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Nonlinear superposition operators. (English) Zbl 0701.47041
Cambridge Tracts in Mathematics, 95. Cambridge etc.: Cambridge University Press. 311 p. £35.00/hbk; $ 59.50/hbk (1990).
With a function f: \(\Omega\times {\mathbb{R}}\rightarrowtail {\mathbb{R}}\), where \(\Omega\) is a measure space, one associates the mapping \((Fu)(x)=f(x,u(x))\) called the superposition or Nemytskij operator. This operator describes the primary structure of nonlinearities in many branches of analysis and depends basically on the function space to which the argument u belongs. Thus, through the book the function u ranges over various frameworks: measurable functions over \(\Omega\), Lebesgue spaces \(L_ p\), Orlicz spaces \(L_ M\), Lorentz \(\Lambda_ p\) and Marcinkiewicz \(M_ p\) spaces, the space C(\(\Omega\)) of continuous functions over \({\bar \Omega}\), the space BV of functions of bounded variation, Hölder spaces \(H^{\alpha}\), certain classes of finite or infinite differentiable functions and finally Sobolev spaces \(W^ k_ p.\)
A systematic study of the superposition operator F as a mapping from a function space to another involves some acting conditions expressed by restrictions upon f. Moreover, all properties of the superposition operator F are characterized in terms of the generating function f. The book provides a comprehensive account of the analytical and topological peculiarities concerning boundedness, continuity, Lipschitz or weak continuity, compactness, differentiability or analyticity, asymptotic linearity of the nonlinear superposition operator. These qualitative results are scattered in research papers or books on particular subjects and require, in general, elaborate proofs, several of which became simpler in time.
At first glance, the volume looks like an enlarged version of the first author’s survey [Expo. Math. 6, 209-270 (1988; Zbl 0648.47041)]. However, by introducing in Chapter II the so-called ideal spaces, which are Banach spaces of measurable functions with monotone norm, the treatment in the rest of the book becomes unified. Ideal spaces and their applications to the study of nonlinear integral operators belong, apparently, to the second author’s field of interest. The bibliography, covering the period from 1918 and 1988, contains about 400 items, half of them in Russian. Thus, it may serve as a guide to Soviet contributions.
The monograph displays in a self-contained form the up-to-date state of knowledge of a basic topic of nonlinear analysis and will be of great help for the people interested in differential and integral equations, probability theory, variational calculus, optimization theory in many other fields of present mathematics as well as a valuable manual in advanced analysis courses.
Reviewer: D.Pascali

47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems