A sharp weak type $(p,p)$ inequality $(p>2)$ for martingale transforms and other subordinate martingales

Abstract

If (d n ) n≥o is a martingale difference sequence, (e n ) n>0 a sequence of numbers in {1, -1}, and n a positive integer, then P( max 0≤m≤n | m Σ k=0 e k d k | ≥ 1) ≤ α p ∥ n Σ k=0 d k ∥ p p . Here α p denotes the best. constant. If 1 2, and that p p-1 /2 is also the best constant in the analogous inequality for two martingales M and N indexed by [0,∞), right continuous with limits from the left, adapted to the same filtration, and such that [M, M] t - [N, N] t is nonnegative and nondecreasing in t. In Section 7, we prove a similar inequality for harmonic functions.