CHAPTER 2
Main result
2.1. Definitions and main results. In this paper we need several types of
pseudo-differential operators, depending on the parameter α 0. For the symbol
a = a(x, ξ), amplitude p = p(x, y, ξ), and any function u from the Schwartz class
on Rd we define
(2.1) (Opαp)u(x)
a
=
α

d
eiα(x−y)ξp(x,
y; ξ)u(y)dξdy,
(2.2) (Opαa)u(x)
l
=
α

d
eiα(x−y)ξa(x,
ξ)u(y)dξdy,
(2.3) (Opαa)u(x)
r
=
α

d
eiα(x−y)ξa(y,
ξ)u(y)dξdy.
If the function a depends only on ξ, then the operators Opα(a),
l
Opα(a)
r
and
Opα(a)a
coincide with each other, and we simply write Opα(a). Later we formulate condi-
tions on a and p which ensure boundedness of the above operators uniformly in the
parameter α 1. Let Λ, Ω be two domains in
Rd,
and let χΛ(x), χΩ(ξ) be their
characteristic functions. We always use the notation
PΩ,α = Opα(χΩ).
We study the operator
(2.4) Tα(a) = Tα(a;Λ, Ω) =
χΛPΩ,αOpα(a)PΩ,αχΛ,l
and its symmetrized version:
Sα(a) = Sα(a;Λ, Ω) = χΛPΩ,α Re Opα(a)
l
PΩ,αχΛ.
Note that Tα(a) differs from the operator (1.1) by the presence of an extra projection
PΩ,α on the left of Opα(a).
l
As we shall see later in Chapter 4, this difference does
not affect the first two terms of the asymptotics (1.2).
Let us now specify the class of symbols and amplitudes used throughout the
paper. We denote by
S(n1,n2,m)
the set of all (complex-valued) functions p =
p(x, y, ξ), which are bounded together with their partial derivatives up to order n1
w.r.t. x, n2 w.r.t. y and m w.r.t. ξ. It is convenient to define the norm in this
class in the following way. For arbitrary numbers 0 and ρ 0 define
(2.5)
N(n1,n2,m)(p;
, ρ) = max
0≤n≤n1
0≤k≤n2
0≤r≤m
sup
x,y,ξ
n+k ρr|∇x∇y∇ξp(x, n k r
y, ξ)|.
5
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