10 A. V. SOBOLEV
An elementary calculation gives:
A
(
χI ; a(x, ξ)
)
=
1
(2π)2
log
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
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 gp(t) =
tp,p
= 1, 2,...,
tr gp(Tα(a)) =
αdW0(gp(a);
Λ, Ω)
+
αd−1
log α W1(A(gp; a); ∂Λ,∂Ω) +
o(αd−1
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 [13], [14] using the
abstract version of the Szeg˝ o formula with a remainder estimate obtained in [23]
(see also [24]).
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
g(z) =

m=0
ωmzm
with a radius of convergence R 0. Let
g(1)(t)
=

m=2
(m
1)|ωm|tm−1,
|t| R.
Then for any b, |b| R the following estimate holds:
(2.24) |A(g; b)|
1
(2π)2
|b|g(1)(|b|).
Proof. Consider first gm(z) = zm,m 2, so that
(2π)2|A(gm;
b)|
|b|m
1
0
t tm
t(1 t)
dt (m
1)|b|m.
Thus
|A(g; b)|

m=2
|ωm||A(gm; b)|
1
(2π)2

m=2
(m
1)|ωm||b|m,
which leads to (2.24).
Lemma 2.11. Suppose that g
C1(R)
and g(0) = 0. Then for any b R the
following estimate holds:
(2.25) |A(g; b)|
1
π2
|b|g L∞(−|b|,|b|).
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