EQUATIONS OF ELASTIC HYPERSURFACES

4. BOUNDARY VALUE PROBLEMS: GENERAL RESULTS 119
is the characteristic polynomial or the symbol of the operator P(x, D). The equality (4.80)
demonstrates a similarity of PDOs and convolution operators (4.49) and justifies the following
definition of a pseudodifferential operator
a(x, D)u(x):=F −1
ξ↦→x {a(x, ξ)F y↦→ξ[u(y)]}
= (2π) n ∫R n e −ixξ a(x, ξ)(F u)(ξ) dξ , u ∈ S(R n ) , (4.82)
which holds under certain restrictions on thy symbol a(x, ξ) and the function u.
Definition 4.22 For r > 0 The Hörmander class S r 1,0(R n × R n ) = S m (R n × R n ), consists
of those functions a(x, ξ) which admit the following estimate
for all x, ξ ∈ R n and all multi-indices α, β ∈ N n 0.
Definition 4.23 An operator on manifold
|∂ α ξ ∂ β x a(x, ξ)| ≤ C 0 α,β〈ξ〉 r−|α|| (4.83)
A : C ∞ (M ) → D ′ (M ) (4.84)
is called pseudodifferential A = a(x, D x ) with the symbol on the cotangent manifold
a ∈ S m (T ∗ M ) if:
i. χ 1 Aχ 2 I : H θ (M ) → C ∞ (M ) are continuous for all θ ∈ R and all pairs χ 1 , χ 2 ∈
C ∞ (M ) of smooth functions with disjoint supports supp χ 1
⋂ supp χ2
= ∅; in other words, χ 1 Aχ 2 I has order −∞;
ii. consider the pull back operator
κ j,∗ ψ j Aκ −1
j,∗ [ψ ju] = a (j) (x, D)u, u ∈ C ∞ 0 (R n ), j = 1, ..., l ,
where κ −1
j : X j → X j is the inverse diffeomorphism and κ j,∗ [ψ j ϕ] ∣ . These pull back
operators are ΨDOs with the “pull back” symbols
a (j) (x, ξ) := ψ j (κ j (x))a(κ j (x), [κ ′ j(x)] ⊤ ξ) , x ∈ Y j , ξ ∈ R n (4.85)
belong to the class S m (R n × R n ) for all j = 1, . . . , l.
An m × m matrix symbol a(x, ξ) from the class (S r ) m×m (T ∗ M ) is called elliptic if
M 1 |ξ| mr ≤ |a(x, ξ|| ≤ M 2 |ξ| mr for some constants 0 < M 1 < M 2 < ∞, all x ∈ M and all
|ξ| ≥ R where R > 0 is sufficiently large.
Theorem 4.24 Let 1 < p < ∞ and θ, r ∈ R. Let a(X , D x ) be a ΨDO on a manifold M
with a symbol a ∈ S r (T ∗ M ) = S r 1,0(T ∗ M ). Then the operator
is continuous.
a(X , D) : X θ p(M ) → X θ−r
p (M )