Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sharp bilinear eigenfunction estimate, $L^\infty_{x_2}L^p_{t,x_1}$-type Strichartz estimate, and energy-critical NLS

Yangkendi Deng, Yunfeng Zhang, Zehua Zhao

Abstract

We establish sharp bilinear eigenfunction estimates for the Laplace-Beltrami operator on the standard three-sphere $\mathbb{S}^3$, eliminating the logarithmic loss that has persisted in the literature since the pioneering work of Burq, Gérard, and Tzvetkov over twenty years ago. This completes the theory of multilinear eigenfunction estimates on the standard spheres. Our approach relies on viewing $\mathbb{S}^3$ as the compact Lie group $\mathrm{SU}(2)$ and exploiting its representation theory. Motivated by applications to the energy-critical nonlinear Schrödinger equation (NLS) on $\mathbb{R} \times \mathbb{S}^3$, we also prove a refined anisotropic Strichartz estimate on the cylindrical space $\mathbb{R}_{x_1} \times \mathbb{T}_{x_2}$ of $L^\infty_{x_2}L^4_{t,x_1}$-type, adapted to certain spectrally localized functions. The argument relies on multiple sharp measure estimates and a robust kernel decomposition method. Combining these two key ingredients, we derive a refined bilinear Strichartz estimate on $\mathbb{R} \times \mathbb{S}^3$, which in turn yields small-data global well-posedness for the above mentioned NLS in the energy space.

Sharp bilinear eigenfunction estimate, $L^\infty_{x_2}L^p_{t,x_1}$-type Strichartz estimate, and energy-critical NLS

Abstract

We establish sharp bilinear eigenfunction estimates for the Laplace-Beltrami operator on the standard three-sphere , eliminating the logarithmic loss that has persisted in the literature since the pioneering work of Burq, Gérard, and Tzvetkov over twenty years ago. This completes the theory of multilinear eigenfunction estimates on the standard spheres. Our approach relies on viewing as the compact Lie group and exploiting its representation theory. Motivated by applications to the energy-critical nonlinear Schrödinger equation (NLS) on , we also prove a refined anisotropic Strichartz estimate on the cylindrical space of -type, adapted to certain spectrally localized functions. The argument relies on multiple sharp measure estimates and a robust kernel decomposition method. Combining these two key ingredients, we derive a refined bilinear Strichartz estimate on , which in turn yields small-data global well-posedness for the above mentioned NLS in the energy space.

Paper Structure

This paper contains 19 sections, 19 theorems, 191 equations, 2 tables.

Table of Contents

  1. Introduction and main results
  2. Statement of main results
  3. Organization of the paper
  4. Notation
  5. Preliminaries
  6. Spectral theory, and Littlewood--Paley projectors
  7. The Bernstein and Sobolev inequalities
  8. Analysis on the group $\mathrm{SU}(2)$
  9. Irreducible representations and their tensor products
  10. Schur orthogonality relations
  11. Eigenfunctions and their products
  12. Sharp bilinear eigenfunction estimate on $\mathbb{S}^3$: Proof of Theorem \ref{['mainthm: 1']}
  13. Refined anisotropic Strichartz estimate on $\mathbb{R}\times\mathbb{T}$: Proof of Theorem \ref{['mainthm: 2']}
  14. Refined bilinear Strichartz estimate and well-posedness of the energy-critical NLS
  15. Bilinear Strichartz estimate
  16. ...and 4 more sections

Key Result

Theorem 1.1

For $m,n\in\mathbb{Z}_{\geq 0}$, let $f,g$ be eigenfunctions of the Laplace--Beltrami operator $\Delta_{\mathbb{S}^3}$ on $\mathbb{S}^3$ such that Assume that $m\geq n$. Then

Theorems & Definitions (50)

  • Theorem 1.1: Sharp bilinear eigenfunction estimate
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2: Refined anisotropic Strichartz estimate
  • Remark 1.3
  • Theorem 1.3: Refined bilinear Strichartz estimate
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.4: Well-posedness
  • Remark 1.6: On the large-data global well-posedness
  • ...and 40 more