Theory of Pseudo-Differential Operators
In this chapter we present a brief description of the basic concepts and results of the theory of pseudo-differential operators — a modern theory of potentials — which will be used in the subsequent chapters. In recent years there has been a trend in the
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Theory of Pseudo-Differential Operators
In this chapter we present abrief description of the basic concepts and results of the theory of pseudo-differential operators - a modern theory of potentials - which will be used in the subsequent chapters. In recent years there has been a trend in the theory of partial differential equations towards constructive methods. The development of the theory of pseudo-differential operators has made possible such an approach to the study of elliptic differential operators. For detailed studies of pseudo-differential operators, the reader might be referred to Chazarain-Piriou [CP], Eskin [Es], Hörmander [H03], Kumano-go [Ku], Rempel-Schulze [RS], Schulze [Su] and Taylor [Ty].
5.1 Function Spaces
n be a
bounded domain of Euclidean space Rn with smooth boundary Its closure n = nu an is an n-dimensional, compact smooth manifold with boundary. We may assume that: Let
an.
(a) The domain n is a relatively compact, open subset of an n-dimensional compact smooth manifold M without boundary (see Fig. 5.1): (b) In a neighborhood W of an in M anormal coordinate t is chosen so that the points of Ware represented as (Xl, t), Xl E an, -1 < t < 1; t > 0 in n, t < 0 in M \ TI and t = 0 only on an (see Fig. 5.2). (c) The manifold M is equipped with a strictly positive density {L which, on W, is the product of a strict1y positive density w on an and the Lebesgue measure dt on (-1, 1). This manifold M is called the double of n. The function spaces we shall treat are the following (see Bergh-Löfström [BL], Calderon [Ca], Stein [Sn2], Taibleson [Tb], Triebel [Tr]): (i) The generalized Sobolev spaces H·,p(n) and H',P(M), consisting of all potentials of order S of LP functions. When s is integral, these spaces coincide with the usual Sobolev spaces w·,p(n) and W"P(M), respectively.
K. Taira, Semigroups, Boundary Value Problems and Markov Processes © Springer-Verlag Berlin Heidelberg 2004
98
5 Theory of Pseudo-Differential Operators
(ii) The Besov spaces BS,p(a.n). These are functions spaces defined in terms of the V modulus of continuity, and enter naturally in connection with boundary value problems.
an
Fig. 5.1.
w······. ··•···········....···{Ö· D 2 a Coo diJJeomorphism. 1f A E L;;:o(Dd, where 1 - P ::::: c5 < p ::::: 1, then the mapping A x :C(f'(D2 )
v
f---->
A( v
--+ 0
X)
Coo(D2 ) 0
X-I
is in L;;:c5(n2 ), and we have the asymptotic expansion
a(A x )(y,1]) '"
L
~!
(8fa(A)) (x,tX'(x) '1])'
D~ (e iT (X,Z,7)))lz=x
(5.6)
0:::':0
with r(x,Z,1])
= (X(z)
- X(x) - X'(x)· (z - x), 1])
Here x = X-I (y), X' (x) is the derivative of X at x and t X' (x) its transpose. The situation may be represented by the following diagram:
COO(fh)
A)
COO(Ol)
Here X'v = v 0 X is the pull-back of v by X and X*U = u forward of u by X, respectively.
0
X-I is the push-
Remark 5.1. Formula (5.6) shows that a(A x )(y,1]) == a(A)
(X,i X'(x) '1])
modS;'c5-(p-c5).
Note that the mapping D2
X
Rn
'3
(y,1])
f---->
(X,i X'(x) . 1])
E DI X
Rn
is just a transition map of the cotangent bundle T
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