Lemma: Frequency control through the origin

For every $R>0$, there is $C_R$ such that

This includes $\kappa =0$. Every frequency derivative of order $k$ has a corresponding bound using $\Vert \psi {\Vert }_{k+1}$.

Proof. On fixed compact intervals, the principal-value subtraction and the delta term give

The same bound holds uniformly for shifts by $\pm \rho$, $0\le \rho \le R$, after enlarging the fixed interval.

The global distributional interpretation of (eq:partial-fractions) is therefore

The shifted tests have a common compact temporal support. The integrand is bounded by $C_R\Vert \psi {\Vert }_1/|\xi |$. Its spatial integral is finite:

The expression is the Fourier transform of the retarded kernel, as follows by the temporal transform of its spatial sine representation. It thus defines the same distribution at the origin, rather than an extra extension chosen there. This proves (eq:MR-order). For derivatives, transfer the derivatives to the test.