In this lecture we began the study of the analysis which leads to the Hodge theorem and other key results about elliptic operators.  In subsequent lectures we’ll make use of the work of specialists on pseudodifferential operators (Kohn, Nirenberg, Hormander, Taylor) to prove the estimates we discuss in this lecture.  As Bott said, “If you have an electrical problem, you consult an electrician.  If you have an elliptical problem, you consult an elliptician.”  (Hormander was the go-to elliptician for Atiyah and Bott.)

We began by discussing the Sobolev spaces.   The Sobolev space on a compact manifold \(M\) is the space of functions whose first \(k\) derivatives (considered as distributions) belong to the Hilbert space \(L^2(M)\).  (In order to avoid an explicit discussion of distribution theory we gave in class an alternative, but equivalent, description which is based on Fourier analysis.)  These form a scale of spaces  (that is, \( W^{k+1}\subseteq W^k\) ) with the following key properties.

• \( W^{k+s}\subseteq C^k \) whenever \(s> {\frac12}\dim M\);
• A differential operator of order \(s\) extends to a bounded linear map \(W^{k+s}\to W^k\), for every \(k\);
• The inclusion \(W^{k+\epsilon}\to W^k \) is a compact linear operator for every \(\epsilon>0\) (Rellich-Kondrakov theorem).

As a result of the second bullet point we can define an operator of order \(\le s\) (not necessarily differential!) to be any linear map on smooth functions that extends to a continuous map \(W^{k+s}\to W^k \) for every \(k\).   Every differential operator of order \(\le s\) is an operator of order \(\le s\), but the converse is of course false.  In particular there exist “operators” of negative order.  That makes the following statement sensible.

Fundamental Elliptic Theorem  If \(D\) is elliptic differential of order \(s\), there exists \(Q\) of order \(-s\) such that

\[ I-DQ = R_1,\quad I-QD = R_2 \]

are smoothing operators (i.e. of order \(-\infty\)).

In the next couple of lectures we will see how the pseudodifferential calculus is used to construct the parametrix \(Q\).   You could read about this in the Kohn-Nirenberg papers (see Atiyah-Bott’s bibliography) or in M. Taylor,Pseudodifferential operators, Chapter II (very terse).