On strongly norm attaining Lipschitz maps

Abstract

We study the set SNA(M, Y ) of those Lipschitz maps from a (complete pointed) metric space M to a Banach space Y which (strongly) attain their Lipschitz norm (i.e. the supremum defining the Lipschitz norm is a maximum). Extending previous results, we prove that this set is not norm dense when M is a length space (or local) or when M is a closed subset of R with positive Lebesgue measure, providing new examples which have very different topological properties than the previously known ones. On the other hand, we study the linear properties which are sufficient to get Lindenstrauss property A for the Lipschitz-free space F (M ) over M , and show that all of them actually provide the norm density of SNA(M, Y ) in the space of all Lipschitz maps from M to any Banach space Y . Next, we prove that SNA(M, R) is weakly sequentially dense in the space of all Lipschitz functions for all metric spaces M . Finally, we show that the norm of the bidual space of F (M ) is octahedral provided the metric space M is discrete but not uniformly discrete or M is infinite.J. Lindenstrauss extended such study to general linear operators, showed that this is not always possible, and also gave positive results. If we say that a Banach space X has (Lindenstrauss) property A when NA(X, Y ) = L(X, Y ) for every Banach space Y , it is shown in [36] that reflexive spaces have property A. This result was extended by J. Bourgain [9] showing that Banach spaces X with the RNP also have Lindenstrauss property A, and he also provided a somehow reciprocal result. We refer the interested reader to the survey paper [3] for a detailed account on norm attaining linear operators.

The Bishop–Phelps–Bollobás property for Lipschitz maps

“…The first statement of the following remark, which is a routine application of Lemma 1.3 of [11] analogous to what is done in [11,Lemma 3.12], gives a reformulation of the Lip-BPB property. We will use both equivalent formulations without giving any explicit reference.…”

“…Now, given p < q ∈ N, ifĝ ∈ L(F(N), R) with g L = 1 attains its norm at a molecule m q,p such that m q,p − m 3n,n < ε, Lemma 1.3 in [11] implies that [2n, 2n + 1] ⊆ [p, q]. Indeed, if we assume that p > 2n or q < 2n + 1, then by applying that lemma we obtain…”

“…By [17,Theorem 5.4], M is Gromov concave if and only if every molecule is a strongly exposed point of the unit ball of F(M ). By [11,Proposition 3.6], M is uniformly Gromov concave if this happens uniformly:…”

“…The extension of the Lip-BPB property from (M, R) to some Banach spaces Y and some results for compact Lipschitz maps are also discussed.Recently, the problem of deciding for which metric spaces M the set LipSNA(M, Y ) is (norm) dense in Lip 0 (M, Y ) has been studied. We refer the reader to [11], [16], [18], and [21], as references for this study. Let us collect here some of the known results.…”

Emerging notions of norm attainment for Lipschitz maps between Banach spaces

“…In fact, SNA(X, R) fails to be dense in Lip 0 (X, R) for every Banach space X (see [16,Theorem 2.3]) and, therefore, SNA(X, Y ) cannot be dense in Lip 0 (X, Y ) for any Banach space Y by [8,Proposition 4.2]. We refer the interested reader to the recent papers [6,8] for the study of the denseness of strongly norm attaining Lipschitz maps defined in general metric spaces. Definition 1.5 ([16]).…”