The content of this proposal concerns a broad array of questions on singular integral operators, a central area of interest in harmonic analysis. The operator which distinguishes positive and negative frequencies of a function is the prototype of a singular integral operator. Good estimates for it in a variety of spaces and the tools developed in the process enjoy applications to many parts of modern harmonic analysis that have experienced rapid growth and widespread recognition in recent years. The investigator is going to tackle the famous problem concerning sharp bounds for the Beurling operator in Lebesgue spaces as well as dimension-free bounds for its multidimensional analog on forms. A broad array of weighted questions will be considered as well, both in the classical setting of kernel oparators as well as in the non-kernel case.
Finally, the PI conducts research in the area of harmonic analysis in product domains and its applications. This area has opened up a broad array of questions and has enjoyed significant growth in recent years. Only recently have harmonic analysis tools been brought to bear these difficult questions. The range of questions pursued by the investigator will require the development of new techniques in Harmonic Analysis.
