nLab
properly supported peudo-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

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}{}& ʃ &\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 07:49:27. See the history of this page for a list of all contributions to it.