Extremal problems in Bounded Mean Oscillation

The concept of mean oscillation in analysis is akin to that of variance in probability theory and statistics. It tells us how much the function deviates (in the mean) from its mean value. Of special interest are the functions of bounded mean oscillation, and the space populated by them is called BMO. While originally introduced by John and Nirenberg in the context of elasticity theory, it later reappeared -- unexpectedly enough -- in connection with Hardy spaces and soon became one of the main characters in harmonic analysis.If F is a function living on the unit circle, and if its "mean values" are understood as the values of its harmonic extension (alias Poisson integral) at points of the unit disk D, the condition for F to be in BMO is that the associated "variance" be bounded on D. The square root of the supremum that arises is known as the Garsia norm of F. Hence every bounded function is in BMO, and the Garsia norm does not exceed the uniform norm.Now, we are concerned with those (bounded) functions F for which the two norms agree; in a sense, this means that F oscillates as much as possible. We call such functions G-extremal, and we uncover many of their amusing properties. In the analytic case, the canonical (inner-outer) factorization of G-extremal functions is studied in detail. We also discuss the extreme points of the unit ball in BMO with respect to the Garsia norm. Finally, we find out what happens if BMO gets replaced by VMO, the space of functions of vanishing mean oscillation.