In Sogge's Lectures on Nonlinear Wave equations, Chapter II, Section 1, Proposition 1.1, is about expressing the radial vector field $\partial_r$ as a linear combination of conformal Killing fields in Minkowski spacetime. It is claimed that $$ (t - r)\partial_r = a_0(t, x)S + \sum_{i = 1}^n a_i(t, x) \Omega_{0i}, $$ where $S = x^\mu \partial_\mu$ is the scaling field and $\Omega_{0i} = x_0 \partial_i - x_i \partial_0$ is a Lorentz boost. The exact form of the $a_\mu$ is $$ a_0(t, x) = \frac{-r}{r + t}, \ \ \ \ a_i(t, x) = \frac{tx_i}{r(r + t)}. $$ Here $r = |x|$.
Question: It is claimed that the $a_\mu$ satisfy bounds of the form: $$ |\partial^\alpha a_\mu(t, x)| \leq C_\alpha (t + |x|)^{-|\alpha|} $$ for all $\alpha$ if $|x| > \delta t$ for some fixed $\delta > 0$. How would one go about proving this? It seems like one must find some inductive formula for the derivatives, but this quickly gets incredibly messy. Is there a simpler way of going about this? (One possibility might be exploiting the radial symmetry, although $a_i$ is not completely radially symmetric?) Is there a standard reference for this type of inverse polynomial bounds on derivatives?