Doob's inequalities revisited: A maximal \(H^ 1\)-embedding
type: Item
instance of: scholarly article
title: Doob's inequalities revisited: A maximal \(H^ 1\)-embedding
zbMATH Open document ID: 0653.60038
DOI: 10.1016/0304-4149(88)90043-9
author: S. D. Jacka
published in: Stochastic Processes and their Applications
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60G07
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 60E99
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B30
zbMATH DE Number: 4066045
zbMATH Keywords: maximal inequality
zbMATH Keywords: embedding into Brownian motion
MaRDI profile type: Publication
full work available at URL: 0304-4149(88)90043-9
OpenAlex ID: W1966652723
cites work: Q3049602
cites work: Q3049603
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cites work: A converse to the dominated convergence theorem
cites work: Q5822308
cites work: Every nonnegative submartingale is the absolute value of a martingale
cites work: Optimal stopping and best constants for Doob-like inequalities. I: The case \(p=1\)
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description: scientific article; zbMATH DE number 4066045