Boundary value problems and sharp inequalities for martingale transforms (Q760963)

Let \(p^*=\max (p,q)\) where \(1<p<\infty\) and \(1/p+1/q=1\). Let \(d=(d_ 1,d_ 2,...)\) be a martingale difference sequence in real \(L^ p(0,1)\), \(\epsilon =(\epsilon_ 1,\epsilon_ 2,...)\) be a sequence of numbers in \(\{\)-1,1\(\}\) and \(n\geq 1\). The author proves that \[ \| \sum^{n}_{k=1}\epsilon_ kd_ k\|_ p\leq (p^*-1)\| \sum^{n}_{k=1}d_ k\|_ p \] and that the constant \((p^*-1)\) is the best possible. He shows further that strict inequality holds iff \(p\neq 2\) and \(\| \sum^{n}_{k=1}d_ k\|_ p>0.\) This result is an improvement over an earlier result of the author [Ann. Math. Stat. 37, 1494-1504 (1966; Zbl 0306.60030)] by giving the best constant and the conditions for equality. The paper contains a number of other sharp inequalities for martingale transforms and stochastic integrals. The underlying method in obtaining these results rests on finding an upper or lower solution to a boundary value problem.