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Superposition of functions in Sobolev spaces of fractional order. A survey. (English) Zbl 0792.47062
Bojarski, Bogdan (ed.) et al., Partial differential equations. Part 2. The 36th semester, held at the Stefan Banach International Mathematical Center in Warsaw, Poland, from September 17 to December 17, 1990. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Cent. Publ. 27, Part 2, 481-497 (1992).
When the monograph on nonlinear superposition operators by P. P. Zabrejko and the reviewer appeared [Cambridge Univ. Press, Cambridge (1990; Zbl 0701.47041)], almost nothing was known about the operator $$T_ G(f)= G\circ f$$ in Sobolev spaces. In the last four years, however, a considerable progress was made by many important contributions due to, e.g., G. Bourdaud, T. Kunst, and the author of this survey article. In this article the author gives a complex characterization of the acting, boundedness and continuity properties of the operator $$T_ G$$ in the Sobolev space $$H_ p^ s(\mathbb{R}^ n)$$ $$(1<p<\infty$$, $$s\geq 0)$$, with an emphasis on degeneracy phenomena (e.g. linearity of $$G$$). The reviewer hopes that this interesting review article will be appear in an extended version as a monograph which should be of interest to a large readership.
For the entire collection see [Zbl 0771.00022].

MSC:
 47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems