Partial differential equations Definition

Microlocal analysis

Builds on Fourier transform Wave front set Pseudo-differential operator

Definition

Let $X$ be an open subset of $\R^n$ , or a smooth manifold, with cotangent bundle $T^*X$ . Microlocal analysis studies distributions and linear operators on $X$ through properties localized at points $(x, \xi) \in T^*X \setminus 0$ . Here $x$ is a position and $\xi \neq 0$ is a frequency direction. A property holds microlocally near $(x_0, \xi_0)$ if it holds after cutting off near $x_0$ and keeping only frequencies in a conic neighbourhood of $\xi_0$ . The basic notions are:

The wave front set $\mathrm{WF}(u) \subset T^*X \setminus 0$ of a distribution $u$ : the set of $(x_0, \xi_0)$ near which $u$ fails to be smooth microlocally.
Properly supported pseudo-differential operators, which act microlocally and satisfy $\mathrm{WF}(Pu) \subseteq \mathrm{WF}(u)$ .

Remarks

Example: let $u = H(x_1)$ on $\R^2$ , where $H$ is the Heaviside function. Then $u$ is smooth off the line $x_1 = 0$ , and $\mathrm{WF}(u) = \set{(x, \xi) : x_1 = 0, \ \xi = (\xi_1, 0), \ \xi_1 \neq 0}$ . The check: take a cutoff $\chi(x)$ near a point of the line. The Fourier transform of $\chi u$ decays rapidly in every closed cone avoiding the $\xi_1$ -axis, but only like $\abs{\xi_1}^{-1}$ along it. Non-example: classical local regularity, such as " $u$ is smooth near $x_0$ ", is not a microlocal statement, since it does not distinguish frequency directions. The projection of $\mathrm{WF}(u)$ to $X$ is the singular support: $\pi(\mathrm{WF}(u)) = \operatorname{sing\,supp} u$ .

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Tag	`JJ5XQEVBZ4`
URI	`m2://JJ5XQEVBZ4`
Kind	definition
Writing	W0 stub
MSC	35A27, 35S05, 58J40
Defines	Microlocal analysis
Uses	Fourier transform, Wave front set, Pseudo-differential operator

Microlocal analysis

Definition

Remarks

Reading path

Further reading

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