Topics in Phase Plane Analysis

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

Proposal Number: DMS-0200241 PI: Michael Lacey ABSTRACT Research will be conducted on several aspects of phase plane analysis. The specific questions come from three different areas. One is to study a conjecture of E.M. Stein on a certain extension of Carleson's theorem. This conjecture seeks to place Carleson in a larger context. Consider this theorem as a supremum over all linear choices of phase functions. Stein's conjecture concerns a supremum over all polynomial choices of phase functions, with however the degree of the polynomial fixed. Advances on this question already suggest that it might be fruitful to consider certain discrete analogues of Carleson's theorem. Here, arithmetic aspects of Number Theory, as exemplified by the Hardy-Littlewood Circle Method should predominate. A third area is continued investigation into the question of the boundedness of the Hilbert transform on smooth families of lines. These investigations seek to deepen our understanding of phase plane analysis: The study of functions in the temporal and oscillatory variables simultaneously. Past successes in this area have lead to a broadening of the synthetic and analytic techniques that can be brought to bear on a wide range of questions. This is especially true for the bilinear Hilbert transform and Carleson's theorem. These new questions are far more subtle than those addressed in the past, and so the new techniques that we seek to develop are hoped to be more powerful than previous ones.