10 A. V. SOBOLEV
An elementary calculation gives:
χI ; a(x, ξ)
λ2(a(x, ξ) − λ1)
λ1(a(x, ξ) − λ2)
The proof of Theorems 2.3 and 2.4 splits in two unequal parts. The crucial and
the most diﬃcult part is to justify the asymptotics for a polynomial g:
Theorem 2.9. Let Λ, Ω ⊂
d ≥ 2 be bounded domains in
such that Λ
and Ω is
Let a = a(x, ξ) be a symbol satisfying (2.20) with a compact
support in both variables. Then for gp(t) =
= 1, 2,...,
tr gp(Tα(a)) =
log α W1(A(gp; a); ∂Λ,∂Ω) +
log α), (2.23)
as α → ∞. If Tα(a) is replaced with Sα(a), then the same formula holds with the
symbol a replaced by Re a on the right-hand side.
Once this theorem is proved, the asymptotics can be closed with the help of
the sharp bounds (12.11) and (12.17), which were derived in ,  using the
abstract version of the Szeg˝ o formula with a remainder estimate obtained in 
(see also ).
2.2. Asymptotic coeﬃcient A(g; b). Here we provide some simple estimates
for the coeﬃcient A defined in (1.5).
Lemma 2.10. Suppose that the analytic function g is given by the series
with a radius of convergence R 0. Let
|t| ≤ R.
Then for any b, |b| R the following estimate holds:
(2.24) |A(g; b)| ≤
Proof. Consider first gm(z) = zm,m ≥ 2, so that
t − tm
t(1 − t)
dt ≤ (m −
|A(g; b)| ≤
|ωm||A(gm; b)| ≤
which leads to (2.24).
Lemma 2.11. Suppose that g ∈
and g(0) = 0. Then for any b ∈ R the
following estimate holds:
(2.25) |A(g; b)| ≤