These notes address the connection between two subjects, and they are thus intended to form an introduction to both but to be about neither. The discoveries of Fefferman and Stein about HP and BMO have interacted fruitfully with a great deal of work on the analogous ideas in martingale theory; the main goal of the following pages is an explanation of the fundamental result of Burkholder, Gundy, and Silver stein, which forms the bridge between these two areas of investigation. The exposition is at as elementary a level as possible, and it is intended in particular to be accessible to graduate students with a basic knowledge of measure theory, complex analysis and functional analysis. For the sake of those not familiar with probability theory, many probabilistic results are introduced and proved as needed, and there is a chapter without proofs on Brownian motion. Again, for those not on everyday terms with classical function theory, a survey of results on the maximal, square, and Littlewood-Paley functions is included, and function-theoretic arguments are given and estimates made in considerable detail. The discussion is restricted mainly to the case of the unit disk in the complex plane. I hope that one who reads these notes will find that GarsiaTs book, the papers of Fefferman and Stein, and the writings of Burkholder, Davis, Gundy, Herz, Silverstein, et al. on these topics are easily approachable.

The purpose of this chapter is to define and construct Brownian motion and to list enough of its properties so that we will be able to carry out the subsequent arguments which rely on them. We omit most proofs and suggest that the interested reader refer to [14, 18, 20, 27, 34, 36, 40, 43, 44, or 58] for more details.

The strange spontaneous movements of small particles suspended in liquid were first studied by Robert Brown, an English botanist, in 1828, although they had apparently been noticed much earlier by other scientists, including even Leeuwenhoek. L. Bachelier gave the first mathematical description of the phenomenon in 1900, going so far as to note the Markov property of the process. In 1905 A. Einstein and, independently and around the same time, M. v. Smoluchowski developed physical theories of Brownian motion which could be used, for example, to determine molecular diameters. It is interesting that Einstein says that at that time he had never heard of the actual observations of Brownian motion, and that he happened to deduce the existence of such a process in the course of some purely theoretical work on statistical mechanics and thermodynamics [50, p. 47], The mathematical theory of Brownian motion was invented in 1923 by N. Wiener, and accordingly the Brownian motion that we will be working with is frequently called the Wiener process.

In 1915 G. H. Hardy, answering a question of Bohr and Landau, investigated properties of the mean over a circle of the modulus |F| of an analytic function F which were similar to those of the maximum value of |F| over a disk. He found that his results applied also to |F|P for p < 0, and thus was founded the theory of HP spaces. Since then these Hardy spaces have been the object of much research, and their connections with such diverse subjects as classical function theory (especially the boundary behavior of analytic functions), potential theory (including the theory of harmonic functions and partial differential equations), Fourier series, functional analysis, and operator theory (for example Beurling's work on invariant subspaces of the shift operator) have been developed in considerable detail.

An entirely new line of investigation for the Hardy spaces was uncovered in 1971 by Burkholder, Gundy, and Silverstein when they showed that for 0 < p < ∞ an analytic function F = u + iũ is in HP if and only if the maximal function of u is in LP. Surprisingly, their arguments were probabilistic in nature, being carried out by manipulation of Brownian motion in the complex plane. Their result showed that the Hardy spaces could be characterized in real-variable terms and thus Hp theory could be easily extended to higher dimensions and more general kinds of spaces.

The sets of functions which form the subject matter of this book are to be considered as sequences in metric spaces. Actually we shall be almost exclusively concerned with various Lp spaces, particularly the case p = 2, and with subspaces of such spaces. Although the notes which follow in §1.1 and §1.2 contain sufficient metric space theory for an understanding of the rest of the book, the reader who is new to metric spaces may wish to fill in from a good text such as Copson (1967). For general background reading Simmons (1963) is also highly recommended.

Throughout the book an effort has been made to present theorems which are sufficiently general to be ‘useful’, but in a small introductory book of this kind a great deal of detail has to be left out; adequate references are given for those who want to consult more advanced sources.

1.1 Notes on metric spaces

1.1.1 Vector space It is assumed that the reader is familiar with elementary set theory. The word ‘space’ is used in mathematics to mean a set with some internal ‘structure’. Let V be a set whose elements are to be called vectors, and let F be a field (the field of scalars; we will usually take it to be the field of complex numbers). The basic structure that we shall require for V is that it be closed under an operation of addition of two vectors u and v, denoted by u + v, and an operation of multiplication of a vector u by a scalar f of F, denoted by fu.

Some years ago I came across the need for precise information concerning the basis properties of sets of special functions, and the methods available for testing for such properties. This material proved to be rather widely scattered, so I began a collection of notes on the subject which have formed the foundations of the present little book.

I hope that the book will prove useful to graduate students of mathematics, particularly those whose research interests are developing in the direction of bases in Hilbert and Banach spaces: it could bridge the gap that exists between the scant treatment this topic usually receives in standard texts on functional analysis on the one hand, and the rather formidable specialist books such as Marti (1969) and Singer (1970) on the other. There is no harm in having some experience on the practical side of the business before aiming to become managing director!

I hope the book will appeal to workers in other scientific fields as well. An appendix has been included which lists many of the standard results, and this may help to make the book useful as a source of reference.

It is assumed that the reader's education will have included the usual first courses in real variable (including Lebesgue integration) and complex variable.