Uniform Convolution and Fourier Restriction estimates for complex polynomial curves in $\mathbb{C}^3$

Abstract

We establish optimal (p, q) ranges for two types of estimates associated to three dimensional complex polynomial curves. These are the estimates for the weighted restriction of the Fourier Transform to a complex polynomial curve, and the weighted Convolution Operator associated to a complex polynomial curve. Establishing these estimates comes down to establishing a lower bound for the Jacobian of a mapping associated to the complex curve in question.