Proposition: Scale and dilation identities

For positive scales $\mu ,\nu$,

For $F_{\ell }(z)={\ell }^{-4}F(z/\ell )$, $\ell >0$,

Scale conversion is additive and every test in (eq:actual-annihilator) is insensitive to it.

Proof. The Riesz identity at $\alpha =4$ gives

Subtract the logarithms in (eq:reference-response) to obtain (eq:scale-conversion). For (eq:dilation), change variables $z=\ell w$ in the weak integral. Four differentiations give the factor ${\ell }^{-4}$, while $\log ({\mu }^{2}s(\ell w))=\log ({\mu }^{2}s(w))+2\log \ell$. The same delta identity completes the calculation.