Bilinear Singular Integrals

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ABSTRACT Lacey DMS-9706884 Lacey, in collaboration with C. Thiele, has made substantial progress towards settling the conjecture of Alberto Calderon concerning the boundedness of the bilinear Hilbert transform from L^2 times L^2 into L^1. Lacey and Thiele will work on developing a more complete understanding of the theory of bilinear singular integrals. Their method of proof can be refined to the point at which three significant aspects of harmonic analysis can be understood from the same group of basic principles. They are the Cauchy integral on Lipschitz curves, Carleson's Theorem on the pointwise convergence of Fourier series and bilinear singular integrals and maximal functions. The method of proof is based on a novel use of phase plane tecniques. The applications of these techniques to other questions will be explored. A musical score is a representation of the music built around temporal and frequency aspects. A phase plane analysis is an adaptive counterpart of the musical score. The hallmark of such an analysis is precisely that of a musical score, a high degree of localization in time and frequency variables. This proposal pursues the study of phase plane analysis and its applications to operators in Harmonic Analysis which are difficult to compute due to their non--local nature. Harmonic Analyisis provides the mathematical foundation for Signal Processing, most recently popularized by the theory of wavelets.