as a Cambridge Tract.

In preparing the first edition I had tried to make the exposition as accessible as I could by concentrating on what I thought were the main ideas in the subject rather than on including as many results as possible. The readers I had in mind were those with some training in analysis who were trying to gain a secure foothold in the theory of Hp spaces, whether with the aim of eventually doing serious work in that subject or for the purpose of understanding its applications in other areas (e.g. in operator theory – some of the material is now even used in electrical engineering). I have been guided by the same concern while working on the second edition and have for that reason tried to preserve the book's original character.

These are the lecture notes for a course I gave on the elementary theory of Hp spaces at the Stockholm Institute of Technology (tekniska högskolan) during the academic year 1977–78. The course concentrated almost exclusively on concrete aspects of the theory in its simplest cases; little time was spent on the more abstract general approach followed, for instance, in Gamelin's book. The idea was to give students knowing basic real and complex variable theory and a little functional analysis enough background to read current research papers about Hp spaces or on other work making use of their theory. For this reason, more attention was given to techniques and to what I believed were the ideas behind them than to the accumulation of a great number of results.

The lectures, about Hp spaces for the unit circle and the upper half plane, went far enough to include interpolation theory and BMO, but not as far as the corona theorem. That omission has, however, been put to rights in an appendix, thanks to T. Wolff's recent work. His proof of the corona theorem given there is a beautiful application of some of the methods developed for the study of BMO.

For Carleson's original proof of the corona theorem the reader may consult Duren's book. I have not included the more recent applications of the geometric construction Carleson devised for that proof, such as Ziskind's.