Wave map null form estimates via Peter-Weyl theory

Abstract

We study spacetime estimates for the wave map null form $Q_0$ on $\mathbb{R}\times {\mathbb{S}}^3$. By using the Lie group structure of ${\mathbb{S}}^3$ and Peter-Weyl theory, combined with the time-periodicity of the conformal wave equation on $\mathbb{R}\times {\mathbb{S}}^3$, we extend the classical ideas of Klainerman and Machedon to estimates on $\mathbb{R}\times {\mathbb{S}}^3$, allowing for a range of powers of natural (Laplacian and wave) Fourier multiplier operators. A key difference in these curved space estimates as compared to the flat case is a loss of an arbitrarily small amount of differentiability, attributable to a lack of dispersion of linear waves on $\mathbb{R}\times {\mathbb{S}}^3$. This arises in Fourier space from the product structure of irreducible representations of $\mathrm{SU}(2)$. We further show that our estimates imply weighted estimates for the null form on Minkowski space.

36 pages, proof of Lemma B.1 simplified and typos fixed. Version accepted in Journal of Functional Analysis

References in corpus (2)

• The global nonlinear stability of Minkowski space for self-gravitating massive fields. The wave-Klein-Gordon model
• Wave maps on (1+2)-dimensional curved spacetimes