  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.
