Some inequalities for the Fourier transform and their limiting behaviour

2023/06/03 by Garofalo, Nicola
#33C10 #42B10 #42B15 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2306.02209

Abstract

We identify a one-parameter family of inequalities for the Fourier transform whose limiting case is the restriction conjecture for the sphere. Using Stein's method of complex interpolation we prove the conjectured inequalities when the target space is L², and show that this recovers in the limit the celebrated Tomas-Stein theorem.

Related

• L² restriction estimates from the Fourier spectrum
• Pitt inequalities and restriction theorems for the Fourier transform
• A note on maximal Fourier Restriction for spheres in all dimensions
• Improved Stein inequalities for the Fourier transform
• The Stein-Tomas inequality under the effect of symmetries