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. According to , the space of functions of bounded mean oscillation was introduced by in connection with his studies of mappings from a bounded set Ω belonging to Rn into Rn 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 , 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 of the duality between BMO and the Hardy space H1, 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. The mean oscillation of a locally integrable function u over a hypercube Q in Rn is defined as the value of the following integral: where |Q| is the volume of Q, i.e. its Lebesgue measure uQ is the average value of u on the cube Q, i.e. A BMO function is a locally integrable function u whose mean oscillation supremum, taken over the set of all cubes Q contained in Rn, is finite. Note 1. The supremum of the mean oscillation is called the BMO norm of u. and is denoted by ||u||BMO (and in some instances it is also denoted ||u||∗). Note 2. The use of cubes Q in Rn as the integration domains on which the is calculated, is not mandatory: uses balls instead and, as remarked by , in doing so a perfectly equivalent definition of functions of bounded mean oscillation arises.