THE INDUCED PROCESS ON THE DIFFEOMORPHISM GROUPS

(A) We shall use some results about the diffeomorphism groups of compact manifolds to obtain information about the flow Ft of an S.O.S. and the uniform convergence of piecewise li.near approximations. This was described, for compact manifolds, i.n (Elworthy 1978). The method mimics that used by Ebin & Marsden (1970) for ordinary differential equations.

Instead of using diffeomorphism groups of compact manifolds some readers might prefer to consider groups of diffeomorphisms of Rn which are the identity outside the unit ball. Many of the properties are easier to see for these groups and they can be used in the compact manifold case by embedding the manifold in some open unit ball and then extending the S.O.S. so that it is the identity in a neighbourhood of the boundary of the ball. This is described in detail in (Carverhill & Elworthy 1982) where the method is used to give a unified treatment of many recent results, e.g. the generalized Itô formula (Bismut 1981a, b) and the criterion for the flow to be a diffeomorphism (Kunita 1981, 1982). A similar technique, using nuclear spaces is employed in (Ustunel).

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.

This Tract has now been out of print for a number of years. Since there still seems to be some demand for it, the Syndics of the Cambridge University Press have judged it desirable to publish a new edition.

However, owing to the vigorous development of Mathematical Probability Theory since 1937, any attempt to bring the book up to date would have meant rewriting it completely, a task that would have been utterly beyond my possibilities under present conditions. Thus I have had to restrict myself in the main to a number of minor corrections, otherwise leaving the work—including the Bibliography—where it was in 1937.

Besides the minor corrections, most of which are concerned with questions of terminology, there are, in fact, only two major alterations. In the first place, a serious error in the statement and proof of Theorem 11 has been put right. Further, the contents of Chapter IV, § 4, which are fundamental for the theory of asymptotic expansions, etc., developed in Chapter VII, have been revised and simplified. This permits a new formulation of the important Lemma 4, on which the proofs of Theorems 24-26 are based. Finally a brief list of recent works on the subject in the English language has been added.

When this Tract was first published in 1937, an important part of it was Chapter VII, containing Liapounoff's classical inequality for the remainder in the Central Limit Theorem, as well as the theory of the related asymptotic expansions. For the Third Edition, this chapter has been partly rewritten, and now brings a proof of the sharper inequality due to Berry and Esseen. Moreover, several minor changes have been made, and the terminology has been somewhat modernized.