Proof. Denote w = g L∞(−|b|,|b|). Since |g(t)| w|t| for all |t| |b|, we have
g(bt) tg(b)
t(1 t)
dt 2w|b|
1 t
dt 2w|b|.
g(bt) tg(b)
t(1 t)
dt 2w|b|
dt 2w|b|.
Adding up the two estimates, one gets (2.25).
2.3. Plan of the paper. We begin with estimates for norms and trace norms
of the pseudo-differential operators with smooth symbols, see Chapter 3. Informa-
tion about various classes of compact operators can be found in [16], [3], [34]. The
first trace norm estimate for PDO’s was obtained in [32], and later reproduced in
[33], Proposition 27.3, and [30], Theorem II-49. The fundamental paper [2] con-
tains estimates in various compact operators classes for integral operators in terms
of smoothness of their kernels. There are also publications focused on conditions
on the symbol which guarantee that a PDO belongs to an appropriate Neumann-
Schatten ideal, see e.g. [29], [1], [38], [8] and the references therein. In spite of
a relatively large number of available literature, these results are not sufficient for
our purposes. We need somewhat more detailed information about trace norms.
In particular, we derive an estimate for the trace norm of a PDO with weights, i.e.
of h1Opα(a)h2,
where the supports of h1 and h2 are disjoint. The most useful was
paper [31], which served as a basis for our approach. Although our estimates are
quite elementary, and, probably not optimal, they provide bounds of correct orders
in α and the scaling parameters.
Chapters 4, 5 are devoted to trace-class estimates for PDO’s with various jump
discontinuities. The most basic estimates are those for the commutators
χΛ], [Opα(a),PΩ,α]
where Λ and Ω are graph-type domains, see Lemmas 4.3 and
4.5. Similar bounds were derived in [42] for the case when one of the domains
Λ, Ω is a half-space. The new estimates which play a decisive role in the proof, are
collected in Chapter 5. Here the focus is on the PDO’s with discontinuous symbols
sandwiched between weights having disjoint supports. As in the case of smooth
symbols in Chapter 3, it is important for us to control the dependence of trace
norms on the distance between the supports. The simplest result in Chapter 5,
illustrating this dependence is Lemma 5.1. The estimates culminate in Lemma 5.5
which bounds the error incurred when replacing Λ by a half-space in the operator
Tα(a;Λ, Ω).
The estimate (6.1) obtained in Chapter 6 is used only for closing the asymp-
totics in Chapter 12. In contrast to the results obtained in Chapters 4 and 5, which
estimate norms of commutators of smooth symbols with χΛ or PΩ,α, the bound
(6.1) is essentially an estimate for the commutator [χΛ,PΩ,α]. Thus it is not sur-
prising that apart from the standard term
the estimate acquires the factor
log α.
Chapter 7 shows that using appropriate cut-offs, one can reduce the study of
arbitrary smooth domains to graph-type domains. This observation is conceptually
straightforward, but technically important.
As explained in the Introduction, the asymptotics (2.21) and (2.22) are even-
tually derived from the appropriate asymptotics in the one-dimensional case. All
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