Emerging notions of norm attainment for Lipschitz maps between Banach spaces

Abstract

We classify several notions of norm attaining Lipschitz maps which were introduced previously, and present the relations among them in order to verify proper inclusions. We also analyze some results for the sets of Lipschitz maps satisfying each of these properties to be dense or not in Lip 0 (X, Y ). For instance, we characterize a Banach space Y with the Radon-Nikodým property in terms of the denseness of norm attaining Lipschitz maps with values in Y . Further, we introduce a property called the local directional Bishop-Phelps-Bollobás property for Lipschitz compact maps, which extends the one studied previously for scalar-valued functions, and provide some new positive results.Recently, a few papers dealing with alternative types of norm attainment for Lipschitz maps defined on Banach spaces have appeared. Kadets, Martín and Soloviova [16, Definition 4.2] introduced another possible definition called (locally) directionally norm attaining Lipschitz function. On the other hand, Godefroy [14] defined other two ways in which a Lipschitz map can attain its norm. We also refer to section 8.8 of the very recent book [7] for an exposition of the results of the two aforementioned papers [14,16]. Our first aim in this paper is to introduce some variations of these definitions of norm attainment and study the possible denseness of the set of Lipschitz maps attaining each of such norms. We first provide with the definitions used throughout the paper. Definitions 1.2 and 1.3 were first introduced in [14], and Definitions 1.4 and 1.5 were first considered in [16] only for Lipschitz (real-valued) functions, which are easily extensible to the general (vector-valued) Lipschitz maps. We will use the usual notation of B X , S X , X * for the closed unit ball, unit sphere, and topological dual, respectively, of a Banach space X.Definition 1.2 ([14]). We say that f ∈ Lip 0 (X, Y ) attains its norm at x ∈ X through a derivative in the direction e ∈ S X if f (x, e) = lim t→0

On a new space consisting of norm maintaining functions

“…As promised, we now generalise a definition of norm attainment given in [1] and we prove a similar result to Theorem 2.3. Definition 2.6.…”

“…We start by the following generalised version of norm attainment. For a detailed discussion on norm attainment for linear functionals and the different kinds of norm attainment for Lipschitz maps, see [1,8].…”

“…We also consider the following weaker version of norm attainment. We notice that the definitions of norm attainment given in [1] can be generalised as we did for strong norm attainment; we will do it later for one case presented in the cited paper (see Definition 2.6). Definition 2.3.…”

“…For a more detailed discussion on Lip 0 , see for instance [1][2][3][4][5][6][7]. Furthermore, we can also define (for a fixed β > 0) the space Höl β 0 (S 1 , S 2 ) (where S 1 , S 2 are Banach) consisting of the β-Hölder maps f from S 1 to S 2 such that f (0) = 0.…”

“…The theory of operators that attain their norm has started to be developed many decades ago (see, for instance, [8]); recently, it has been started the study of norm attainment for Lipschitz maps. There are many important kinds of norm attainment in Lip 0 (which can actually be defined in a similar way also in Höl β 0 ); see [1] for a detailed discussion. We will see in Section 2 that some kinds of norm attainment (in a generalised sense) are strictly related to our new space.…”