# nLab symbol order

### Context

#### 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

(symbol order)

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

$\left\vert \frac{\partial^\alpha}{\partial k_\alpha} \frac{\partial^\beta}{\partial x^\beta} q(x,k) \right\vert \;\leq\; C_{\alpha,\beta,K}\left( 1+ {\vert k\vert}\right)^{m - {\vert \alpha\vert}}$

for all $x \in K$ and all cotangent vectors $k$ to $x$.

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

1. it is of the form

$Q f (x) \;=\; \int \int e^{i k \cdot (x - y)} \hat f(x,y,k) f(y) \, d y d k$
2. its principal symbol $q$ is of order $m$, in the above sense.

## Examples

###### Example

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