Boundary Value Problems and Sharp Inequalities for Martingale Transforms

Abstract

Let p* be the maximum of p and q where and 1/p+ 1/q= 1. If d=(d1, d2,⋯) is a martingale difference sequence in real Lp (0, 1), ε=(ε1, ε2,⋯) is a sequence of numbers in {-1, 1}, and n is a positive integer, then|∑ nk= 1 εkd k| p≤(p*-1)|∑ nk= 1 dk| p and the constant p*-1 is ...