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 ∞.
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