Atomic decompositions, two stars theorems, and distances for the Bourgain–Brezis–Mironescu space and other big spaces

Abstract

Given a Banach space E with a supremum-type norm induced by a collection of operators, we prove that E is a dual space and provide an atomic decomposition of its predual. We apply this result, and some results obtained previously by one of the authors, to the function space B introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomic decomposition of the predual B * , the biduality result that B * 0 = B * and B * * = B, and a formula for the distance from an element f ∈ B to B 0 .

Modulars from Nakano onwards

“…More generally, other examples are all the proper, closed Banach subspaces of Banach function spaces: for instance, if Ω ⊂ R n , the Banach space exp, defined as the closure in L ∞ (Ω) in EXP , is not a Banach function space. We quote also the John-Nirenberg BMO space and other BMO-like spaces like the recent ones introduced in Bourgain, Brezis, Mironescu [27] (see also D'Onofrio, Greco, Perfekt, Sbordone, Schiattarella [44]), which are Banach spaces whose elements are measurable functions modulo constants, and only representatives from each equivalence class belong to L 0 .…”

Atomic decomposition of finite signed measures on compacts of R^n

“…To obtain a decomposition of elements of M 0 (K) c -which will induce a decomposition of elements of M 0 (K)-we generalize the approach of [3], which relies on the o-O structure of (c 0,α , C 0,α ), by using results contained in [9], which allow us to remove the dependence on the "little o" space, because for Lip and Lip 0 it is trivial. We start by writing Lip 0 in a suitable way.…”

“…We start by writing Lip 0 in a suitable way. Indeed we want to make use of [9,Theorem 3] and, to do this, we have to characterize Lip 0 by means of linear bounded operators L : X → Y where X is a reflexive Banach space containing Lip 0 and Y is some other Banach space. In particular, we want to find a countable family F = {L j } j∈N of such operators such that…”

“…In this paper we will give an atomic decomposition of the spaces M(K) and M 0 (K) by restriction of the decomposition of their completions, seen as preduals of Lipschitz spaces. We recall that the description of atomic decompositions of Hölder spaces on compact spaces was given in [20] and [2], following different approaches; in particular, in [20] the atomic decomposition is closer to other "classical" examples [7,9], while in [2] a more abstract atomic decomposition is obtained. We decided to follow this second approach, based on techniques from [9], which are inspired by the o-O construction in [24].…”

“…We recall that the description of atomic decompositions of Hölder spaces on compact spaces was given in [20] and [2], following different approaches; in particular, in [20] the atomic decomposition is closer to other "classical" examples [7,9], while in [2] a more abstract atomic decomposition is obtained. We decided to follow this second approach, based on techniques from [9], which are inspired by the o-O construction in [24]. In particular, in the case of the distance d α (x, y) = |x − y| α on a compact set of R n , the o-O construction has already been shown in [24], while in the general framework of doubling compact metric-measure spaces it has been achieved, under some approximation hypotheses, in [2], where the atomic decomposition in this case has been already deduced.…”