UCLA

The first part of the thesis focuses on the uniformity of harmonic analysis estimates on curves. We first show a decomposition theorem for polynomial curves on local fields as a bounded number of perturbations of monomial curves. Using this theorem, we extend uniform restriction estimates for real curves to the endpoint case, show uniform decoupling for those curves, and show novel uniform restriction estimates for curves over C, and Qp. We then show uniform estimates for the discrete analog to this problem in a restricted range of exponent.

The second part focuses on decoupling estimates for sets with a product or self-similar structure. A recurring phenomenon for those sets is that functions with constant Fourier transform on their support are far from extremizers. As applications we will show a de- coupling estimate for fractal subsets of the parabola, and study subsets of cubes with high additive energy compared to their cardinality.