We study the extremal structure of Banach spaces of continuous functions with the diameter norm.

Under some mild assumptions, non-linear diameter-preserving bijections between (vector-valued) function spaces are characterized with the help of a well-known theorem of Ulam and Mazur. A necessary and sufficient condition for the existence of a diameter-preserving bijection between function spaces in the complex scalar case is derived, and a complete description of such maps is given in several important cases.

Assume that L p,q, $L^{p_1,q_1},...,L^{p_n,q_n}$ are Lorentz spaces. This article studies the question: what is the size of the set $E=\{(f_1,...,f_n)\in L^{p_{1,}{q}_1}\times \cdots \times L^{p_n,q_n}:f_1\cdots f_n\in L^{p,q}\}$. We prove the following dichotomy: either $E=L^{p_1,q_1}\times \cdots \times L^{p_n,q_n}$ or E is σ-porous in $L^{p_1,q_1}\times \cdots \times L^{p_n,q_n}$, provided 1/p ≠ 1/p 1 + … + 1/p n. In general case we obtain that either $E=L^{p_1,q_1}\times \cdots \times L^{p_n,q_n}$ or E is meager. This is a generalization of the results for classical L p spaces.

A bounded linear operator between Banach spaces is called a Dieudonné operator ( = weakly completely continuous operator) if it maps weakly Cauchy sequences to weakly convergent sequences. Let (Ω,Σ,μ) be a finite measure space, and let X and Y be Banach spaces. We study Dieudonné operators T: L¹(X) → Y. Let $i_{\infty }:L^{\infty }\left(X\right)\to L¹\left(X\right)$ stand for the canonical injection. We show that if X is almost reflexive and T: L¹(X) → Y is a Dieudonné operator, then $T\circ i_{\infty }:L^{\infty }\left(X\right)\to Y$ is a weakly compact operator. Moreover, we obtain that if T: L¹(X)...

In the geometries of stratified groups, we provide differentiability theorems for both functions of bounded variation and Sobolev functions. Proofs are based on a systematic application of the Sobolev-Poincaré inequality and the so-called representation formula.

We study Lebesgue points for Sobolev functions over other collections of sets than balls. Our main result gives several conditions for a differentiation basis, which characterize the existence of Lebesgue points outside a set of capacity zero.

Soient ${𝒦}_1$ et ${𝒦}_2$ deux espaces de Krein de fonctions analytiques dans le disque unité invariants pour l’opérateur de déplacement à gauche $R_0(R_{0}f\left(z\right)=\left(f\right(z)-f(0\left)\right)/z)$ et soit $A$ un opérateur linéaire continu de ${𝒦}_1$ dans ${𝒦}_2$ dont l’adjoint commute avec $R_0$. Nous étudions les dilatations $B$ de $A$ qui conservent cette propriété de commutation et pour lesquelles les formes hermitiennes définies par $I-A{A}^{*}$ et $I-B{B}^{*}$ ont le même nombre de carrés négatifs. Nous obtenons ainsi une version du théorème de dilatation des commutants d’opérateurs dans le cadre...

We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. We show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space. For foliations of a metric space X defined by a David–Semmes regular mapping Π : X → W, we quantitatively estimate, in terms of Hausdorff dimension in W, the...

The notion of dimensionally compact class in a biequivalence vector space is introduced. Similarly as the notion of compactness with respect to a $\pi$-equivalence reflects our nonability to grasp any infinite set under sharp distinction of its elements, the notion of dimensional compactness is related to the fact that we are not able to measure out any infinite set of independent parameters. A fairly natural Galois connection between equivalences on an infinite set $s$ and classes of set functions $s\to Q$...

We survey recent dimension-invariant imbedding theorems for Sobolev spaces.