2. MAIN RESULT 7

Then a straightforward calculation gives for any ρ 0:

(2.10) W Opα(b)W

a −1

= Opβ(b,ρ),

a

b,ρ(x, y, ξ) = b( x, y,ρξ), β = αρ.

In particular,

(2.11) W χΛW

−1

= χΛ , Λ =

−1Λ,

It is important that the norm (2.5) is invariant under certain linear transformations.

Note first of all that

(2.12)

N(n1,n2,m)(p;

, ρ) =

N(n1,n2,m)(p

1,ρ1

;

−1,ρρ1

1

−1),

for arbitrary positive ,

1

, ρ, ρ1. Moreover, the norm is also invariant under Eu-

clidean isometries:

(2.13)

N(n1,n2,m)(p;

, ρ) =

N(n1,n2,m)(pO,k,k1

; , ρ).

Sometimes we refer to and ρ as scaling parameters.

Now we can specify the classes of domains which we study. We always assume

that Λ and Ω are domains with smooth boundaries in the standard sense. However,

for the reference convenience and to specify the precise conditions on the objects

involved, we state our assumptions explicitly.

Definition 2.1. We say that a domain Γ ⊂

Rd,

d ≥ 2, is a

Cm-graph-type

domain, with some m ≥ 1, if one can find a real-valued function Φ ∈

Cm(Rd−1),

with the properties

(2.14)

⎧

⎪Φ(ˆ)

⎨

⎪

⎩

0 = 0,

∇Φ is uniformly bounded on

Rd−1,

∇Φ is uniformly continuous on

Rd−1,

and some transformation E = (O, k) ∈ E(d) such that

E−1Γ

= {x : xd Φ(ˆ)}, x ˆ x = (x1,x2,...,xd−1).

In this case we write Γ = Γ(Φ; O, k) or Γ = Γ(Φ), if the omission of the dependence

on E does not lead to confusion.

We often use the notation

(2.15) MΦ = ∇Φ

L∞

.

It is clear that Γ(Φ; O, k) = EΓ(Φ; I, 0) with E = (O, k). The point k is on the

boundary of the domain Γ(Φ; O, k).

If Λ = Γ(Φ; O, k), then the domain Λ (see definition (2.11)) has the form

(2.16) Λ = Γ(Φ ; O,

−1k),

Φ (ˆ) x =

−1Φ(

ˆ). x

Note that the value (2.15) is invariant under scaling:

(2.17) ∇Φ

L∞

= ∇Φ

L∞

.

In what follows we extensively use the relations (2.6), (2.7) and (2.8) in order to

reduce the domains or symbols to a more convenient form. For these purposes, let

us make a note of the following elementary property for the domain Λ = Γ(Φ; I, 0).

According to (2.7), for any k ∈ ∂Λ we have

(2.18) ΛI,k = Γ(Φk; I, 0), where Λ = Γ(Φ; I, 0), Φk(ˆ) x = Φ(ˆ x +

ˆ)

k − kd.

Clearly,

Φk(ˆ)

0 = 0 and ∇Φ

L∞

= ∇Φk

L∞

.