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.