Proof of Fourier Extension Conjecture for Paraboloid
Analysis
This paper provides a proof of the Fourier extension conjecture for the paraboloid in dimensions greater than 2. The authors leverage a decomposition technique and trilinear equivalences to tackle the problem. The core of the proof involves converting a complex exponential sum into an oscillatory integral, enabling localization on the Fourier side. The paper extends the argument to higher dimensions using bilinear analogues.
Key Takeaways
- •Proves the Fourier extension conjecture for the paraboloid in dimensions greater than 2.
- •Employs a decomposition technique and trilinear equivalences.
- •Uses oscillatory integrals to achieve localization on the Fourier side.
- •Extends the argument to higher dimensions using bilinear analogues.
Reference
“The trilinear equivalence only requires an averaging over grids, which converts a difficult exponential sum into an oscillatory integral with periodic amplitude.”