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Analytic double-fibration transform and canonical relation

Definition

An analytic double fibration is a diagram of real-analytic manifolds and real-analytic submersions

G \overset{π_{G}}{\leftarrow} Z \overset{π_{X}}{\to} X

such that $Z$ is embedded in $G \times X$ , both projections are submersions, and

\dim G + \dim X > \dim Z > \dim G \geq \dim X .

Write $C_{Z} = {(g, η; x, ξ) : (g, x; η, - ξ) \in N^{*} Z ∖ 0}$ for the incidence cotangent relation. Assume $π_{G}$ is proper on $π_{X}^{- 1} (K)$ for every compact $K \subset X$ . Let $ω_{Z}$ be a nowhere vanishing real-analytic smooth density on $Z$ and let $κ ⁣ : Z \to C$ be real analytic and nowhere zero. For a compactly supported distribution $u$ on $X$ , the weighted transform is the distributional density

R u = (π_{G})_{*} (κ ω_{Z} π_{X}^{*} u),

where pullback, multiplication by a density, proper pushforward, and $C_{Z}$ use the conventions of the preceding definition. In local fiber coordinates this is integration of $u$ over the incidence fibers with weight $κ$ and density $ω_{Z}$ . The hypotheses on proper support ensure this transform is defined for every compactly supported input. This is the class treated on this page; no noisy-data or stability assertion is included.

Definition references

Incidence kernel, distribution operations, and relation

Smooth manifolds

Smooth maps and submersions

Smooth density

Real-analytic structure

Power-series convention

Facts & Assumptions

Given: The real-analytic double fibration, its density and nonvanishing analytic weight, and a compactly supported distribution $u$ on $X$ .

[F1]

Distributions, conormal kernels, and cotangent relations: submersion pullback is defined by transposed fiber integration; density multiplication turns it into a distributional density; pushforward is defined when proper on support; the canonical relation uses $(η, - ξ)$ .

[F2]

Smooth manifolds and their smooth charts: local charts define the smooth manifold structures.

[F3]

$C^{r}$ and smooth maps between smooth manifolds: smooth submersions have smooth local coordinate representatives.

[F4]

Density bundle and smooth density fields: densities transform by absolute Jacobians and can be integrated without orientations.

[F5]

Real analytic germs in several variables: the analytic maps and coefficients have holomorphic local complexifications.

[F6]

Multi-indexed power series in $C^{m}$ and their absolute convergence: those complexifications are represented by convergent power series on smaller polydiscs.

Proof

Given: The manifolds $G, Z, X$ , maps $π_{G}, π_{X}$ , density $ω_{Z}$ , weight $κ$ , and input $u$ .

1.1givenF2F3F5F6

The chart representatives are smooth submersions by [F2]–[F3], and their analytic representatives have local holomorphic extensions by [F5]–[F6]. Hence the fiber and incidence data in the statement are well-defined in the stated smooth and analytic categories.

1.2F1given

The pullback $π_{X}^{*} u$ exists by [F1]. Multiplying it by the nowhere vanishing analytic density $ω_{Z}$ gives a distributional density; multiplication by the smooth function $κ$ preserves that type. If $supp u \subset K$ with $K$ compact, the support of this product lies in $π_{X}^{- 1} (K)$ , on which $π_{G}$ is proper by hypothesis.

2.1F1F3F4step 1.1step 1.2∎

The proper pushforward in [F1] therefore defines $R u$ as a distributional density on $G$ . In coordinates, transposing fiber integration in the two submersions shows that it pairs a data test function with the integral over the corresponding incidence fiber, weighted by $κ ω_{Z}$ ; the density Jacobians cancel by [F4]. The cotangent relation of the kernel is precisely $C_{Z}$ by the sign convention in [F1].

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Sources

Marco Mazzucchelli, Mikko Salo, and Leo Tzou, A general support theorem for analytic double fibration transforms (standard reference, not scraped)
Johannes Sjöstrand, Singularités analytiques microlocales (standard reference, not scraped)