Mathematical Sciences: Bilinear Problems in Analysis

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9423678 Lacey Bilinear operators are a source of very difficult questions in harmonic analysis and ergodic theory. A major focus in the current project is the bilinear Hilbert transform. In particular, the investigator will study the difficult problem of Calderon involving the square integrability of the value of this transform. The approach will involve a synthesis of ergodic and Fourier transform techniques. Mathematical analysis involves the study of qualitative and quantitative information about solutions of equations. Frequently, real world equations can be viewed as operator equations and the analysis of solutions involves estimates on operators both globally and locally. This project involves estimation on the value of bilinear operators, that is, operators associated with pairs of functions. The approach to these difficult analytical questions will involve a synthesis of several deep branches of analysis.