Doob's inequalities revisited: A maximal \(H^ 1\)-embedding (Q1107901)

With the aim of finding best constants in Doob's \(L^ p\)-inequalities for nonnegative submartingales the author first establishes a maximal inequality and then an embedding into Brownian motion, with the aid of which he shows that his inequality is sharp. An important fact used is that every nonnegative submartingale can be written as the absolute value of a martingale, first proved by ly continuous spectrum. Theorem 4.1. If for \(p\in {\mathcal P}\) \([c_ 1,\infty)\subset N(w_ p)\subset [c_ 2,\infty)\), \(I(\lambda)=2^{-1}\int ((\xi -\lambda)^{- 1}-\xi (1+\xi)^{-1}d\xi\), \(u=\exp I\), then for \(q\in \sup p p\) there exists a singular measure \(\nu\) on \([c_ 1,c_ 2]\) and solutions \(f_{1,2}\) of \(L(q)p=\xi f\) for a.e. \(\xi\) with respect to \(\nu\) such that if \(g_{\lambda}(x,y)\) is the Green function of L(q), then \[ u(\lambda)g_{\lambda}(x,x)=\int (f^ 2_ 1+f^ 2_ 2)(\xi - \lambda)^{-1}d\nu,\quad \int (f^ 2_ 1+f^ 2_ 2)d\nu =1. \] This theorem extends results by Moser for finite zone potentials to the general ergodic case.