We construct, under Axiom ♢, a family ${\left(C\left(K_{\xi }\right)\right)}_{\xi <2^{\left(2^{\omega }\right)}}$ of indecomposable Banach spaces with few operators such that every operator from $C\left(K_{\xi }\right)$ into $C\left(K_{\eta }\right)$ is weakly compact, for all ξ ≠ η. In particular, these spaces are pairwise essentially incomparable. Assuming no additional set-theoretic axiom, we obtain this result with size $2^{\omega }$ instead of $2^{\left(2^{\omega }\right)}$.

For each ordinal α < ω₁, we prove the existence of a Banach space with a basis and Szlenk index ${\omega }^{\alpha +1}$ which is universal for the class of separable Banach spaces with Szlenk index not exceeding ${\omega }^{\alpha }$. Our proof involves developing a characterization of which Banach spaces embed into spaces with an FDD with upper Schreier space estimates.

For each ordinal α < ω₁, we prove the existence of a separable, reflexive Banach space W with a basis so that Sz(W), $Sz(W*)\le {\omega }^{\alpha +1}$ which is universal for the class of separable, reflexive Banach spaces X satisfying Sz(X), $Sz(X*)\le {\omega }^{\alpha }$.

It is shown that for every k ∈ ℕ and every spreading sequence eₙₙ that generates a uniformly convex Banach space E, there exists a uniformly convex Banach space $X_{k+1}$ admitting eₙₙ as a k+1-iterated spreading model, but not as a k-iterated one.

Let $X$ be a non-reflexive real Banach space. Then for each norm $|·|$ from a dense set of equivalent norms on $X$ (in the metric of uniform convergence on the unit ball of $X$), there exists a three-point set that has no Chebyshev center in $(X,|·\left|\right)$. This result strengthens theorems by Davis and Johnson, van Dulst and Singer, and Konyagin.

We study the notion of fractional $L^p$-differentiability of order $s\in (0,1)$ along vector fields satisfying the Hörmander condition on ${ℝ}^n$. We prove a modified version of the celebrated structure theorem for the Carnot-Carathéodory balls originally due to Nagel, Stein and Wainger. This result enables us to demonstrate that different $W^{s,p}$-norms are equivalent. We also prove a local embedding $W^{1,p}\subset W^{s,q}$, where q is a suitable exponent greater than p.

The notion of functions dependent locally on finitely many coordinates plays an important role in the theory of smoothness and renormings on Banach spaces, especially when higher smoothness (C∞) is involved. In this note we survey most of the main results in this area, and indicate many old as well as new open problems.

A classical result of Cembranos and Freniche states that the C(K,X) space contains a complemented copy of c₀ whenever K is an infinite compact Hausdorff space and X is an infinite-dimensional Banach space. This paper takes this result as a starting point and begins a study of conditions under which the spaces C(α), α < ω₁, are quotients of or complemented in C(K,X). In contrast to the c₀ result, we prove that if C(βℕ ×[1,ω],X) contains a complemented copy of $C\left({\omega }^{\omega }\right)$ then X contains a copy of c₀. Moreover,...

Let X and Y be Banach spaces. An operator G: X → Y is a Daugavet center if ‖G +T‖ = ‖G‖+‖T‖ for every rank-1 operator T. For every Daugavet center G we consider a certain set of operators acting from X, so-called G-narrow operators. We prove that if J is the natural embedding of Y into a Banach space E, then E can be equivalently renormed so that an operator T is (J ○ G)-narrow if and only if T is G-narrow. We study G-rich subspaces of X: Z ⊂ X is called G-rich if the quotient map q: X → X/Z is...

We introduce higher order spreading models associated to a Banach space X. Their definition is based on ℱ-sequences ${\left(x_s\right)}_{s\in ℱ}$ with ℱ a regular thin family and on plegma families. We show that the higher order spreading models of a Banach space X form an increasing transfinite hierarchy ${\left({ℳ}_{\xi }\left(X\right)\right)}_{\xi <\omega ₁}$. Each ${ℳ}_{\xi }\left(X\right)$ contains all spreading models generated by ℱ-sequences ${\left(x_s\right)}_{s\in ℱ}$ with order of ℱ equal to ξ. We also study the fundamental properties of this hierarchy.