8 A. V. SOBOLEV

In the next definition we introduce general

Cm-domains.

Let B(x,r) = {v ∈

Rd

: |x − v| r} be the ball of radius r 0 centered at x.

Definition 2.2. Let Λ ⊂ Rd, d ≥ 2 be a domain, w ∈ Rd be a vector and

R 0 be a number.

(1) For a w ∈ ∂Λ we say that in the ball B(w,R) the domain Λ is represented

by the

Cm-graph-type

domain Γ = Γ(Φ; O, w), m ≥ 1, if there is a number

R = Rw 0, such that

(2.19) Λ ∩ B(w,R) = Γ ∩ B(w,R).

(2) For a w ∈ Λ we say that in the ball B(w,R) the domain Λ is represented

by

Rd,

if there is a number R = Rw 0, such that

Λ ∩ B(w,R) = B(w,R).

(3) The domain Λ is said to be

Cm,

m ≥ 1, if for each point w ∈ ∂Λ there

is a number R = Rw 0, such that in the ball B(w,R) the domain

Λ is represented by a

Cm-graph-type

domain Γ(Φ; O, w) with some

Cm-

function Φ = Φw : Rd−1 → R, satisfying (2.14), and some orthogonal

transformation O = Ow. In this case we also say that the boundary ∂Λ

is a

Cm-surface.

The next two theorems represent the main results of the paper:

Theorem 2.3. Let Λ, Ω ⊂

Rd,

d ≥ 2 be bounded domains in

Rd

such that Λ is

C1

and Ω is

C3.

Let a = a(x, ξ) be a symbol with the property

(2.20) max

0≤n≤d+2

0≤m≤d+2

sup

x,ξ

|∇x∇ξ

n ma(x,

ξ)| ∞,

supported on the set B(z, ) × B(μ,ρ) with some z, μ ∈ Rd and , ρ 0. Let g be a

function such that g(0) = 0, analytic in the disk of radius R =

C1N(d+2,d+2)(a;

, ρ)

with some constant C1 0. There exists a constant C0 0 independent of Λ, Ω

and of the symbol a such that for any C1 ≥ C0 the following formula holds:

tr g(Tα(a)) =

αdW0(g(a);

Λ, Ω)

+

αd−1

log α W1(A(g; a); ∂Λ,∂Ω) +

o(αd−1

log α), (2.21)

as α → ∞.

For the self-adjoint operator Sα(a) we have a wider choice of functions g:

Theorem 2.4. Let Λ, Ω ⊂ Rd, d ≥ 2 be bounded domains in Rd such that Λ

is

C1

and Ω is

C3.

Let a = a(x, ξ) be a symbol satisfying (2.20) with a compact

support in both variables. Then for any function g ∈

C∞(R),

such that g(0) = 0,

one has

tr g(Sα(a)) =

αdW0(g(Re

a); Λ, Ω)

+

αd−1

log α W1(A(g;Re a); ∂Λ,∂Ω) +

o(αd−1

log α), (2.22)

as α → ∞.