Rademacher functions in symmetric spaces

Abstract

The Rademacher system or the Bernoulli sequence of independent, identically and symmetrically distributed random variables taking values ±1 is a classical object of orthogonal series theory and probability theory. It has many applications in other fields as well, first of all, in the geometric theory of Banach spaces, theory of operators, harmonic analysis, and mathematical statistics. The topic of this monograph is more specific: the study of the Rademacher system in symmetric (rearrangement invariant) functional spaces. The first work devoted to this problem (apart from the classical results for L p -spaces such as the Khintchine inequality) is an article by Rodin and Semenov that was published in 1975. A number of principal problems related to the behavior of Rademacher systems in symmetric spaces was posed and the path for further investigations was indicated. We recall that the initial works on general symmetric spaces (first of all, works of G. Lorentz and Semenov) appeared in 50s-60s of the last century. Then [92,96,97] summarized a certain stage of the investigations.Investigations of the Rademacher systems in symmetric spaces received renewed impetus in 90th after the works by S. Montgomery-Smith and P. Hitchenko, where they refined the Khintchine inequality. The main role was played by the Peetre K-functional.

Structure of Cesàro function spaces: a survey

“…From the classical Khintchine inequality it follows that these functions span an isomorphic copy of l 2 in L p for every 0 < p < ∞. Investigations of Rademacher sums in general rearrangement invariant spaces rather than L p are well presented in a series of papers and in the books by Lindenstrauss-Tzafriri [61], Krein-Petunin-Semenov [53] and Astashkin [4]. However, not so much we know on the behaviour of the Rademacher functions in general Banach lattices.…”

Józef Marcinkiewicz (1910–1940) – on the centenary of his birth

“…We can find the proof of the inequalities (17) and (18) In 2000 Astashkin [As00] proved that the system of independent random variables satisfying (16) is even equivalent in the distribution sense to the Rademacher system (see also [As09], Theorem 8.4 and Corollary 8.3).…”

Pointwise products of some Banach function spaces and factorization

“…More information about Banach ideal spaces, quasi-Banach ideal spaces, symmetric Banach and quasi-Banach spaces can be found, for example, in [39,48,35,33,43,7,63,4].…”