By using the Holder space theory of pseudo-differential operators as in the proof
of Theorem 1, we can show that if condition (^4) is satisfied, then the operator LHa
is bijective in the framework of Holder spaces.
Therefore, we find that a unique solution u of problem (**) can be expressed as
u = G"J- Ha
This formula allows us to verify all the conditions of the generation theorems of
Feller semigroups, especially the density of the domain D(2t) in the space Co(D\M).
If we use instead of Gva the Green operator G°a for the Dirichlet problem as in
the proof of Theorem 1, our proof would break down.
We do not prove Theorems 3 and 4, since their proofs are essentially the same
as those of Theorems 1 and 2, respectively.