nLab properly supported peudo-differential operator

Redirected from "properly supported pseudo-differential operator".
Contents

Context

Functional analysis

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 QQ on a manifold XX is called properly supported if for each compact subset KXK \subset X there exists a compact subset KXK' \subset X such that for uu a distribution with support in KK it follows that the derivative of distributions QuQ u has support in KK'

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

and such that

u| K=0AA(Qu)| K=0. 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 DD, 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(Df)supp(f). supp(D f) \subset supp(f) \,.

References

Last revised on November 23, 2017 at 12:49:27. See the history of this page for a list of all contributions to it.