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Interpolation of \(H^p\)-spaces: Some recent developments. (English) Zbl 0956.46018

Brudnyi, Yuri (ed.) et al., Function spaces, interpolation spaces, and related topics. Proceedings of the workshop, Haifa, Israel, June 7-13, 1995. Ramat Gan: Bar-Ilan University/distr. by the American Mathematical Society, Isr. Math. Conf. Proc. 13, 102-140 (1999).
Based on the Introduction: At the end of the 80’s, the theory of interpolation of \(H^p\) spaces seemed to be finished in the main. However, some more recent events have led to revision (at least partial) of the theory. Surprisingly, the new methods have turned out to be interesting and meaningful already for the classical Hardy spaces of functions analytic in the disc. Probably, the main feature of this revised attitude towards the interpolation of \(H^p\)-spaces is the understanding that a right question here is the question of \(K\)-closedness rather than of various specific interpolation theorems. For the first time, this was clearly articulated by G. Pisier [Pac. J. Math. 155, No. 2, 341-368 (1992; Zbl 0782.46055); Rev. Mat. Iberoam. 9, No. 2, 281-291 (1993; Zbl 0788.46071); Mich. Math. J. 39, No. 3, 475-484 (1992; Zbl 0787.47021)]. Roughly speaking, in this survey I want to convince the reader of two theses:
1) for \(H^p\)-like spaces, \(K\)-closedness is an easy matter, and
2) there is good evidence that for such spaces there should remain not too much interpolation in general beyond the property of \(K\)-closedness.
It should also be noted that the material has been selected in accordance with the author’s personal taste, so that many (old and recent) things pertaining to the interpolation of Hardy spaces are even not mentioned here.
For the entire collection see [Zbl 0919.00055].

MSC:

46B70 Interpolation between normed linear spaces
46M35 Abstract interpolation of topological vector spaces
42B30 \(H^p\)-spaces
30D55 \(H^p\)-classes (MSC2000)
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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