Wave map null form estimates via Peter-Weyl theory

TL;DR

This paper develops new spacetime estimates for wave map null forms on the three-sphere using Peter-Weyl theory, extending classical flat-space techniques to curved space with a small differentiability loss.

Contribution

It introduces a novel approach combining Peter-Weyl theory with classical estimates to analyze null forms on a3^3, accounting for curvature effects.

Findings

Establishes null form estimates on a3^3 using Lie group structure.

Identifies a small differentiability loss due to lack of dispersion.

Derives weighted estimates for Minkowski space from curved space results.

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.