Noncommutative strong maximals and almost uniform convergence in several directions
| Autor: | Adrián M. González-Pérez, Javier Parcet, José M. Conde-Alonso |
|---|---|
| Přispěvatelé: | European Commission, Ministerio de Economía y Competitividad (España) |
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Statistics and Probability
Pure mathematics Uniform convergence 01 natural sciences Theoretical Computer Science 0103 physical sciences Classical Analysis and ODEs (math.CA) FOS: Mathematics Discrete Mathematics and Combinatorics Ergodic theory Almost everywhere 0101 mathematics Operator Algebras (math.OA) Commutative property Mathematical Physics Mathematics Sequence Algebra and Number Theory Semigroup 010102 general mathematics Probability (math.PR) Mathematics - Operator Algebras Length function Noncommutative geometry Functional Analysis (math.FA) Mathematics - Functional Analysis Computational Mathematics Mathematics - Classical Analysis and ODEs 010307 mathematical physics Geometry and Topology Analysis Mathematics - Probability |
| Zdroj: | Digital.CSIC. Repositorio Institucional del CSIC instname |
| Popis: | Our first result is a noncommutative form of the Jessen-Marcinkiewicz-Zygmund theorem for the maximal limit of multiparametric martingales or ergodic means. It implies bilateral almost uniform convergence (a noncommutative analogue of almost everywhere convergence) with initial data in the expected Orlicz spaces. A key ingredient is the introduction of the Lp -norm of the lim¿sup of a sequence of operators as a localized version of a ¿¿/c0 -valued Lp -space. In particular, our main result gives a strong L1 -estimate for the lim¿sup ¿as opposed to the usual weak L1,¿ -estimate for the sup ¿with interesting consequences for the free group algebra. Let LF2 denote the free group algebra with 2 generators, and consider the free Poisson semigroup generated by the usual length function. It is an open problem to determine the largest class inside L1(LF2) for which the free Poisson semigroup converges to the initial data. Currently, the best known result is Llog2L(LF2) . We improve this result by adding to it the operators in L1(LF2) spanned by words without signs changes. Contrary to other related results in the literature, this set grows exponentially with length. The proof relies on our estimates for the noncommutative lim¿sup together with new transference techniques. We also establish a noncommutative form of Córdoba/Feffermann/Guzmán inequality for the strong maximal: more precisely, a weak (¿,¿) inequality¿as opposed to weak (¿,1) ¿for noncommutative multiparametric martingales and ¿(s)=s(1+log+s)2+¿ . This logarithmic power is an ¿ -perturbation of the expected optimal one. The proof combines a refinement of Cuculescu¿s construction with a quantum probabilistic interpretation of M. de Guzmán¿s original argument. The commutative form of our argument gives the simplest known proof of this classical inequality. A few interesting consequences are derived for Cuculescu¿s projections. Supported by the European Research Council. Consolidator Grant 614195- RIGIDITY.†Partially supported by CSIC GrantPIE-201650E030and ICMAT Severo Ochoa Grant SEV-2015-0554. |
| Databáze: | OpenAIRE |
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