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bounded mean oscillation
In harmonic analysis in mathematics, a function of bounded mean oscillation, also known as a BMO function, is a real-valued function whose mean oscillation is bounded (finite). The space of functions of bounded mean oscillation (BMO), is a function space that, in some precise sense, plays the same role in the theory of Hardy spaces ''Hp'' that the space ''L''∞ of essentially bounded functions plays in the theory of ''Lp''-spaces: it is also called John–Nirenberg space, after Fritz John and Louis Nirenberg who introduced and studied it for the first time.
== Historical note ==
According to ,〔Aside with the collected papers of Fritz John, a general reference for the theory of functions of bounded mean oscillation, with also many (short) historical notes, is the noted book by .〕 the space of functions of bounded mean oscillation was introduced by in connection with his studies of mappings from a bounded set belonging to R''n'' into R''n'' and the corresponding problems arising from elasticity theory, precisely from the concept of elastic strain: the basic notation was introduced in a closely following paper by ,〔The paper just precedes the paper in volume 14 of the Communications on Pure and Applied Mathematics.〕 where several properties of this function spaces were proved. The next important step in the development of the theory was the proof by Charles Fefferman〔Elias Stein credits only Fefferman for the discovery of this fact: see .〕 of the duality between BMO and the Hardy space ''H''1, in the noted paper : a constructive proof of this result, introducing new methods and starting a further development of the theory, was given by Akihito Uchiyama.〔See his proof in the paper .〕
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