nLab symbol order

Contents

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

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

(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. .

References

Last revised on November 23, 2017 at 15:53:48. See the history of this page for a list of all contributions to it.