Topics in Harmonic Analysis

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ABSTRACTRecent years have seen a burst of activity in the interface of harmonicanalysis and its related fields (such as partial differential equations, combinatorics, number theory, and mathematical physics) with beautiful and mutually beneficial consequences. Long-standing open problems in harmonic analysis have been solved using combinatorial techniques, while tools from harmonic analysis have helped in proving exciting results in number-theory and mathematical physics. In this project, the PI outlines future goals in three areas of harmonic analysis: (a) Fourier estimates for cones and their applications (section 2), involving a local smoothing estimate of Wolff (b) analysis of linear and monomial polyhedra, and their applications to (i) associated maximal operators and (ii) Bergman kernel estimates for weakly pseudoconvex domains (section 3), and (c) decay properties of oscillatory integral operators with degenerate phases section 4).Projects 1, 2 and 3 address a class of problems that may be approachedusing Wolff's microlocal estimate for light cones (arising in his proof of the local smoothing conjecture), and would answer a variety of questions regarding averaging and Fourier integral operators and partial differential equations. They necessitate a deeper understanding and extension of the delicate combinatorial arguments employed in his proof. Results from project 4 would mark nontrivial progress on a much-studied open question on Bergman kernel estimates, and would be of significant interest in complex analysis of several variables. Project 5 addresses a problem of which very little is known at the moment, but which has deep ties not only to numerous problems in harmonic analysis (regarding degenerate Fourier integral operators), but also to an extremely rich and thriving area of algebraic geometry, namely resolution of singularities.