# nLab properly supported peudo-differential operator

Contents

## Topics in Functional Analysis

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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:

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

## References

Last revised on November 23, 2017 at 07:49:27. See the history of this page for a list of all contributions to it.