Wave Front Set - Introduction

Introduction

In more familiar terms, WF(f) tells not only where the function f is singular (which is already described by its singular support), but also how or why it is singular, by being more exact about the direction in which the singularity occurs. This concept is mostly useful in dimension at least two, since in one dimension there are only two possible directions. The complementary notion of a function being non-singular in a direction is microlocal smoothness.

Intuitively, as an example, consider a function ƒ whose singular support is concentrated on a smooth curve in the plane at which the function has a jump discontinuity. In the direction tangent to the curve, the function remains smooth. By contrast, in the direction normal to the curve, the function has a singularity. To decide on whether the function is smooth in another direction v, one can try to smooth the function out by averaging in directions perpendicular to v. If the resulting function is smooth, then we regard ƒ to be smooth in the direction of v. Otherwise, v is in the wavefront set.

Formally, in Euclidean space, the wave front set of ƒ is defined as the complement of the set of all pairs (x₀,v) such that there exists a test function with φ(x₀) ≠ 0 and an open cone Γ containing v such that the estimate

holds for all positive integers N. Here denotes the Fourier transform. Observe that the wavefront set is a conical set, since if (x,v) ∈ Wf(ƒ), then (x,λv) ∈ Wf(ƒ) for all λ > 0. In the example discussed in the previous paragraph, the wavefront set is the set-theoretic complement of the image of the tangent bundle of the curve inside the tangent bundle of the plane.

Because the definition involves cutoff by a compactly supported function, the notion of a wave front set can be transported to any differentiable manifold X. In this more general situation, the wave front set is a closed conical subset of the cotangent bundle T*(X), since the ξ variable naturally localizes to a covector rather than a vector. The wave front set is defined such that its projection on X is equal to the singular support of the function.