DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Local Fourier definition of a wavefront set

Definition

Let $Ω \subseteq R^{n}$ be open and let $u \in D^{'} (Ω)$ , using the local distribution convention of Scalar distributions on smooth manifolds. For $p \in Ω$ and $ξ \in R^{n} ∖ {0}$ , call $(p, ξ)$ regular for $u$ if there are $χ \in C_{c}^{\infty} (Ω)$ with $χ = 1$ near $p$ and an open cone $Γ \subseteq R^{n} ∖ {0}$ containing $ξ$ such that for every integer $N \geq 0$ , $\sup_{η \in Γ} (1 + ∣ η ∣)^{N} ∣ ⟨ u, χ (x) e^{- 2 π i x \cdot η} ∣ d x^{1} \dots d x^{n} ∣ ⟩ ∣ < \infty .$ The local wavefront set ${WF}_{Ω} (u)$ is the complement of these regular pairs in $T^{*} Ω ∖ 0$ , with the standard cotangent coordinates $(p, ξ)$ . A cone is invariant under multiplication by positive scalars. This is a chart-level definition; this item makes no assertion about its behavior under a change of coordinates.

Local Fourier definition of a wavefront set

Definition

Definition references

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