Contents

Idea

This page is about the concept of a wavefront set of generalized functions like hyperfunctions in the context of microlocal analysis. The wavefront set is used to further classify the kind of singularity that a generalized function exhibits in a point.

Definition

Motivation

The definition of wavefront sets is motivated by a version of a Paley-Wiener theorem that characterizes smooth compactly supported functions ($\mathbb{R}^n \to \mathbb{R}$) by a growth condition on their Fourier transform $\mathcal{F}$:

Theorem

(Paley-Wiener for $C^{\infty}_0$)

The vector space of smooth compactly supported functions is (algebraically and topologically) isomorphic, via the Fourier transform, to the space of entire functions $F$ which satisfy the following estimate: there is a positive constant $B$ such that for every integer $n \gt 0$ there is a constant $C_n$ such that:

$F(z) \le C_n (1 + |z|)^{-n} \exp{ (B \; |\operatorname{Im}(z)|)}$

Smoothness

We call a smooth compactly supported function that is identical $1$ in a neighbourhood of a point $x_0$ a cutoff function at $x_0$. Let $U \subset \mathbb{R}$ be open, we identify the cotangent bundle of $U$ with $U \times \mathbb{R}^n$. A subset of $U \times \mathbb{R}^n$ is said to be conic if it stable under the transformation

$(x, \zeta) \mapsto (x, \rho \zeta) \quad \text{with} \; \rho \gt 0$
Definition

Let $f$ be a distribution and $(x_0, \zeta_0)$ with $\zeta_0 \neq 0$ be a point of the cotangent bundle of $U$. f is smooth in $(x_0, \zeta_0)$ if there is a cutoff function $\chi$ in $x_0$ and an open cone $\Gamma_0$ in $\mathbb{R}^n$ containing $\zeta_0$ such that for every $m \gt 0$ there is a nonnegative constant $C_m$ such that for all $\zeta \in \Gamma_0$:

$\| \mathcal{F}(\chi f) (\zeta) \| \le C_m (1 + \| \zeta \|)^{-m}$

where $\mathcal{F}(\chi f)$ is the Fourier transform (of the variable $\zeta$) of the function $\chi f$ (of the variable $x$).

Definition

A distribution $f$ is smooth in a conic subset $\Gamma$ of the cotangent bundle of $U$ if $f$ is smooth in a neighbourhood of every point in $\Gamma$.

Wavefront set

Let $U \subseteq \mathbb{R}^n$ be an open subset, $T^* U$ its cotangent bundle and $f$ be a distribution on $U$. The complement of the union of all conic subsets of $T^* U$ where $f$ is smooth is the wavefront set $WF(f)$.

Examples

Onedimensional Examples

We take a brief look at distributions on $\mathcal{D} \mathbb{R}$ with singular support consisting of the origin. In one dimension, at the origin, there are of course exactly two directions along which a distribution could be smooth, namely $\zeta \le 0$ and $\zeta \gt 0$.

We define the Fourier transform to be

$\mathcal{F}(f)(\zeta) = \hat f (\zeta) = \int_{- \infty}^{\infty} f(x) \exp(-2 \pi i x \zeta)$

For the delta distribution $\delta$, we have $\mathcal{F}(\delta) = 1$, which does not satisfy the decay condition of smoothness.

References

Revised on May 3, 2010 10:16:45 by Tim van Beek (192.76.162.8)