The subset of the cotangent bundle (with the zero section removed) that records both the points of a manifold where a distribution is singular and the cotangent directions in which that singularity is not microlocally smooth; usually denoted WF(u).

Principle

Microlocal regularity: a distribution is smooth at a point in a particular cotangent direction exactly when that cotangent pair is not in the wavefront set. The wavefront set organizes singularities by location and frequency direction.

Demonstration

For the Dirac delta at the origin in R^n, WF(δ) = { (0, ξ) : ξ ≠ 0 }, reflecting that the distribution is singular at 0 in every cotangent direction. For the Heaviside step function on R, the wavefront set contains (0, ξ) with ξ pointing to the positive frequency side corresponding to the jump.

Misapplication

Treating the wavefront set as identical to the singular support (which loses directional information), or attempting to use it as a subset of the tangent bundle instead of the cotangent bundle; both discard the microlocal directional data essential to WF.

Consequence

When correctly identified, the wavefront set permits precise statements about propagation of singularities under differential and pseudodifferential operators and is fundamental for microlocal solutions of PDEs and scattering analysis.

Reversal

The complementary viewpoint is micro-regularity: the set of cotangent directions at which a distribution is microlocally smooth (the complement of WF in the cotangent bundle minus the zero section); this highlights directions where singularities do not occur.

Boundary

Defined only for distributions (or hyperfunctions) on manifolds and as a subset of the cotangent bundle with the zero covectors removed; it does not apply unchanged to pointwise-defined classical functions whose regularity is already smooth, nor to arbitrary tangent directions.

Semantic Tension

Close concepts include singular support (location-only singularities) and spectral support or frequency content (global Fourier support); unlike those, WF encodes local position plus cotangent direction, creating tension when one seeks simpler 'location-only' descriptions.

Synthesis

The wavefront set is the microlocal fingerprint of a distribution: it refines singular support by attaching to each singular point the cotangent directions in which smoothness fails, enabling directional propagation and regularity analysis in PDE and harmonic analysis.