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.