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Characterizations of Besov-Hardy-Sobolev spaces: A unified approach. (English) Zbl 0644.46017
From the introduction. This paper deals with the Besov-Hardy-Sobolev spaces \(B^ s_{p,q}(R_ n)\) and \(F^ s_{p,q}(R_ n)\) on the euclidean n-space \(R_ n\). It is a self-containing survey about a special aspect, the theory of equivalent norms and quasi-norms in these spaces. In [Theory of function spaces (1983; Zbl 0546.46027)] we dealt extensively with characterizations of the spaces considered via rather different means: differences \((\Delta^ M_ hf)(x)\) and derivatives \((D^{\alpha}f)(x)\) of functions, several types of mean values of differences of functions, traces of harmonic functions, and temperatures in \(R^+_{n+1}=\{(x,t)| x\in R_ n,t>0\}\) on the hyperplane \(t=0\), etc. In [loc. cit.] we used specific tools for different characterizations. One aim of the present paper is to demonstrate that these different characterizations can be obtained from a unified point of view. The second aim of this paper is to extend the previous results. For that purpose we introduced weighted means of differences and derivatives of functions and distributions, which give also the possibility to characterize spaces \(B^ s_{p,q}(R_ n)\) and \(F^ s_{p,q}(R_ n)\) with negative smoothness s and to establish a localization principle for all these spaces in a rather easy way.
Reviewer: J.Wloka

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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