On Martingale Spaces and Inequalities

Abstract

In the past twenty years, the H _p -BMO theory on ℝ ⁿ has undergone a flourishing development, which should partly give the credit to the applications of some martingale ideas and methods. A number of examples can be taken to show this. The famous Calderón-Zygmund decomposition, one of the key parts of Calderón-Zygmund’s real method, may be said to be a counterpart of stopping time argument in Probability Theory; the atomic decomposition of H _p spaces, the constructive proof of the Fefferman-Stein decomposition of BMO spaces, and the good λ-inequality technique etc. were all first germinated in martingale setting. In addition, there ard also many applications of Martingale Theory to Harmonic Function Theory and many recent applications to Analysis and especially to Harmonic Analysis. Among them, two examples are worth to be mentioned. One is that D. Burtholder described an important kind of Banach spaces (called UMD spaces) by using martingales, another is that by using martingales as a tool, a much more simplified proof of the important T(b) theorem in Calderón-Zygmund Singular Integral Theory was given. From the above-Cited examples we can see what an important role Martingale Theory has played in the development of Analysis, especially of Harmonic Analysis. In this topic, i.e. Martngale Spaces and Inequalities, our Chinese Mathematicians made some contributions too. In what follows, we will list some of them without proofs.