1. INTRODUCTION 3

2. Analysis of the problem for d = 1,

3. A geometrical estimate,

4. Reduction of the initial problem to the case d = 1.

The most important and diﬃcult is Step 4. Here we divide the domain Λ into a

boundary layer, which contributes to the first and second terms in (1.2), and the

inner part, which matters only for the first term. Then we construct a suitable

partition of unity subordinate to this covering, which is given by the functions

q↓ = q↓(x) and q↑ = q↑(x) respectively. An interesting feature of the problem is

that the thickness of the boundary layer does not depend on α. The asymptotics

of tr

(

q↑g(Tα)

)

, α → ∞, do not feel the boundary, and relatively standard quasi-

classical considerations lead to the formula

(1.8) tr

(

q↑g(Tα)

)

=

αdW0(q↑g(a))

+

O(αd−1),α

→ ∞.

To handle the trace tr

(

q↓g(Tα)

)

we construct an appropriate covering of the bound-

ary layer by open sets of a specific shape. For each of these sets the boundary ∂Λ

is approximated by a hyperplane, which makes it possible to view the operator Tα

as a PDO on the boundary hyperplane, whose symbol is an operator of the same

type, but in dimension one. This reduction brings us to Step 2 of the plan. The

1-dim situation was studied in H. Widom’s paper [41]. Although its results are

not directly applicable, the method developed there allows us to get the required

asymptotics. These ensure that

(1.9)

tr

(

q↓g(Tα)

)

=

αdW0(q↓g(a))

+

αd−1

log α W1

(

A(g; a)

)

+

o(αd−1

log α),α → ∞.

Adding up (1.8) and (1.9), gives (1.2).

In the reduction to the 1-dim case an important role is played by a result

of geometrical nature, which is listed above as the third ingredient. It is loosely

described as follows. Representing ξ ∈

Rd

as ξ =

(ˆ,t),

ξ with

ˆ

ξ = (ξ1,ξ2,...,ξd−1),

define for each

ˆ

ξ ∈

Rd−1

the set

Ω(ˆ)

ξ = {t :

(ˆ,t)

ξ ∈ Ω} ⊂ R.

If it is non-empty, then it is at most countable union of disjoint open intervals in

R whose length we denote by ˜j, ρ j = 1, 2,... . The important observation is that

under appropriate restrictions on the smoothness of the boundary ∂Ω, the function

(1.10) ˜ m

δ(ˆ)

ξ =

j

˜j ρ

−δ

,

belongs to

L1(Rd−1)

for all δ ∈ (0, 2). The precise formulation of this result is given

in Appendix 1.

From the technical viewpoint Steps 2 and 4 are based on the trace class es-

timates derived at Step 1. In order to work with discontinuous symbols we also

establish convenient estimates for smooth ones. The emphasis is on the estimates

which allow one to control explicitly the dependence on the parameter α, and on

the scaling properties of the symbols.

At this point it is appropriate to compare our proof with H. Widom’s paper

[42], where (1.2) was justified for the case when Λ (or Ω) was a half-space. In fact,

our four main steps are the same as in [42]. However, in [42] the relative weight of

these ingredients was different. If Λ is a half-space, then the reduction to the 1-dim

case (i.e. Step 4) is almost immediate whereas in the present paper, for general Λ