8

KAZUAKI TAIRA

used in subsequent chapters. In particular, we introduce the notion of transmission

property, due to Boutet de Monvel [2], which is a condition about symbols in the

normal direction at the boundary. Furthermore, we prove an existence and unique-

ness theorem for a class of pseudo-differential operators which enters naturally in

the construction of Feller semigroups.

Chapter III is devoted to the proof of Theorem 1. We reduce the problem

of construction of Feller semigroups to the problem of unique solvability for the

boundary value problem

J (a - W)u = f in D,

\ (A — L)u = p on dD,

and then prove existence theorems for Feller semigroups. Here a 0 and A 0.

The idea of our approach is stated as follows (cf. [1], [9], [12]).

First we consider the following Dirichlet problem:

( (a - W)v = / in D,

X

V\QD

= 0 on dD.

The existence and uniqueness theorem for this problem is well established in the

framework of Holder spaces. We let

v = G°af.

The operator G° is the Green operator for the Dirichlet problem. Then it follows

that a function u is a solution of the problem

Ua-W)u = f in A

W 1 Lu = 0 on dD

if and only if the function w — u — v is a solution of the problem

J (a - W)w = 0 in D,

{ Lw = -Lv = ~LG°J on dD.

But we know that every solution w of the equation

(a - W)w = 0 in D

can be expressed by means of a single layer potential as follows:

w = Hat/).

The operator Ha is the harmonic operator for the Dirichlet problem. Thus, by

using the Green and harmonic operators, one can reduce the study of problem (*)

to that of the equation:

LHai = -LG°af.