Wavefront set - Imported Knowlpedia Content

Definition

Let $T\in\mathcal D'(\Omega)$ be a distribution, where $\Omega\subseteq\mathbb R^n$ is open. A pair $(x_0,\xi_0)\in\Omega\times(\mathbb R^n\setminus\{0\})$ is absent from the wavefront set $\operatorname{WF}(T)$ if some $\chi\in C_c^\infty(\Omega)$ , with $\chi(x_0)\ne0$ , and some open conic neighborhood $\Gamma$ of $\xi_0$ satisfy the following condition. The product $\chi T$ is a compactly supported distribution, hence tempered, and its distributional Fourier transform obeys

\sup_{\xi\in\Gamma}(1+\lVert\xi\rVert)^N \lvert\widehat{\chi T}(\xi)\rvert<\infty \quad\text{for every }N\geq0.

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

Relation to singular support

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

Behavior under operations

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

\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. These microlocal criteria, and the transformation law under diffeomorphisms, are developed in Hörmander, Chapter 8.

Examples and geometric meaning

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

\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

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.
J. J. Duistermaat, Fourier Integral Operators, Birkhäuser, 1996. DOI record. Relevant: Chapter I, wavefront sets and microlocal operations.