Φ-inequalities on Martingales

The L p -inequalities introduced in the preceding chapter, concerning various operators on martingales (such as maximal operator, square operator and conditioned square operator, etc.), could be extended to so-called Φ-inequalities. In this chapter, we will study them according to Φ’s different situations: Φ is convex, or convex in a strict sense, or concave, or merely of moderate growth (we will call such Φ “general Φ” for the simplicity). §3.1 will be devoted to some elementary properties of convex functions, especially to the moderate growth property, and two related indices. §3.2–3.5 will be devoted to the general theory of Φ -inequalities in four cases, which may be useful in some domains other than martingale, then to -inequalities on martingales. The last section §3.6 will be devoted to the rearrangement technique in martingale setting. This technique is almost equivalent to “distribution” one with slight differences between them. Roughly speaking, by means of the latter, we could get general Φ-inequalities, and by means of the former we lose some generality of Φ, but as a compensation, we get better constants. In this section, we keep the assumptions imposed on (Ω, F, μ, {F n }n≥0) as before.