Article
Mathematics
Matthew Alexander, Matthieu Fradelizi, Luis C. Garcia-Lirola, Artem Zvavitch
Summary: This paper studies the geometric and extremal properties of the convex body B-F(M), which is the unit ball of the Lipschitz-free Banach space associated with a finite metric space M. It investigates l(1) and l(infinity)-sums, characterizes metric spaces where B-F(M) is a Hanner polytope, and discusses extreme properties of the volume product P(M). Furthermore, it examines the conditions under which all triangle inequalities in M are strict and B-F(M) is simplicial when P(M) is maximal among all metric spaces with the same number of points.
JOURNAL OF FUNCTIONAL ANALYSIS
(2021)
Article
Mathematics, Applied
Luis C. Garcia-Lirola, Guillaume Grelier
Summary: We study the ultrapower MU of a metric space M and its properties and applications. We prove that the Lipschitz-free space F(MU) is finitely representable in F(M). Moreover, we characterize the metric spaces that are finitely Lipschitz representable in a Banach space as those embeddable into an ultrapower of the Banach space. Using these results, we show that if M is finitely Lipschitz representable in a Banach space X, then F(M) is finitely representable in F(X). We also apply these findings to investigate cotype in Lipschitz-free spaces and the stability of Lipschitz-free spaces and spaces of Lipschitz functions under ultraproducts.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Mathematics
Ramon J. Aliaga, Colin Petitjean, Antonin Prochazka
Summary: This paper explores the embedding relations between the Lipschitz-free space F(M) of a separable and complete metric space M and l(1), as well as the connection between M being a subset and the length measure of an R-tree. It also discusses the preservation of extreme points of the unit ball in subspaces of L-1 spaces, as well as the characterization of extreme points of the unit ball of F(M) when M is a subset of an R-tree.
JOURNAL OF FUNCTIONAL ANALYSIS
(2021)
Article
Mathematics
Giuliano Basso, Stefan Wenger, Robert Young
Summary: This article investigates Lipschitz k-connectivity, Euclidean isoperimetric inequalities, and coning inequalities. It shows that Lipschitz connectivity implies Euclidean isoperimetric inequalities, and Euclidean isoperimetric inequalities imply coning inequalities, in spaces with finite Nagata dimension. Additionally, it proves that Lipschitz k-connectivity in such spaces can be approximated by Lipschitz chains in total mass.
ADVANCES IN MATHEMATICS
(2023)
Article
Mathematics
Petr Hajek, Ruben Medina
Summary: In this note, a construction of a Schauder basis for the Lipschitz free space F(N) over a net N in any separable infinite dimensional L-infinity-space X is given based on a retractional argument. This yields the first example of an infinite dimensional Banach space X not containing c(0) with such a property.
REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS
(2023)
Article
Mathematics, Applied
Petr Hajek, Ruben Medina
Summary: In this note, we present two explicit constructions based on a retractional argument, for the construction of a Schauder basis for the Lipschitz free space F(N) over certain uniformly discrete metric spaces N. The first construction applies to every net N in a finite dimensional Banach space, resulting in a basis constant independent of the dimension. The second construction applies to grids in Banach spaces with an FDD structure.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Leandro Candido, Pedro L. Kaufmann
Summary: The paper proves that the Lipschitz-free spaces over a Banach space X which is isomorphic to its hyperplanes are isomorphic to the ones over its sphere.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Leondro Candido, Hector H. T. Guzman
Summary: We prove that the Lipschitz-free space F(X) over a Banach space X of density kappa is linearly isomorphic to its l(1)-sum (circle plus k F(X)). This extends a previous result by Kaufmann in non-separable Banach spaces. Furthermore, we provide a complete classification of the spaces of real-valued Lipschitz functions that vanish at 0 over a Lp-space. Specifically, if X is a Lp-space of density kappa with 1 <= p <= infinity, then Lip(0)(X) is isomorphic to either Lip(0)(lp(kappa)) if p < infinity or Lip(0)(c(0)(kappa)) if p = infinity.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics, Applied
Alexandre N. Carvalho, Arthur C. Cunha, Jose A. Langa, James C. Robinson
Summary: We provide a simple proof of a result by Mane (1981) that states a compact subset A of a Banach space, which is negatively invariant under a map S, is finite-dimensional if DS(x) = C(x) + L(x) where C is compact and L is a contraction. Additionally, we demonstrate that if S is both compact and differentiable, A is finite-dimensional. Furthermore, we present some results concerning the (box-counting) dimension of such sets assuming a 'smoothing property' and the Kolmogorov epsilon-entropy of the embedding of Z into X. Finally, we apply these results to an abstract semilinear parabolic equation and the two-dimensional Navier-Stokes equations on a periodic domain.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Ramon J. Aliaga, Camille Nous, Colin Petitjean, Antonin Prochazka
Summary: In this paper, we prove a general principle regarding weakly precompact sets in Lipschitz-free spaces, which allows us to reduce certain infinite-dimensional phenomena to free spaces over compact subsets. As a result, we obtain new results such as F(X) being weakly sequentially complete for every superreflexive Banach space X, and F(M) having the Schur property and the approximation property for every scattered complete metric space M.
PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS
(2021)
Article
Mathematics, Applied
Richard J. Smith, Filip Talimdjioski
Summary: This article discusses the metric approximation property of the pointed metric space (M, d), where M is a closed C1-submanifold of RN and d is the metric given by d(x, y) = ilx - yil, x, y E M.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Mathematics, Applied
Jaan Kristjan Kaasik, Triinu Veeorg
Summary: We construct a Lipschitz-free space that is locally almost square but not weakly almost square, which is the first example of such a Banach space. We also prove a result indicating that geodesic metric spaces have the potential to characterize weakly almost square Lipschitz-free spaces. Lastly, we prove that a Lipschitz-free space cannot have the symmetric strong diameter 2 property.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Mathematics, Applied
Ramon J. Aliaga
Summary: We prove that all extreme points of the unit ball of a Lipschitz-free space over a compact metric space have finite support. Combined with previous results, this completely characterizes extreme points and implies that all of them are also extreme points in the bidual ball. For the proof, we develop some properties of an integral representation of functionals on Lipschitz spaces originally due to K. de Leeuw.
MEDITERRANEAN JOURNAL OF MATHEMATICS
(2022)
Article
Mathematics
Fernando Albiac, Jose L. Ansorena, Marek Cuth, Michal Doucha
Summary: We investigate the isomorphism conditions of Lipschitz-free spaces over metric spaces to their infinite direct l(1)-sum and present several applications, such as isomorphism of Lipschitz-free spaces over balls and spheres of the same finite dimensions, over Z(d) to its l(1)-sum, and over any snowflake of a doubling metric space to l(1). Additionally, we provide a self-contained proof that Lipschitz-free spaces over doubling metric spaces are complemented in Lipschitz-free spaces over their superspaces and have BAP, building on new ideas from Bru`e et al. (J. Funct. Anal. 280 (2021), pp. 108868, 21).
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics
Marek Cuth, Michal Doucha
Summary: This paper investigates the problem of compact group G acting continuously on a metric space M, and shows that the Lipschitz-free space over the space of orbits M/G is complemented in the Lipschitz-free space over M, given uniformly bounded norms of bi-Lipschitz bijections. It also explores the more general case when G is amenable, locally compact, or SIN and its action has bounded orbits, and provides conditions for complementation of Lipschitz functions in Lip0(M/G) in Lip0(M). The paper also discusses applications and includes preliminaries on projections induced by actions of amenable groups on general Banach spaces.
MATHEMATISCHE NACHRICHTEN
(2023)