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

Shuanglin Shao, Ming Wang

TL;DR

This paper investigates the conditional existence of maximizers for the Tomas-Stein adjoint restriction inequality on the sphere, establishing existence under the hypothesis $\mathcal{R}>\mathcal{R}_{\mathbf{P}}$. It develops a refined cap-based Tomas-Stein estimate and a two-stage profile decomposition that blends sphere-restriction analysis with paraboloid Strichartz via Tao's bilinear restriction, enabling orthogonality of profiles and decoupling of mass and norms. The main result shows that extremizers exist and extremizing sequences are precompact modulo the natural symmetries, provided the sphere constant strictly exceeds the paraboloid Strichartz constant. This links restriction phenomena on the sphere to Schrödinger-type inequalities on the paraboloid, deepening the understanding of extremizers for restriction inequalities on manifolds and contributing to the broader framework of concentration-compactness in harmonic analysis.

Abstract

The Tomas-Stein inequality for a compact subset $Γ$ of the sphere $S^d$ states that the mapping $f\mapsto \widehat{fσ}$ is bounded from $L^2(Γ,σ)$ 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 \cite{Frank-Lieb-Sabin:2007:maxi-sphere-2d}, where they used the method of the missing mass.

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

TL;DR

This paper investigates the conditional existence of maximizers for the Tomas-Stein adjoint restriction inequality on the sphere, establishing existence under the hypothesis . It develops a refined cap-based Tomas-Stein estimate and a two-stage profile decomposition that blends sphere-restriction analysis with paraboloid Strichartz via Tao's bilinear restriction, enabling orthogonality of profiles and decoupling of mass and norms. The main result shows that extremizers exist and extremizing sequences are precompact modulo the natural symmetries, provided the sphere constant strictly exceeds the paraboloid Strichartz constant. This links restriction phenomena on the sphere to Schrödinger-type inequalities on the paraboloid, deepening the understanding of extremizers for restriction inequalities on manifolds and contributing to the broader framework of concentration-compactness in harmonic analysis.

Abstract

The Tomas-Stein inequality for a compact subset of the sphere states that the mapping is bounded from to . 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 \cite{Frank-Lieb-Sabin:2007:maxi-sphere-2d}, where they used the method of the missing mass.

Paper Structure

This paper contains 8 sections, 17 theorems, 155 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The refined Tomas-Stein estimate
  3. Profile decomposition: the first decompostion
  4. The second decomposition for each piece
  5. Key propositions
  6. The full decomposition
  7. A synthesis
  8. Conditional existence

Key Result

Theorem 1.2

Assume that $\mathcal{R}>\mathcal{R}_{\mathbf{P}}$. Then there exists an extremal function $f\in L^2(\Gamma, \sigma)$ for eq-1 by showing that any extremising sequence $\{f_\nu\}$ is precompact in $L^2(\Gamma, \sigma)$. This extremizer can be chosen to be a smooth function.

Figures (2)

  • Figure 1: The Whitney type decomposition.
  • Figure 2: Distant caps interact weakly.

Theorems & Definitions (32)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Lemma 2.2
  • Theorem A
  • Theorem 2.3
  • Theorem B
  • Theorem C
  • Theorem D
  • ...and 22 more