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420 8. Pseudodifferential and Boundary Integral Operators

σ
Re (w, Λ ΘΛ −σ Aw) % p =1 H (t −s )/2 (Γ )
Γ
Γ
2
≥ β 0
w
2 % p H (Γ ) − β 1
w
% p H −1 (Γ ) (8.1.20)
t
t
=1 =1
p
% σ s j
t
for all w ∈ H (Γ),where Λ =((Λ δ j )) p×p . The last lower order
=1 Γ Γ p
term in (8.1.20) defines a linear compact operator C : % =1 H (Γ) →
t
p
% −s
=1 H (Γ) which is given by
(v, Cw) % p H (t −s )/2 (Γ ) = β 1 (v, w) % p H −1 (Γ ) .
t
=1 =1
With this compact operator C, the G˚arding inequality (8.1.20) takes the
form
σ −σ
Re w, (Λ ΘΛ ≥ β 0
w
2 % p . (8.1.21)
t
Γ Γ A + C)w % p H (t −s )/2 (Γ ) H (Γ )
=1 =1
Remark 8.1.2: As a consequence, any strongly elliptic pseudodifferential
operator or any strongly elliptic system of pseudodifferential operators defines
a Fredholm operator of index zero since for the corresponding bilinear form
σ
a(v, w):=(v, Λ ΘΛ −σ Aw) % ℘ =1 H (t −s )/2 (Γ )
Γ
Γ
one may apply Theorem 5.3.10 which means that the classical Fredholm
∗
alternative holds implying dim N(A) = dim N(A ), hence, index (A)=0.
8.1.2 Schwartz Kernels on Boundary Manifolds
m
For the representation of A ∈L (Γ) in terms of the Schwartz kernel we
m
∈L (U r ) which has the representation
consider the local operator A χ r
(r) α

χ r∗ v)( )= a ( )(D χ r∗ v)( )

α
(A χ r
|α|≤m
(8.1.22)

+p.f. k (r) ( , − τ )(χ r∗ v)(τ )dτ .
U r
with the Schwartz kernel k (r) given in Theorems 6.1.2, 7.1.1 and 7.1.8.
For two charts (O r ,U r ,χ r ) , (O t ,U t ,χ t ) with O rt = O r ∩ O t = ∅ let
(−1)
Φ rt = χ t ◦ χ r be the associated diffeomorphism from χ r (O rt )to χ t (O rt ).
Then the representation (8.1.22) is transformed into
(t) α

χ t∗ v)(τ )= a (τ )(D χ t∗ v)(τ )

α τ
(A χ t
|α|≤m

+p.f. k (t) (τ ,τ − λ )(χ t∗ v)(λ )dλ

U t