Corollary: Relative strength away from transitions

At a nonzero diffractive coefficient and away from geometric fronts, the transverse Fourier amplitude of $W_{\beta }$ has order $-1/2$. At a single noncoincident geometric front away from other fronts it has order $0$. Thus the strictly diffracted covariance is one-half derivative smoother in its normal variable.

Equivalently, at such elliptic points the local normal Sobolev thresholds are $s<0$ for the diffracted front and $s<-1/2$ for the geometric front. No uniform half-order assertion at their transition is made.

Proof. The square-root model has Fourier amplitude ${\lambda }^{-1/2}$ on a half-line. The simple-pole geometric model has amplitude of order $0$. After localization, the corresponding one-dimensional Sobolev integrals have thresholds

and

The nonzero leading coefficients make these thresholds sharp.