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