Sequences of independent functions and structure of rearrangement invariant spaces

S. V. Astashkin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 79 (2024) no. 3, pp. 375-457

Résumé

The main aim of the survey is to present results of the last decade on the description of subspaces spanned by independent functions in $L_p$-spaces and Orlicz spaces on the one hand, and in general rearrangement invariant spaces on the other. A new approach is proposed, which is based on a combination of results in the theory of rearrangement invariant spaces, methods of the interpolation theory of operators, and some probabilistic ideas. The problem of the uniqueness of the distribution of a function such that a sequence of its independent copies spans a given subspace is considered. A general principle is established for the comparison of the complementability of subspaces spanned by a sequence of independent functions in a rearrangement invariant space on $[0,1]$ and by pairwise disjoint copies of these functions in a certain space on the half-line $(0,\infty)$. As a consequence of this principle we obtain, in particular, the classical Dor–Starbird theorem on the complementability of subspaces spanned by independent functions in the $L_p$-spaces. Bibliography: 103 titles.

Détail

Keywords: independent functions, $L_p$-space, rearrangement invariant space, Orlicz function, Orlicz space, $p$-convex function, $p$-concave function, Boyd indices, Rosenthal's inequalities, Kruglov property, $\mathcal K$-functional, complemented subspace, projection.