Paraproducts With Flag Singularities

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DMS - 0355360Camil Muscalu Cornell UniversityParaproducts with Flag SingularitiesAbstractOver the last few years we have been working on developing furtherthe theory of multilinear singular integrals and their Carleson-type maximal analogs, in connection with the spectral theory of one-dimensionalSchroedinger operators. We understood many of its beautiful features, but still much more work needs to be doneThe present Project is focused on creating a consistent multi-parametertheory for these multilinear operators. The motivation for such a studyis not only our previous work, but also the desire of analystsof having a general Kato-Ponce theory, so useful in PDE.Some of these operators have a very special multi-parameterstructure not seen among the previously studied operators inharmonic analysis. Understanding their combinatorics is a deep and challenging study. Our plan is to handle not only large classes ofsuch paraproducts with "flag singularities", but also "analytic series"of such operators. We believe that these new techniques will have a broadimpact to several problems in nonlinear analysis.