Related papers: On the action of Lipschitz functions on vector-val…

In this paper, we consider the characterization of norm--parallelism problem in some classical Banach spaces. In particular, for two continuous functions $f, g$ on a compact Hausdorff space $K$, we show that $f$ is norm--parallel to $g$ if…

The classical theorems of Banach and Stone, Gelfand and Kolmogorov, and Kaplansky show that a compact Hausdorff space $X$ is uniquely determined by the linear isometric structure, the algebraic structure, and the lattice structure,…

To describe a set of functions, which forms a reflexive subspace B of the classical Banach space L a special function that characterizes their average integral growth is introduced. It is shown that this function essentially depends on the…

The following theorem is the main result of this note. Theorem 1. Let $(E, \|\cdot\|_E) $ be a rearrangement invariant Banach function space on the interval $[0, 1]$. If $E$ is isometric to $\L_p [0, 1]$ for some $1\le p<\infty$, then $E$…

In this article we consider the following equivalence relation on the class of all functions of two variables on a set $X$: we will say that $L,M: X\times X\to \mathbb{C}$ are rescalings if there are non-vanishing functions $f,g$ on $X$…

We prove various notions of uniform continuity for compact-quantum-group representations on Hilbert or Banach spaces equivalent to having finite spectrum, i.e. finitely many isotypic components. This generalizes the classical analogue for…

Suppose that $B$ is a $G$-Banach algebra over $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$, $X$ is a finite dimensional compact metric space, $\zeta : P \to X$ is a standard principal $G$-bundle, and $A_\zeta = \Gamma (X, P \times_G B)$ is the…

We prove that for any constant $K\geq1$, the value functions for time homogeneous stochastic differential games in the whole space can be approximated up to a constant over $K$ by value functions whose second-order derivatives are bounded…

We study absolute summability of inclusions of r.i. function spaces. It appears that such properties are closely related, or even determined by absolute summability of inclusions of subspaces spanned by the Rademacher system in respective…

We give a complete description of homomorphisms between two Banach algebras of Lipschitz functions with values in continuous functions. We also characterize the compactness of those homomorphisms.

We prove two theorems about differentiable functions on the Banach space C(K), where K is compact. (i) If C(K) admits a non-trivial function of class C^m and of bounded support, then all continuous real-valued functions on C(K) may be…

Given an arbitrary $1$-Lipschitz function $f$ on the torus $\mathbb{T}^n $, we find a $k$-dimensional subtorus $M \subseteq \mathbb{T}^n$, parallel to the axes, such that the restriction of $f$ to the subtorus $M$ is nearly a constant…

Let $A,X,Y$ be Banach spaces and $A\times X\to Y$, $(a,x)\mapsto ax$, be a continuous bilinear function, called a *Banach action*. We say that this action *preserves unconditional convergence* if for every bounded sequence…

We denote the local ``little" Lipschitz constant of a function $f: {{\mathbb R}}\to { {\mathbb R}}$ by $ {\mathrm{lip}}f$. In this paper we settle the following question: For which sets $E {\subset} { {\mathbb R}}$ is it possible to find a…

Given a continuous function $f: {{\mathbb R}}\to {{\mathbb R}}$ we denote the so-called "big Lip" and "little lip" functions by $ {{\mathrm {Lip}}} f$ and $ {{\mathrm {lip}}} f$ respectively}. In this paper we are interested in the…

A classical result of Milman roughly states that every Lipschitz function on $\mathbb{S}^n$ is almost constant on a sufficiently high-dimensional sphere $\mathbb{S}^m\subset \mathbb{S}^n$. In this paper we extend the result by proving that…

We obtain an optimal deviation from the mean upper bound \begin{equation} D(x)\=\sup_{f\in \F}\mu\{f-\E_{\mu} f\geq x\},\qquad\ \text{for}\ x\in\R\label{abstr} \end{equation} where $\F$ is the class of the integrable, Lipschitz functions on…

Given a probability measure space $(X,\Sigma,\mu)$, it is well known that the Riesz space $L^0(\mu)$ of equivalence classes of measurable functions $f: X \to \mathbf{R}$ is universally complete and the constant function $\mathbf{1}$ is a…

Let $C_0(K, X)$ denote the space of all continuous $X$-valued functions defined on the locally compact Hausdorff space $K$ which vanish at infinity, provided with the supremum norm. If $X$ is the scalar field, we denote $C_0(K, X)$ by…

Given a unital algebra $\mathscr A$ of locally Lipschitz functions defined over a metric measure space $({\mathrm X},{\mathsf d},\mathfrak m)$, we study two associated notions of function of bounded variation and their relations: the space…