2019/10/23 by José M. Conde‐Alonso, Conde-Alonso, José M., Adrián M. González‐Pérez +3 · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Advanced Operator Algebra Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1910.10551
openalex publication_date 2019/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Our first result is a noncommutative form of Jessen/Marcinkiewicz/Zygmund\ntheorem for the maximal limit of multiparametric martingales or ergodic means.\nIt implies bilateral almost uniform convergence with initial data in the\nexpected Orlicz spaces. A key ingredient is the introduction of the Lp-norm\nof the limsup of a sequence of operators as a localized version of a\n\ℓ_\∞/c0-valued Lp-space. In particular, our main result gives a\nstrong L1-estimate for the limsup, as opposed to the usual weak\nL1,\∞-estimate for the \sup.\n Let \L \F2 denote the free group algebra and consider the\nfree Poisson semigroup generated by the usual length function. It is an open\nproblem to determine the largest class inside L1(\L \F2)\nfor which this semigroup converges to the initial data. Currently, the best\nknown result is L \log2 L(\L \F2). We improve this by\nadding to it the operators in L1(\L \F2) spanned by words\nwithout signs changes. Contrary to other related results in the literature,\nthis set has exponential growth. The proof relies on our estimates for the\nnoncommutative limsup together with new transference techniques.\n We also establish a noncommutative form of C 'ordoba/Feffermann/Guzm 'an\ninequality for the strong maximal. More precisely, a weak (\Φ,\Φ)\ninequality for noncommutative multiparametric martingales and \Φ(s) = s (1 +\n\log+ s)2 + \ε. This logarithmic power is an\n\ε-perturbation of the expected optimal one. The proof combines a\nrefinement of Cuculescu's construction with a quantum probabilistic\ninterpretation of de Guzm 'an's argument. The commutative form of our argument\ngives the simplest known proof of this classical inequality. A few interesting\nconsequences are derived for Cuculescu's projections.\n