Wave map null form estimates via Peter-Weyl theory

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 $\mathrm{S}\mathrm{U}(2)$. We further show that our estimates imply weighted estimates for the null form on Minkowski space.