Interpolation of linear operators / by S. G. Kreĭn, Ju. I. Petunin, E.M. Semenov ; [translated from the Russian by J. Szűcs]

Bibliogr. p. 355-371. Index

The Russian original (containing Chapters I-IV) was published in 1978 ["Nauka'', Moscow, 1978; MR0506343 (81f:46086)], and the manuscript had actually been ready a few years earlier. This eagerly awaited translation is very good (except for some infelicities in the translation of the Foreword). Several mathematical expressions were translated with their commonly used English counterparts, rather than literally; explanatory footnotes are supplied.
The translation contains a new chapter, Chapter V: Interpolation in spaces of smooth functions, written by Kreĭn alone. This chapter was written at the same time as the other four but was "not included in the Soviet edition for technical reasons''. The Notes on the Literature and the Bibliography have of course been expanded to take account of the addition. In Chapter V, Kreĭn gives an original treatment of Sobolev and Besov spaces from an abstract (interpolation-theoretic) point of view. There are three sections.
Section 1 is entitled "Interpolation spaces constructed from an unbounded operator and a smoothing approximation process'', and further subdivided as follows: 1. Banach couples associated with unbounded operators, 2. Construction of intermediate spaces, 3. Interpolation intermediate spaces, 4. Subordinate operators, 5. Smoothing by means of the resolvent, 6. Smoothing by means of a semigroup, 7. A system of operators with commuting resolvents, 8. A commuting system of semigroups, 9. Unbounded operators in a couple of Banach spaces, imbedding theorem. In this section, intermediate and interpolation spaces in a Banach space E are constructed from an unbounded operator A and what the author calls a "smoothing approximation process'' in E associated with A (i.e. a certain, suitable one-parameter family of operators Qt commuting with A). The abstract approach to this construction is published here for the first time; concrete realizations have been studied by many authors. One of several starting points for this section is a paper by P. Grisvard [J. Math. Pures Appl. (9) 45 (1966), 143–206; MR0221309 (36 #4361)].
Section 2 (with subsections 1. Generalized functions with locally summable derivatives, and traces, 2. Some simple operators in spaces of differentiable functions, 3. Spaces of traces, 4. Complete theorem on traces, 5. Functions smooth in t and smooth relative to an unbounded operator) develops trace theory in an abstract setting. As is well known, this theory originated with J.-L. Lions and was developed by Lions and J. Peetre, and further by Grisvard. The exposition here is based on papers of Grisvard [Math. Scand. 13 (1963), 70–74; MR0164236 (29 #1535); Rend. Mat. (6) 5 (1972), 657–729; MR0341059 (49 #5809)].
In the final Section 3 the author presents the theory of (concrete) Sobolev and Besov spaces in Rn or on "regular'' domains of Rn, as an application of the abstract exposition in the first two sections. Subheadings: 1. Sobolev spaces, 2. Spaces with intermediate smoothness, 3. Imbedding theorem, 4. Function spaces on regular domains, 5. Function spaces on the boundary and trace theorems. (MathSciNet)