Wavelet characterization of functions with conditions on the mean oscillation

The space BMO of those real functions defined on IRn for which the mean oscillation over cubes is bounded, appears in the pioneer works of J. Moser [13] and John-Nirenberg [9] in the early sixties as a tool for the study of regularity of weak solutions of elliptic and parabolic differential equations. Their main result, known today as John-Nirenberg Theorem, provides a characterization of BMO in terms of the exponential decay of the distribution function on each cube. Although the depth of this result, the space BMO only became well known in harmonic analysis after the celebrated Fefferman-Stein theorem of duality for the Hardy spaces: BMO is the dual of the Hardy space H1. Since H1 was already known to be the good substitute of L1 for many questions in analysis, the space BMO was realized as the natural substitute of L∞ in the scale of the Lebesgue spaces. Also due to Fefferman and Stein is the Littlewood-Paley type characterization of BMO in terms of the derivatives of the harmonic extension and Carleson measures (see for example [16]). This result has a discrete version: the Lemarié-Meyer characterization of BMO using wavelets, [10].