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H 1 Two-weight inequalities meet $R$-boundedness

Auteurs : Hytönen, Tuomas P. (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : One of my recent main interests has been the characterization of boundedness of (integral) operators between two $L^p$ spaces equipped with two different measures. Some recent developments have indicated a need of "Banach spaces and their applications" also in this area of Classical Analysis. For instance, while the theory of two-weight $L^2$ inequalities is already rich enough to deal with a number of singular operators (like the Hilbert transform), the $L^p$ theory has been essentially restricted to positive operators so far. In fact, a counterexample of $F$. Nazarov shows that the common "Sawyer testing" or "David-Journé $T(1)$" type characterization will fail, in general, in the two-weight $L^p$ world. What comes to rescue is what we so often need to save the $L^2$ results in an Lp setting: $R$-boundedness in place of boundedness! Even in the case of positive operators, it turns out that a version of "sequential boundedness" is useful to describe the boundedness of operators from $L^p$ to $L^q$ when $q < p$. - This is about my recent joint work with T. Hänninen and K. Li, as well as the work of my student E. Vuorinen.

    two-weight inequalities - boundedness - singular operators

    Codes MSC :
    42B25 - Maximal functions, Littlewood-Paley theory
    47G40 - Potential operators [See also 31-XX]

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 21/01/15
      Date de captation : 13/01/15
      Collection : Research talks ; Analysis and its Applications
      Format : quicktime ; audio/x-aac
      Durée : 00:55:14
      Domaine : Analysis and its Applications
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2015-01-13_Hytonen.mp4

    Informations sur la rencontre

    Nom de la rencontre : Banach spaces and their applications in analysis / Espaces de Banach et applications à l'analyse
    Organisateurs de la rencontre : Albiac, Fernando ; Casazza, Peter G. ; Godefroy, Gilles ; Lancien, Gilles
    Dates : 12/01/15 - 16/01/15
    Année de la rencontre : 2015

    Citation Data

    DOI : 10.24350/CIRM.V.18665403
    Cite this video as: Hytönen, Tuomas P. (2015). Two-weight inequalities meet $R$-boundedness. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18665403
    URI : http://dx.doi.org/10.24350/CIRM.V.18665403


    Bibliographie

    1. Hänninen, T.S., Hytönen, T.P., & Li, K. (2014). Two-weight $L^p$-$L^q$ bounds for positive dyadic operators: unified approach to $p \leq q$ and $p > q$. - http://arxiv.org/abs/1412.2593

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