nLab
symbol order

Context

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

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& Rh & \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

(symbol order)

A smooth function qq on a cotangent bundle (e.g. the symbol of a differential operator) is of order mm (and type 1,01,0, denoted qS m=S 1,0 mq \in S^m = S^m_{1,0}), for mm \in \mathbb{N}, if on each coordinate chart ((x i),(k i))((x^i), (k_i)) we have that for every compact subset KK of the base space and all multi-indices α\alpha and β\beta, there is a real number C α,β,KC_{\alpha, \beta,K } \in \mathbb{R} such that the absolute value of the partial derivatives of qq is bounded by

| αk α βx βq(x,k)|C α,β,K(1+|k|) m|α| \left\vert \frac{\partial^\alpha}{\partial k_\alpha} \frac{\partial^\beta}{\partial x^\beta} q(x,k) \right\vert \;\leq\; C_{\alpha,\beta,K}\left( 1+ {\vert k\vert}\right)^{m - {\vert \alpha\vert}}

for all xKx \in K and all cotangent vectors kk to xx.

A Fourier integral operator QQ is of symbol class L m=L 1,0 mL^m = L^m_{1,0} if

  1. it is of the form

    Qf(x)=e ik(xy)f^(x,y,k)f(y)dydk Q f (x) \;=\; \int \int e^{i k \cdot (x - y)} \hat f(x,y,k) f(y) \, d y d k
  2. its principal symbol qq is of order mm, in the above sense.

(Hörmander 71, def. 1.1.1 and first sentence of section 2.1)

Examples

Example

The wave operator/Klein-Gordon operator on Minkowski spacetime is of class L 2L^2, according to def. 1.

References

Last revised on November 23, 2017 at 10:53:48. See the history of this page for a list of all contributions to it.