Proposition: Classical pullback preserves the proved tail

If a smooth section $u$ satisfies (eq:dipolebound), then the fixed pullback $u\circ \alpha$, with the corresponding bounded bundle transformation, has the same rate. A rapidly decreasing smooth source remains rapidly decreasing under such a fixed pullback.

Proof. For sufficiently large $|\mathbf{x}|$,

The chain rule expresses each fixed derivative of the pullback as a finite sum of derivatives of $u$ times bounded derivatives of $\alpha$. The displayed asymptotic derivative bounds give the same spatial rates. The argument with arbitrarily large weights proves the Schwartz statement.