By Sergei M. Nikol'skii, J. Peetre, L.D. Kudryavtsev, V.G. Maz'ya, S.M. Nikol'skii

In the half to hand the authors adopt to provide a presentation of the old improvement of the speculation of imbedding of functionality areas, of the interior in addition to the externals causes that have prompted it, and of the present country of paintings within the box, specifically, what regards the equipment hired this present day. The impossibility to hide all of the huge, immense fabric attached with those questions necessarily compelled on us the need to limit ourselves to a constrained circle of principles that are either basic and of relevant curiosity. in fact, this type of selection needed to some degree have a subjective personality, being within the first position dictated via the non-public pursuits of the authors. therefore, the half doesn't represent a survey of all modern questions within the concept of imbedding of functionality areas. as a result additionally the bibliographical references given don't faux to be exhaustive; we in basic terms record works pointed out within the textual content, and a extra entire bibliography are available in acceptable different monographs. O.V. Besov, v.1. Burenkov, P.1. Lizorkin and V.G. Maz'ya have graciously learn the half in manuscript shape. All their serious comments, for which the authors hereby show their honest thank you, have been taken account of within the ultimate modifying of the manuscript.

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Let us formulate, in the case POI = r lZ , a condition for the family {cpro} to be the trace of a function / E WOO{alZ,PIZ}(G). For each N = 0, 1, . D. M. Nikol'skii 44 L a",llf("')II~:, (]N(f) = Ivl=O and for each fixed family of boundary functions {qJ(JJ} d f 0 ~ {j(x) = g(x) +h(x) : g(x) E WN , h(x) E W N, gloG = qJ(JJ, Iwl :::; N -1}. EN Then there exits a unique function (denote it by f N) such that A family of boundary functions {qJ(JJ} is the trace of a function f E WOO{a""PIX}(G) iff the following conditions hold true: a) for each N = 1,2, ...

G. ). e. functions taking their values in a suitable Banach space (cf. L. B. V. ). If G is any nonempty open set in 1Rn then WJI)(G) admits a basis. One can also prove that the spaces WJI)(G), 1 < p < +00, are isomorphic to Lp(O,I). It follows that Sobolev spaces with 1 < p < +00 have an unconditional basis (Nikolsky-Lions-Lisorkin [1965], Kufner-John-Fucik [1977]). All these results have the character of pure existence theorems. Concrete bases for Sobolev spaces over the n-dimensional cube were found by Cisielski.

E. 8). M. Nikol'skii's theorems for H -classes and coincide with the latter if q = +00. Let us give their formulation. D. M. Nikol'skii 56 Theorem 1. ="1 r , 1:::;; P < Po :::;; +00, 1 1 R 1 "1=1-(---)L->0. 10) Theorem 2. 11) (! J r. j=m+1 J Theorem 3. If f E B;,q(RR), a 1 - 'Lj=1 ajlrj > 0, (! = "3r, then = 1:::;; P:::;; +00. (ab ... 13) where the constant c > 0 does not depend on f. 23). In the case of limit indices ("j = 0, j = 1,2,3) one has the following results. K.