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Oscillatory integrals and extremal problems in harmonic analysis


Oscillatory integrals appear naturally in a variety of problems related to harmonic analysis, and have been a part of the subject since its creation. One is generally interested in quantifying how certain degeneracies of the phase affect the behavior of the integral. Numerous generalizations of the so-called stationary phase method have been the focus of past and current research, and the study of multilinear oscillatory forms with rough amplitudes falls into this category.

In the first part of this thesis, in a joint work with Michael Christ, we examine a certain class of trilinear integral operators which incorporate oscillatory factors e^iP , where P is a real-valued polynomial, and prove smallness of such integrals in the presence of rapid oscillations.

Oscillatory integrals provide a link between geometric properties of manifolds and harmonic analysis related to them, as illustrated by a multitude of restriction theorems for the Fourier transform which have been the object of careful investigation since the late 1960's.

In the second part of this thesis, we establish the existence of extremizers for a Fourier restriction inequality on planar convex arcs whose curvature satisfies a natural assumption. More generally, we prove that any extremizing sequence of nonnegative functions has a subsequence which converges to an extremizer. By studying the three-fold convolution of arclength measure on the curve with itself we additionally show that, if the geometric assumption on the curvature fails in a strong sense, then extremizing sequences concentrate at a point on the curve and extremizers do not exist.

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