2004/09/08 by Narcisse Randrianantoanina, Randrianantoanina, Narcisse · 1 citation
Mathematics · #46L52 #46L53 #Advanced Banach Space Theory #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics #math.FA #math.OA #msc:46L52 #msc:46L53
paper · pdf · doi:10.48550/arxiv.math/0409139
38 pages
arxiv created 2004/09/08 · openalex publication_date 2004/09/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a weak-type (1,1) inequality for square functions of non-commutative martingales that are simultaneously bounded in L2 and L1. More precisely, the following non-commutative analogue of a classical result of Burkholder holds: there exists an absolute constant K>0 such that if \calM is a semi-finite von Neumann algebra and (\calMn)∞n=1 is an increasing filtration of von Neumann subalgebras of \calM then for any given martingale x=(xn)∞n=1 that is bounded in L2(\calM)∩ L1(\calM), adapted to (\calMn)∞n=1, there exist two \underlinemartingale difference sequences, a=(an)n=1^∞ and b=(bn)n=1^∞, with dxn = an + bn for every n≥ 1, | (∑^∞n=1 an^*an)^1/2|2 + | (∑^∞n=1 bnbn^*)1/2|2 ≤ 2| x |2, and | (∑^∞n=1 an^*an)^1/2|1,∞ + | (∑^∞n=1 bnbn^*)1/2|1,∞ ≤ K| x |1. As an application, we obtain the optimal orders of growth for the constants involved in the Pisier-Xu non-commutative analogue of the classical Burkholder-Gundy inequalities.