I will show how the Funk-Hecke theorem can be used to easily prove smoothing estimates for a wide class of dispersive
equations, under natural power-like asymptotics on the Fourier transform of the weight, and thus unifying a number of well-known estimates. When the Fourier transform of the weight is positive, the approach allows sharper information regarding the optimal constant and extremisers, extending earlier work of Simon. These observations are closely related to the Mizohata-Takeuchi conjecture; this is known in the radial case, and we give a quick re-proof in three dimensions and higher when the Fourier transform of the weight is positive. Finally, I will show how the Funk-Hecke theorem can be used to prove sharp null form estimates for the wave equation.