The Atiyah-Singer theorem for Dirac-type operators can be proved using the heat kernel and this proof has an advantage over the proof via K-theory, because the first is local but the latter is not. But the K-theoretic proof has the advantage that it proves Atiyah-Singer not only for Dirac-type operators, but much more generally for elliptic, symmetric pseudodifferential operators (PDOs).

I was wondering whether one could do the heat kernel proof also for pseudodifferential operators?

I know that we can conclude Atiyah-Singer for PDOs from the theorem for Dirac operators by using Poincare duality (i.e., by using that on compact spin$^c$-manifolds every K-homology class can be represented by the class of a twisted Dirac operator).

But I'm asking if the proof via the heat kernel directly generalizes to PDOs? One problem that I can see is that it is often used that $e^{itD}$ has finite propagation speed, and I think that this does not hold anymore if D is merely pseudodifferential (right?). But maybe it is possible to work around that? Or what is the reason why I only encounter heat kernel proofs for Dirac type operators but not for PDOs?