*(English)*Zbl 0981.30026

The subject of bounded mean oscillation (BMO) has been of interest for approximately 40 years, and especially since C. Fefferman’s result of about 30 years ago identifying ${\left({H}^{1}\right)}^{*}$ with BMOA, the space of analytic functions of bounded mean oscillation. While there have been a number of short expositions over the years dealing with limited aspects of the subject, this paper is the first in many years to deal in depth with a wide scope of what is known about BMOA-functions. It also brings an up-to-date account of a subject which is still actively under investigation.

The author gives an extensive account of many aspects of BMOA. This paper is the length of a monograph, and could easily be used as a text for a graduate course or seminar. Proofs are provided for all of the basic results, and also for some of those results that are more specialized. While the paper’s basic intention is to present an account of many aspects of BMOA, there are also a few new results presented. Not every aspect of BMOA is presented, but the treatment of the selected topics given is quite complete and quite broad. There are 117 references cited. The table of contents gives a reasonable idea of the kind of topics covered: 1. Preliminaries and notation, 2. the space BMO and the space VMO, 3. BMO as a conformally invariant space, 4. the John-Nirenberg theorem, 5. the space BMOA and the space VMOA, 6. Littlewood-Paley identities and Carleson measures, 7. Fefferman’s duality theorem, 8. distinct characterizations of BMOA-functions, 9. Taylor coefficients of BMOA-functions, 10. Blaschke products, inner functions, and BMOA, 11. BMOA and univalent functions, 12. growth of the derivative of BMOA-functions, 13. BMOA and mean Lipschitz spaces. This paper is a major work of exposition that anyone with an active interest in investigating the space BMOA to any depth should obtain and consult. It is a valuable reference and guide to the subject.

##### MSC:

30D50 | Blaschke products, etc. (MSC2000) |