Mgr. Eva Pernecká, Ph.D.

Článek

10.1090/proc/17019

Katedra aplikované matematiky

The normed spaces of molecules constructed by Arens and Eells allow us to define two natural equivalence relations on the class of complete metric spaces. We say that two complete metric spaces M and N are M at- equivalent if their normed spaces of molecules are isomorphic and we say that they are .F- equivalent if the corresponding completions - the Lipschitz-free Banach spaces .F ( M ) and .F ( N ) - are isomorphic. In this note, we compare these and some other relevant equivalences of metric spaces. Clearly, M at-equivalent spaces are .F-equivalent. Our main result states that M at-equivalent spaces must have the same covering dimension. In combination with the work of Godard, this implies that M at-equivalence is indeed strictly stronger than .F-equivalence. However, M at-equivalent spaces need not be homeomorphic, as we demonstrate through a general construction. We also observe that M at-equivalence does not preserve the Assouad dimension. We introduce a natural notion of a free basis to simplify the notation.

Článek

10.1016/j.jfa.2024.110560

Katedra aplikované matematiky

We introduce convex integrals of molecules in Lipschitz-free spaces F(M) as a continuous counterpart of convex series considered elsewhere, based on the de Leeuw representation. Using optimal transport theory, we show that these elements are determined by cyclical monotonicity of their supports, and that under certain finiteness conditions they agree with elements of F(M) that are induced by Radon measures on M, or that can be decomposed into positive and negative parts. We also show that convex integrals differ in general from convex series of molecules. Finally, we present some standalone results regarding extensions of Lipschitz functions which, combined with the above, yield applications to the extremal structure of F(M). In particular, we show that all elements of F(M) are convex series of molecules when M is uniformly discrete and identify all extreme points of the unit ball of F(M) in that case.

Článek

10.1093/imrn/rnab329

Katedra aplikované matematiky

We analyze the relationship between Borel measures and continuous linear functionals on the space Lip_0(M) of Lipschitz functions on a complete metric space M. In particular, we describe continuous functionals arising from measures and vice versa. In the case of weak* continuous functionals, that is, members of the Lipschitz-free space F(M), measures on M are considered. For the general case, we show that the appropriate setting is rather the uniform (or Samuel) compactification of M and that it is consistent with the treatment of F(M). This setting also allows us to give a definition of support for all elements of Lip_0(M)* with similar properties to those in F(M), and we show that it coincides with the support of the representing measure when such a measure exists. We deduce that the members of Lip_0(M)* that can be expressed as the difference of two positive functionals admit a Jordan-like decomposition into a positive and a negative part.

Článek

10.1017/S147474802100013X

Katedra aplikované matematiky

Let Lip_0(M) be the space of Lipschitz functions on a complete metric space M that vanish at a base point. We prove that every normal functional in Lip_0(M)* is weak* continuous; that is, in order to verify weak* continuity it suffices to do so for bounded monotone nets of Lipschitz functions. This solves a problem posed by N. Weaver. As an auxiliary result, we show that the series decomposition developed by N. J. Kalton for functionals in the predual of Lip_0(M) can be partially extended to Lip_0(M)*.

Článek

10.1007/s13163-020-00383-7

Katedra aplikované matematiky

Given a topological group G that can be embedded as a topological subgroup into some topological vector space (over the field of reals) we say that G has invariant linear span if all linear spans of G under arbitrary embeddings into topological vector spaces are isomorphic as topological vector spaces. For an arbitrary set A let Z(A) be the direct sum of |A|-many copies of the discrete group of integers endowed with the Tychonoff product topology. We show that the topological group Z(A) has invariant linear span. This answers a question from a paper of Dikranjan et al. (J Math Anal Appl 437:1257–1282, 2016) in positive. We prove that given a non-discrete sequential space X, the free abelian topological group A(X) over X is an example of a topological group that embeds into a topological vector space but does not have invariant linear span.

Článek

10.4171/RMI/1191

Katedra aplikované matematiky

For a complete metric space M, we prove that the finitely supported extreme points of the unit ball of the Lipschitz-free space FM are precisely the elementary molecules (δ(p)−δ(q))/d(p,q) defined by pairs of points p,q in M such that the triangle inequality d(p,q)