6 A. V. SOBOLEV
Here we use the notation
|∇lf(t)|2
=
r
j1,j2,...,jl=1
|∂j1 ∂j2 · · · ∂jl
f(t)|2
for a function f of the variable t
Rr.
The presence of the parameters , ρ allows
one to consider amplitudes with different scaling properties.
In the same way we introduce the classes
S(n,m)
(resp.
S(m))
of all (complex-
valued) functions a = a(x, ξ) (resp. a = a(ξ)), which are bounded together with
their partial derivatives up to order n w.r.t. x, and m w.r.t. ξ. The norm
N(n,m)(a; , ρ) (resp. N(m)(a; ρ)) is defined in a way similar to (2.5). Note the
straightforward inequality: if a
S(n,m),b

S(n,m),
then ab
S(n,m)
and
N(m,n)(ab;
, ρ)
Cm,nN(m,n)(a;
,
ρ)N(m,n)(b;
, ρ).
As we show later, if the amplitude p and/or symbol a belong to an appropriate
class S, then the PDO’s (2.1)– (2.3) are bounded.
Let M be a non-degenerate linear transformation, and let k, k1 Rd be some
vectors. By A = (M, k) we denote the affine transformation Ax = M + k. A
special role is played by the Euclidean isometries, i.e. by the affine transformations
of the form E = (O, k), where O is an orthogonal transformation. The set of
all Euclidean isometries on Rd is denoted by E(d). Let us point out some useful
unitary equivalence for the operators (2.1) - (2.3). For the affine transformation
(M, k) define the unitary operator U = UM,k
:
L2(Rd)

L2(Rd)
by
(UM,ku)(x) = | det M| u(Mx + k), u
L2(Rd).
It is straightforward to check that for an arbitrary k1
Rd,
(2.6)

⎨UM,ke−iαx·k1
⎩bM,k,k1
Opα(b)eiαx·k1
a
UM1,k
= Opα(bM,k,k1
a
),
(x, y, ξ) = b
(
Mx + k, My + k, (M
T
)−1ξ
+ k1
)
,
and
(2.7)

⎨UM,kχΛUM1,k
⎩ΛM,k

= χΛM,k ,
UM,ke−iαx·k1 PΩ,αeiαx·k1 UM1,k
= PΩT
M,k1

,
= M
−1(Λ
k), ΩM,k1
T
= M
T
k1).
For the operator Tα(a) this implies that
(2.8)
UM,ke−iαx·k1
Tα(a;Λ,
Ω)eiαx·k1 UM1,k
= Tα(aM,k,k1 ; ΛM,k, ΩM,k1
T
).
Note that the asymptotic coefficients W0 and W1 are invariant with respect to
affine transformations, i.e.
(2.9)

⎨W0(b;Λ,
⎩W1(b;
Ω) = W0(bM,k,k1 ; ΛM,k, ΩM,k1
T
),
∂Λ,∂Ω) = W1(bM,k,k1 ; ∂ΛM,k,∂ΩM,k1
T
).
The first of the relations (2.9) is immediately checked by changing variables under
the integral (1.3). The second one is proved in Appendix 4, Lemma 16.1.
As one particular type of a linear transformation, it is useful to single out the
case when M = I for some 0, and k = 0, i.e. M is a scaling transformation.
In this situation we denote
(
W u
)
(x) =
(
UM,0u
)
(x) =
d
2
u( x).
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