Hello,


I am trying to understand the calculus of pseudodifferential operators on manifolds. All the textbooks I could put my hand on define the principal symbol of a pseudodifferential operator locally, then prove that it transforms well, hence becomes a "global" object. Is there any good way to define the the principal symbol without coordinate patches? Am I asking too much here :)?



Symbol of pseudodiff operator
There is an invariant way of defining pseudodifferential operators, and a (much simpler and quite classical) invariant way of defining symbols.


The latter appears already in the old Atiyah-Singer volume from the early '60's. Choose any point $(x\_0, \xi\_0)$ in the cotangent bundle. Choose a function $\phi \in \mathcal C^\infty(M)$ such that $d\phi(x\_0) = \xi\_0$, and then set
$\sigma\_m(A)(x\_0,\xi\_0) = \lim\_{\lambda\to\infty} \lambda)^{-m} e^{-i\lambda \phi}A( e^{i\lambda \phi})$
(perhaps I am missing a factor of $i$). Here $A$ is a psido of order $m$. This is pretty direct and ``natural''.


As for the coordinate-free definition of pseudodifferential operators, the first step is to define the notion of a conormal (or polyhomogeneous conormal) distribution on a manifold $X$ relative to a (closed, embedded) submanifold $Y$. Such a distribution $u$ lies in some fixed (Banach or Hilbert) space $H$ -- for example, a weighted $L^\infty$ space, $r^s L^\infty$, where $r$ is the distance to $Y$ in $X$ and $s$ is any fixed real number -- and is stably regular in this space, i.e. $V\_1 \ldots V\_k u \in r^s L^\infty$ for all positive integers $k$ and for all vector fields on $X$ which are smooth and unconstrained away from $Y$, but which are tangent to $Y$. 


Finally, a linear operator $A$ on a smooth manifold $M$ (which satisfies some weak continuity requirements) has a Schwartz kernel $K\_A$, which is a distribution on $M \times M$. The operator $A$ is a pseudodifferential operator if $K\_A$ is conormal with respect to the diagonal in $M \times M$. 


A classical, or polyhomogeneous, distribution is conormal and also has an expansion in ascending powers of $r$ and positive integer powers of $\log r$. 


If $K\_A$ satisfies this condition, then one can transfer it to a distribution on the normal bundle of the diagonal in $M \times M$, supported near the zero section (it is smooth elsewhere anyway). Then its Fourier transofrm in the fibres of the normal bundle is a symbol in the usual sense, and vice versa, any symbol on these fibres has F.T. which is conormal to the zero section and hence, by transferal, to the diagonal in $M \times M$.


The one unsatisfactory thing about this definition is that it is not apparent that if $A$ and $B$ are psido's, then so is their composition $A \circ B$, nor does one ``immediately'' get a symbol calculus, i.e. the fact that the symbol mapping is a homomorphism. 


Anyway, this is a down-to-earth and very useful definition of pseudodifferential operators which allows for all sorts of interesting generalizations. This definition, or certainly the emphasis on this formulation, is due to Melrose, but appears already in Vol. 3 of H\"ormander. 


Rafe Mazzeo


