Inversion of Abel's half-line integral equation, smooth families

Let $P$ be a real normed space (a type with a normed additive commutative group structure and a real normed space structure). The assertion is the existence of an operator $T$ carrying functions $\mathbb{R} \to \mathbb{C}$ to functions $\mathbb{R} \to \mathbb{C}$ with the following two properties. First, $T$ is $\mathbb{C}$-linear on smooth compactly supported data: for all $f, g : \mathbb{R} \to \mathbb{C}$ that are $C^\infty$ with compact support and all $a, b \in \mathbb{C}$, one has $T(a f + b g) = a\, T f + b\, T g$ as functions of $\xi$. Second, for every $G : \mathbb{R} \times P \to \mathbb{C}$ that is $C^\infty$ with compact support, three things hold: the function $(\xi, p) \mapsto T(G(\cdot, p))(\xi)$ on $\mathbb{R} \times P$ is $C^\infty$; for every $R \in \mathbb{R}$, if $G(\xi, p) = 0$ for all $p \in P$ and all $\xi \ge R$, then also $T(G(\cdot, p))(\xi) = 0$ for all $p \in P$ and all $\xi \ge R$; and for every $\eta \in \mathbb{R}$ and every $p \in P$, the Bochner integral of $\xi \mapsto T(G(\cdot, p))(\xi) / \sqrt{\xi - \eta}$ over the open half-line $(\eta, \infty)$, with respect to Lebesgue measure, equals $G(\eta, p)$. No constraint is imposed on the values of $T$ at functions outside the smooth compactly supported ones.

This is the solvability of Abel's integral equation with kernel $(\xi - \eta)^{-1/2}$ on a half-line, in a form uniform in an auxiliary parameter $p$ in a normed space: the solution operator is linear, preserves smoothness jointly in $(\xi, p)$, and propagates vanishing on upper half-lines. It is used in the construction of the archimedean splitting transform for automorphic forms on $\mathrm{GL}_2(\mathbb{R})$, via AutomorphicForm.GL2Real.exists_linear_entrySlice_archWeightChar_zero_splitTransform_eq.