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
, α ∞, do not feel the boundary, and relatively standard quasi-
classical considerations lead to the formula
(1.8) tr
To handle the trace tr
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
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
log α W1
A(g; a)
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
as ξ =
ξ with
ξ = (ξ1,ξ2,...,ξd−1),
define for each
the set
ξ = {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
ξ =
˜j ρ
belongs to
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
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