Abstract
If the deviation of a function from its averages over all cubes is bounded, then the function is called of bounded mean oscillation (BMO). Bounded functions are of bounded mean oscillation, but there exist unbounded BMO functions. Such functions are slowly growing, and they typically have at most logarithmic blowup. The space BMO shares similar properties with the space L ∞ , and often serves as a substitute for it. For instance, classical singular integrals do not map L ∞ to L ∞ but L ∞ to BMO. And in many instances interpolation between L p and BMO works just as well between L p and L ∞ . But the role of the space BMO is deeper and more far-reaching than that. This space crucially arises in many situations in analysis, such as in the characterization of the L 2 boundedness of nonconvolution singular integral operators with standard kernels.
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