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Rearrangement of functions and embedding of anisotropic spaces of Sobolev type. (English) Zbl 0917.46019

The paper is a survey on the theory of embedding of function spaces. The main goal of the author is to present the fundamental results of the embedding theory by using of uniform approach. All notions, facts and methods needed for the general theory are explained in details. The theorems are given with complete proofs. That is why the survey can be a useful handbook for the specialists of other fields to find their way in the embedding theory. New, unpublished results are also presented. Special emphasis is placed on the \(L_1\)-theory. It is known that the methods of harmonic analysis, integral transforms and interpolation theory can not be always applied to the case of \(L_1\)-norm. The author presents also methods, based on rearrangement of functions in the embedding theory. The papers of P. L. Ul’yanov have an important role in the development of these methods.
Reviewer: D.Dryanov (Sofia)

MSC:

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
46E15 Banach spaces of continuous, differentiable or analytic functions
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