In this lecture we gave some of the basic definitions related to (linear, partial) differential operators – first on Euclidean space \({\mathbb R}^n \), and then on manifolds.  Key ideas were

1.  The definition of a differential operator of order \(\le m\): a linear map \(D\) on smooth (vector valued) functions having the form

\[ Du(x) = \sum_{|\alpha|\le m} L_\alpha(x) \frac{\partial^\alpha u}{\partial x^\alpha} (x), \]

where the coefficients \(L_\alpha\) are smooth (matrix valued) functions. Differential operators form an algebra (they can be added and composed) which is filtered by the order of the operator.  Note that multiplication in this algebra is non commutative, even for scalar operators (those where the coefficients matrices are multiples of the identity).

2.  The definition of the symbol

\[ \sigma_D(x,\xi) = \sum_{|\alpha|\le m} L_\alpha(x) \xi^\alpha \]

which is a matrix valued function, polynomial in \(\xi\).

3.  The observation that for constant coefficient operators the Fourier transform conjugates such an operator into the operator of multiplication by its symbol (“Fourier turns differentiation into multiplication”).  Thus constant coefficient operators can be completely understood in terms of Fourier analysis.  The symbol of the composite of two constant coefficient operators is exactly the product of their symbols.

4.  When we work with variable coefficient operators the composite of operators does not map exactly to the product of symbols, but it does do so “modulo lower order terms”.

5.  A symbol is elliptic if its top degree part is invertible for all nonzero \(\xi\) (see previous post for discussion of this definition).  Taking into account items 3 and 4, we expect that elliptic operators will be invertible “modulo lower order terms”.   Making this expectation precise leads to the theory of pseudodifferential operators (notice that, whatever kind of beast the inverse of a differential operator may be, it certainly won’t be another differential operator.)