2005/08/24 by Javier Parcet, Parcet, Javier
Mathematics · #42B25 #60G46 #60G50 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR) #math.FA #math.PR #msc:42B25 #msc:60G46 #msc:60G50
paper · pdf · doi:10.48550/arxiv.math/0508447
20 pages
openalex publication_date 2005/08/24 · arxiv created 2005/08/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a probability space (Ω, A, μ), let A1, A2, ... be a filtration of σ-subalgebras of A and let E1, E2, ... denote the corresponding family of conditional expectations. Given a martingale f = (f1, f2, ...) adapted to this filtration and bounded in Lp(Ω) for some 2 ≤ p < ∞, Burkholder's inequality claims that ‖f‖Lp(Ω) ∼cp ‖ (∑k=1^∞ Ek-1(|dfk|2) )1/2 ‖_Lp(Ω) + (∑k=1^∞ ‖dfk‖pp )1/p. Motivated by quantum probability, Junge and Xu recently extended this result to the range 1 < p < 2. In this paper we study Burkholder's inequality for p=1, for which the techniques (as we shall explain) must be different. Quite surprisingly, we obtain two non-equivalent estimates which play the role of the weak type (1,1) analog of Burkholder's inequality. As application, we obtain new properties of Davis decomposition for martingales.