By Sergei Mihailovic Nikol’skii (auth.)
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Additional info for Approximation of Functions of Several Variables and Imbedding Theorems
Thus we have defined the product III for functions IE Lp (1 ~ P ~oo). For p = 00 this definition already does not go through, since a function bounded on 1R cannot be approximated arbitrarily well in the metric of Loo by finite functions. But for our needs the definition of /ll introduced in the preceding section will be fully satisfactory in the case p = 00, when /l = K E L. A multiplicator Il (satisfying property (2)) will be called a Mareinkiewiez multiplier (see further 1. 3) . ,. IlllIllp ~ epllflip (5) for all IE Lp(1 ~ P <(0), where the ep is the same constant as in the corresponding inequality for I E S.
We will not need this in what follows. But there is one case which we shall need-the case of the factor V-lIl V, where V, p, eLand V moreover is a positive infinitely differentiable function of polynomial growth. If ! E L p , then the operation V-1IlVj = V-l(ll(Vj)) has meaning. Indeed, vj may be understood in the sense (1) or (6). This leads to one and the same result. 5. Generalized functions every case be understood in the sense (2). For this we need only remark that p(Vl) E 5', because ;CVl) E LpIt is important that the equation (12) holds for every 1 E Lp.
U(x) is a bounded function measurable on JR = JR n , so that ,U E 5'. We emphasize that if 1 E 5, then 1 E 5 is an infinitely differentiable function of polynomial growth. Accordingly, the product pl E 5' is defined: Gul, cp) (1) = (P,]CP) , which is represented by the measurable function ,ul = p(x) l(x) . -,. ufllp ~ cpll/lip (2) is satisfied, where the constant cp does not depend on I. /k -Izlip ~ 0 (k, l ~ (0). " piz tends. It is natural to denote it by ~ 00, the 42 1. Preparatory information calling /l * t the convolution of the function /l (generally speaking generalized) with t.
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