The Analytic Wave Front Set
The analytic wave front set or singular spectrum WFA(f) of a distribution f (or more generally of a hyperfunction) can be defined in terms of the FBI transform (Hörmander (1983)) as the complement of the conical set of points (x, λ·ξ) (λ > 0) such that the FBI transform satisfies the Bros–Iagolnitzer inequality
for all a > 0, y near x and t = λ·ξ, with |t| sufficiently large. J.M. Bony (Sjöstrand (1982), Hörmander (1983)) proved that this definition coincided with other definitions introduced independently by Sato, Kashiwara and Kawai and by Hörmander. If P is an mth order linear differential operator having analytic coefficients
with principal symbol
and characteristic variety
then
In particular, when P is elliptic, char P = ø, so that
- WFA(Pf) = WFA(f).
This is a strengthening of the analytic version of elliptic regularity mentioned above.
Read more about this topic: FBI Transform
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