Conditional existence of maximizers for the Tomas-Stein inequality for the sphere

2025/09/12 by Shao, Shuanglin, Wang, Ming
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.10754

Abstract

The Tomas-Stein inequality for a compact subset Γ of the sphere S^d states that the mapping f↦ \widehatfσ is bounded from L²(Γ,σ) to L^2+4/d(\R^d+1). Then conditional on a strict comparison between the best constants for the sphere and for the Strichartz inequality for the Schrödinger equations, we prove that there exist functions which extremize this inequality, and any extremising sequence has a subsequence which converges to an extremizer. The method is based on the refined Tomas-Stein inequality for the sphere and the profile decompositions. The key ingredient to establish orthogonality in profile decompositions is that we use Tao's sharp bilinear restriction theorem for the paraboloids beyond the Tomas-Stein range. Similar results have been previously established by Frank, Lieb and Sabin \citeFrank-Lieb-Sabin:2007:maxi-sphere-2d, where they used the method of the missing mass.

Citations

• On existence of extremizers for the Tomas-Stein inequality for S¹
• Global Maximizers for the Sphere Adjoint Fourier Restriction Inequality
• Existence of Extremals for a Fourier Restriction Inequality
• On the extremizers of an adjoint Fourier restriction inequality
• Heat-flow monotonicity of Strichartz norms
• The linear profile decomposition for the Airy equation and the existence of maximizers for the Airy Strichartz inequality
• Maximizers for the Strichartz inequality
• A bilinear approach to the restriction and Kakeya conjectures

Related

• On existence of extremizers for the Tomas-Stein inequality for S¹
• The Stein-Tomas inequality under the effect of symmetries
• A note on maximal Fourier Restriction for spheres in all dimensions
• On smoothness of extremizers of the Tomas-Stein inequality for S¹
• Some inequalities for the Fourier transform and their limiting behaviour