Uniform Schwartz bounds for compact families of affine pullbacks

Let $E$ be a finite-dimensional real normed space, let $E'$ and $V$ be real normed spaces, and let $P$ be a topological space. Let $f : E' \to V$ be smooth (of regularity $\infty$ over $\mathbb{R}$) with compact support, let $Q \subseteq P$ be compact, and let $c : P \to E'$ and $\ell : P \to E \to_{L[\mathbb{R}]} E'$ be maps, continuous on $Q$, such that $\ell(p) : E \to E'$ is injective for every $p \in Q$. Writing $g_p(x) := f(c(p) + \ell(p)x)$ for $x \in E$, the conclusion is the conjunction of two assertions. First, there is a single compact set $S \subseteq E$ with $\operatorname{tsupport} g_p \subseteq S$ for every $p \in Q$; that is, one compact set contains the closures of the supports of all members of the family. Second, for all natural numbers $k$ and $n$ there is a real constant $C$ such that $\|x\|^k \, \|D^n g_p(x)\| \le C$ for every $p \in Q$ and every $x \in E$, where $D^n$ denotes the $n$-th iterated Fréchet derivative over $\mathbb{R}$. The quantifier order is the content of the second part: $C$ depends on $k$ and $n$ but is uniform in $p \in Q$ and in $x$.

This is the uniformity statement that a compact family of affine reparametrisations of a fixed compactly supported smooth function is bounded uniformly in every Schwartz seminorm, with supports confined to one compact set; finite-dimensionality of $E$ and injectivity of the linear parts give the uniform lower bound $m\|x\| \le \|\ell(p)x\|$ that confines the supports. It is used for the archimedean test-function estimates on the geometric side of the trace formula, where unipotent slices of a smooth compactly supported function on $GL_2$ over the archimedean places form such a family.