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

In the half handy the authors adopt to provide a presentation of the ancient improvement of the speculation of imbedding of functionality areas, of the inner in addition to the externals factors that have influenced it, and of the present country of artwork within the box, specifically, what regards the equipment hired at the present time. The impossibility to hide all of the huge, immense fabric hooked up with those questions necessarily pressured on us the need to limit ourselves to a constrained circle of rules that are either basic and of important curiosity. after all, this kind of selection needed to some degree have a subjective personality, being within the first position dictated by means of the private pursuits of the authors. hence, the half doesn't represent a survey of all modern questions within the idea of imbedding of functionality areas. consequently additionally the bibliographical references given don't faux to be exhaustive; we basically checklist works pointed out within the textual content, and a extra whole bibliography are available in applicable 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 severe feedback, for which the authors hereby exhibit their honest thank you, have been taken account of within the ultimate enhancing of the manuscript.

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**Example text**

16) in the opposite case, j = 1, ... , n. 16). In the formulation of the definition there is some ambiguity in the choice of the pairs of numbers kj' Sj corresponding to the numbers rj. 14) of functions f E H; (G), defined with an arbitrary choice of pairs kjo Sj satisfying the conditions kj > rj - Sj > 0, j = 1,2, ... , n, are all equivalent. 4) (cf. Chap. 4). ·,r)(G). n. I. 19) II 1 ::s;; p ::s;; +00, l/rj > O. 21) j=l P = "l r, a = (at. 23) where the constant c > 0 does not depend on I. 16) is a special case of an old result by Hardy and Littlewood.

Xn) E Lp(Rn) of exponential type of degrees V..... , Vn in the variables X ..... ,Xn. n), r = (rl, ... ,rn ), hold the following direct and inverse theorems of the type of the theorems of Bernshteln, Jackson and Zygmund. n), r = (r ..... (f)p =s;; c L If v? 25) with Vj = k = 0,1,2, ... , aj > 1, j = 1,2, ... ,n, then f E H;(Rn). M. K. Potapov [1964] in the study of properties of functions analytic in a strip on the boundary of the strip. e. "). This fact a function has lead mathematicians to investigate the extension of functions to the whole space with presevation of class or, if this is not possible, to the extension to the whole space in such a way that the extended function belongs to a in some sense optimal class among all possible classes (for this cf.

7) = h~1 ... h~, klhl = kdhd + ... + knlhnl. It is of importance that in a certain sense also the converse statement (loc. II ::S;;c, 'rIh eRn,k= (kt. 8) Lp(Gklhl) then the generalized derivative I(k) exists on G and III (k) II Lp(G) ::s;; C. 9) Note that this statement is a generalization of the following fact (corresponding to the case n = 1, p = +(0): if the function I satisfies on the interval (a, b) a Lipschitz condition with constant c > 0 then it has almost everywhere on (a, b) a derivative satisfying the inequality If'(x) I ::s;; c.