In the first part of the Project we intend to revisit some of the most fundamental operators of harmonic analysis, namely Calderon commutators and the Cauchy integral on Lipschitz curves. Our goal is to give new and conceptually simpler proofs to their well known L^p boundedness properties. As a byproduct of our techniques, we should also be able to give a complete affirmative answer to a question of Coifman regarding the possibility of extending these important results to the multi-parameter setting of a polydisc of arbitrary dimension.
The second part aims to continue the study of the AKNS systems of mathematical physics and their deep connection with Fourier analysis. A beautiful but extremely hard conjecture in the field asks to prove that as long as the entries of the potential matrix of such a system belong to the space of square integrable functions on the real line, its corresponding solutions are all bounded functions. It has been recently observed that the simplest particular case of this conjecture is essentially equivalent to Carleson's fameous theorem on almost everywhere convergence of partial sums of Fourier series. Our main task here is to give a positive answer to this open question, for a large class of potential matrices which strictly includes the upper triangular and lower triangular ones.
We believe that the research of the present Proposal will not only extend and deepen our knowledge about many important operators of classical analysis, but will also provide a better mathematical understanding of various processes in quantum mechanics and nuclear physics.