nLab properly supported peudo-differential operator

Definition

Definition

(properly supported peudo-differential operator)

A pseudo-differential operator $Q$ on a manifold $X$ is called properly supported if for each compact subset $K \subset X$ there exists a compact subset $K' \subset X$ such that for $u$ a distribution with support in $K$ it follows that the derivative of distributions $Q u$ has support in $K'$

$$supp(u)\subset K \phantom{A}\Rightarrow \phantom{A} supp(Q u) \subset K'$$

and such that

$$u\vert_{K'} = 0 \phantom{A} \Rightarrow \phantom{A} (Q u)\vert_{K} = 0 \,.$$

(Hörmander 85 (18.1.21) recalled e.g. in Radzikowski 96. p. 8,9)

Examples

Example

(differential operators are properly supported pseudo-differential operators)

Every ordinary differential operator $D$, regarded as a pseudo-differential operator, is properly supported (def. ), since differential operators do not increase the support of the functions they act on:

$$supp(D f) \subset supp(f) \,.$$

Related concepts

• propagation of singularities theorem

References

• Johann Duistermaat, Lars Hörmander, Fourier integral operators II, Acta Mathematica 128, 183-269, 1972 (Euclid)

• Lars Hörmander, The analysis of partial differential operators III, Springer 1985

• Marek Radzikowski, Micro-local approach to the Hadamard condition in quantum field theory on curved space-time, Commun. Math. Phys. 179 (1996), 529–553 (Euclid)