On the generalized Radon transform and currents

Question

Given a family of hypersurfaces $H_{t,p} = $ {$x \in \mathbb{R}^n \mid g(x,p) = t $} one defines a generalized Radon transform $R$ of a function $u \colon \mathbb{R}^n \to \mathbb{R}$ as $$ R[u] (t,p) = \int\limits_{H_{t,p}} u(x) a(x) \omega $$ where $\omega$ is such differential form as $d_{x}g \wedge \omega = dx_1 \wedge ... \wedge dx_n$: $$ \omega = \sum\limits_{k=1}^{n} (-1)^{k-1} \frac{\partial_{x_k} g(x,p)}{|\nabla_{x} g(x,p)|^2} dx_1 \wedge ... \wedge \overline{dx_k} \wedge ... \wedge dx_n $$

1. The first question is why we choose such a form? Which applications provides us this differential form?

2. The second question concerns the generalisation of this transform to the currents. This transform may be considered as integration of some absolutely continious measure. Is there some generalisation to the case of an arbitrary measure (I think it is related with currents).

Thank You.

Answer 1

Here is the simple explanation (credits should go to the late great V.I. Arnold who gave the explanation below in one of his books). The form $\omega$ is sometimemes referred to as the Gelfand-Leray residue.

Place yourself in the case when $t$ is a regular value of $g$. Fix a point p on the fiber $g^{-1}(t)$ so that $dg(p)\neq 0$. The implicit function theorem then shows that we can find local coornidates $y^1,\dotsc, y^n$ near $p$ such that, in these coordinates, $g=y^1$. In these coordinates

$$dx^1\wedge \cdots \wedge dx^n=\rho dy^1 \wedge \cdots \wedge dy^n,$$

$$ \omega = \rho dy^2 \wedge \cdots \wedge dy^n. $$

Using these coordinates, and $a$ is supported in the domain of the coordinates $y^j$ we deduce from the Fubini theorem that

$$\int u a dx^1\wedge \cdots \wedge dy^n = \int_{\mathbb{R}} H_{y^1}[au] dy_1, $$

$$H_{y^1}[au]=\int au \rho dy^2\wedge dy^n= \int au \omega. $$

The relationship with currents is symple. The $n$-form $\eta= udx^1\wedge \cdots \wedge dx^n$ defines a $0$-current. The $1$-form $H_{t}[u]dt$, viewed as a $0$-current, is the pushforward via the map $g$ of the current defined by $\eta$.