Wavefront sets and microlocal regularity

Work with an h-tempered family, for example one whose local distribution norms grow at most polynomially in one over h. A finite phase-space point is absent from WF_h(u_h) if some fixed compact smooth cutoff a, elliptic at that point, satisfies ||a^w u_h||_(L²)=O(h^N) for every N. The cutoff may be chosen equal to one nearby. The constants may depend on N, but the neighbourhood and cutoff are fixed as h tends to zero. Such a family is called microlocally residual there.

An O(h) localized norm does not imply absence: it is only one power of smallness. Conversely an exponentially small family can be residual even when its normalized version has substantial oscillations. The wavefront set records the chosen normalization of the family. Local cutoffs in both position and momentum avoid confusing spatial support with oscillatory direction; a family can remain localized in position while its momentum moves outside every fixed compact set.

The classical wavefront set tests rapid Fourier decay in conic frequency regions for one distribution and excludes the zero covector. It is conic because only the direction at arbitrarily large ordinary frequency matters. The finite semiclassical wavefront set instead resolves an actual rescaled momentum ξ, can include ξ=0, and need not be conic. Multiplying ξ by two generally changes the point being tested. A fixed compact smooth function has no classical wavefront set but can have semiclassical wavefront over its support at zero momentum.

For χ(x)exp(iS(x)/h) with a smooth compact envelope and real smooth phase, nonstationary phase identifies the finite semiclassical momentum graph ξ=S′(x), where the envelope is nonzero. A cutoff away from that graph gives repeated integration by parts and residual size. On the graph, the oscillation survives. Zeros of the amplitude at isolated points do not automatically remove points from the closed wavefront set, so statements are most transparent on open sets where the amplitude does not vanish.

A normalized family with phase exp(ix/h²) has rescaled momentum one over h. Every fixed compact momentum cutoff can miss it as h tends to zero, even though its L² norm is unchanged. Its finite-phase-space wavefront set can therefore be empty while information remains at fibre infinity. A compactified calculus can record that escape, but the finite definition above does not. An empty finite wavefront set is not a global residual conclusion without additional localization controlling the missing infinity.

This distinction is also why a finite-band drawing cannot prove a wavefront assertion. Any fixed finite sum of smooth sine waves is classically smooth, and inspecting a single value or a small plot does not establish an estimate for every power of h. The definition requires an asymptotic family and a neighbourhood test. Numerically visible oscillations may suggest a phase-space location, but the nonstationary-phase estimate or a microlocal operator argument supplies the actual regularity statement.