Knowlpedia

Definition

Wavefront set

The wavefront set records the points and nonzero cotangent directions in which a distribution is not microlocally smooth.

Let T∈D′(Ω)T\in\mathcal D'(\Omega) be a , where Ω⊆Rn\Omega\subseteq\mathbb R^n is open. A pair (x0,ξ0)∈Ω×(Rn∖{0})(x_0,\xi_0)\in\Omega\times(\mathbb R^n\setminus\{0\}) is absent from the wavefront set WF⁡(T)\operatorname{WF}(T) if some χ∈Cc∞(Ω)\chi\in C_c^\infty(\Omega), with χ(x0)≠0\chi(x_0)\ne0, and some open conic neighborhood Γ\Gamma of ξ0\xi_0 satisfy the following condition. The product χT\chi T is a , hence tempered, and its obeys

sup⁡ξ∈Γ(1+∥ξ∥)N∣χT^(ξ)∣<∞for every N≥0.\sup_{\xi\in\Gamma}(1+\lVert\xi\rVert)^N \lvert\widehat{\chi T}(\xi)\rvert<\infty \quad\text{for every }N\geq0.

Thus WF⁡(T)\operatorname{WF}(T) records where the localized transform fails to decay rapidly. It is a closed conic subset of T∗Ω∖0T^*\Omega\setminus0, and different qualifying cutoffs give the same exclusion criterion.

Relation to singular support

The base projection of WF⁡(T)\operatorname{WF}(T) is exactly the of TT. A point may therefore be singular while only some cotangent directions above it are in the wavefront set. The definition uses the because χT\chi T has compact support and hence is tempered.

Behavior under operations

Differentiation and multiplication by smooth functions do not create new wavefront directions:

WF⁡(∂αT)⊆WF⁡(T),WF⁡(aT)⊆WF⁡(T).\operatorname{WF}(\partial^\alpha T)\subseteq\operatorname{WF}(T), \qquad \operatorname{WF}(aT)\subseteq\operatorname{WF}(T).

More delicate transversality conditions on wavefront sets govern whether products and pullbacks of distributions exist. Wavefront sets also transform naturally under diffeomorphisms.

Examples and geometric meaning

A smooth function has empty wavefront set. For the Dirac distribution at the origin,

WF⁡(δ0)={(0,ξ):ξ≠0},\operatorname{WF}(\delta_0) =\{(0,\xi):\xi\ne0\},

so every nonzero cotangent direction is singular. A distribution conormal to a smooth hypersurface instead has wavefront directions normal to that hypersurface, illustrating that the construction detects direction as well as location.

Conventions and scope
References
  1. Lars Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed., Springer, 2003. DOI record. Relevant: Chapter 8, definition and calculus of wavefront sets.
  2. J. J. Duistermaat, Fourier Integral Operators, Birkhäuser, 1996. DOI record. Relevant: Chapter I, wavefront sets and microlocal operations.