Wave map null form estimates via Peter–Weyl theory

Abstract

We study spacetime estimates for the wave map null form $Q_0$ on $R\times S^3$. By using the Lie group structure of $S^3$ and Peter–Weyl theory, combined with the time-periodicity of the conformal wave equation on $R\times S^3$, we extend the classical ideas of Klainerman and Machedon to estimates on $R\times S^3$, 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 $R\times S^3$. This arises in Fourier space from the product structure of irreducible representations of $\mathrm{SU}\left(2\right)$. We further show that our estimates imply weighted estimates for the null form on Minkowski space.