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

TL;DR

This paper derives optimal Fourier restriction and convolution estimates for complex polynomial curves in three dimensions, advancing understanding of harmonic analysis in complex settings.

Contribution

It provides the first optimal $(p,q)$ ranges for restriction and convolution estimates related to complex polynomial curves in $ extbf{C}^3$, including Jacobian bounds.

Findings

Optimal $(p,q)$ ranges established for restriction estimates

Optimal $(p,q)$ ranges established for convolution estimates

Key Jacobian lower bounds derived for complex polynomial curves

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.