Banach spaces and classical harmonic analysis (Q2760186)

This is an interesting and useful paper on the connection between the techniques of harmonic analysis and the known techniques of classical Banach spaces, taking into account that almost every classical Banach space admits a natural action of some group. The points of contact of the two fields considered by the author are the following: projections onto translation invariant subspaces, Cohen's idempotent theorem (the central problem of complemented subspaces), the use of invariant means, \(p\)-summing and \(p\)-integral operators [see \textit{A. Pelczynski}, Stud. Math. 33, 63-70 (1969; Zbl 0189.43701)], Grothendieck theorem and invariant Grothendieck theorem, Stein theorem, \(\Lambda_p\)-sets and Bourgain's results on the \(\Lambda_p\)-problem, Sidon sets [see \textit{N. J. Kalton} and \textit{A. Pelczynski}, Math. Ann. 309, 135-158 (1997; Zbl 0901.46008); \textit{J. Bourgain}, Lect. Notes Math. 1153, 96-127 (1984; Zbl 0583.43008)], Gordon-Lewis local unconditional structure, quasi-Cohen sets, multipliers on spaces of vector-valued functions. NEWLINENEWLINENEWLINEThe entire paper is clearly written and carefully organized, being a good illustration of many remarkable problems of the fields in question. NEWLINENEWLINENEWLINEThe paper has an extensive bibliography of 38 items which gives an excellent survey of certain aspects of the mathematical literature in conjunction with a historical outline of the development of the problems considered by the author.NEWLINENEWLINEFor the entire collection see [Zbl 0970.46001].