Wave Front Set - Definition

Definition

In Euclidean space, the wave front set of a distribution ƒ is defined as

where is the singular fibre of ƒ at x. The singular fibre is defined to be the complement of all directions such that the Fourier transform of f, localized at x, is sufficiently regular when restricted to an open cone containing . More precisely, a direction v is in the complement of if there is a compactly supported smooth function φ with φ(x) ≠ 0 and an open cone Γ containing v such that the following estimate holds for each positive integer N:

Once such an estimate holds for a particular cutoff function φ at x, it also holds for all cutoff functions with smaller support, possibly for a different open cone containing v.

On a differentiable manifold M, using local coordinates on the cotangent bundle, the wave front set WF(f) of a distribution ƒ can be defined in the following general way:

where the singular fibre is again the complement of all directions such that the Fourier transform of f, localized at x, is sufficiently regular when restricted to a conical neighbourhood of . The problem of regularity is local, and so it can be checked in the local coordinate system, using the Fourier transform on the x variables. The required regularity estimate transforms well under diffeomorphism, and so the notion of regularity is independent of the choice of local coordinates.