Exposés de recherche

Collection Exposés de recherche

00:00:00 / 00:00:00
56 380

Two-weight inequalities meet $R$-boundedness

By Tuomas P. Hytönen

Also appears in collection : Banach spaces and their applications in analysis / Espaces de Banach et applications à l'analyse

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

Information about the video

Citation data

  • DOI 10.24350/CIRM.V.18665403
  • Cite this video Hytönen, Tuomas P. (13/01/2015). Two-weight inequalities meet $R$-boundedness. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.18665403
  • URL https://dx.doi.org/10.24350/CIRM.V.18665403

Bibliography

  • 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 $p > q$. <arXiv:1412.2593 > - http://arxiv.org/abs/1412.2593

Last related questions on MathOverflow

You have to connect your Carmin.tv account with mathoverflow to add question

Ask a question on MathOverflow




Register

  • Bookmark videos
  • Add videos to see later &
    keep your browsing history
  • Comment with the scientific
    community
  • Get notification updates
    for your favorite subjects
Give feedback