12 A. V. SOBOLEV
the necessary information on the one-dimensional model problem is assembled in
Chapter 8. The analysis here is based on ideas from [41] and [42].
The proof of Theorem 2.9 starts in Chapter 9. Here we divide the domain Λ into
a boundary layer and inner part. In order to implement the idea of approximating
the domain Λ by a half-space, we construct a partition of unity subordinate to an
appropriate covering of the boundary layer by open sets of specific shape, and for
each set perform the approximation of Λ individually. A convenient covering is
a diadic-type covering, defined in Subsect. 9.1. Lemma 9.4 describes the sought
approximation of Λ.
From the technical point of view, Chapter 10 is the most demanding: here we
find the local asymptotics on each of the covering sets described in Chapter 9. The
calculations are based on the asymptotics in the one-dimensional case obtained in
Chapter 8.
All the intermediary results are put together in Chapter 11 where the proof of
Theorem 2.9 is completed.
The asymptotics (1.2) is extended from polynomials to more general functions
in Chapter 12. The argument is based on the sharp bounds of the type (1.6) derived
in [13], [14]. They lead to Theorems 2.3 and 2.4.
Appendices 1-4 contain some technical material. In particular, Appendix 1
contains the geometrical lemma about the function (1.10) mentioned in the Intro-
duction.
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