We investigate a scale of $BM{O}_{\psi }$-spaces defined with the help of certain Lorentz norms. The results are applied to extrapolation techniques concerning operators defined on adapted sequences. Our extrapolation works simultaneously with two operators, starts with $BM{O}_{\psi }$-$L_{\infty }$-estimates, and arrives at $L_p$-$L_p$-estimates, or more generally, at estimates between K-functionals from interpolation theory.

This paper is meant as a (short and partial) introduction to the study of the geometry of Carnot groups and, more generally, of Carnot-Carathéodory spaces associated with a family of Lipschitz continuous vector fields. My personal interest in this field goes back to a series of joint papers with E. Lanconelli, where this notion was exploited for the study of pointwise regularity of weak solutions to degenerate elliptic partial differential equations. As stated in the title, here we are mainly concerned...

We investigate Baire classes of strongly affine mappings with values in Fréchet spaces. We show, in particular, that the validity of the vector-valued Mokobodzki result on affine functions of the first Baire class is related to the approximation property of the range space. We further extend several results known for scalar functions on Choquet simplices or on dual balls of L₁-preduals to the vector-valued case. This concerns, in particular, affine classes of strongly affine Baire mappings, the...

Let $X$ be a complex $L_1$-predual, non-separable in general. We investigate extendability of complex-valued bounded homogeneous Baire-$\alpha$ functions on the set $\mathrm{ext}{B}_{X^{*}}$ of the extreme points of the dual unit ball $B_{X^{*}}$ to the whole unit ball $B_{X^{*}}$. As a corollary we show that, given $\alpha \in [1,{\omega }_1)$, the intrinsic $\alpha$-th Baire class of $X$ can be identified with the space of bounded homogeneous Baire-$\alpha$ functions on the set $\mathrm{ext}{B}_{X^{*}}$ when $\mathrm{ext}{B}_{X^{*}}$ satisfies certain topological assumptions. The paper is intended to be a complex counterpart to the same authors’...

We characterize Baire-like spaces Cc(X,E) of continuous functions defined on a locally compact and Hewitt space X into a locally convex space E endowed with the compact-open topology.

Results on singular products of the distributions $x_{\pm }^{-p}$ and $x^{-p}$ for natural $p$ are derived, when the products are balanced so that their sum exists in the distribution space. These results follow the pattern of a known distributional product published by Jan Mikusiński in 1966. The results are obtained in the Colombeau algebra of generalized functions, which is the most relevant algebraic construction for tackling nonlinear problems of Schwartz distributions.

Let U, V be two symmetric convex bodies in ${ℝ}^n$ and |U|, |V| their n-dimensional volumes. It is proved that there exist vectors $u_1,...,u_n\in U$ such that, for each choice of signs ${\epsilon }_1,...,{\epsilon }_n=\pm 1$, one has ${\epsilon }_1{u}_1+...+{\epsilon }_n{u}_n\notin rV$ where $r={\left(2\pi e^2\right)}^{-1/2}{n}^{1/2}\left(\right|U|/{\left|V\right|)}^{1/n}$. Hence it is deduced that if a metrizable locally convex space is not nuclear, then it contains a null sequence $\left(u_n\right)$ such that the series ${\sum }_{n=1}^{\infty }{\epsilon }_n{u}_{\pi \left(n\right)}$ is divergent for any choice of signs ${\epsilon }_n=\pm 1$ and any permutation π of indices.

The notion of ball proximinality and the strong ball proximinality were recently introduced in [2]. We prove that a closed * subalgebra A of C(Q) is strongly ball proximinal in C(Q) and the metric projection from C(Q), onto the closed unit ball of A, is Hausdorff metric continuous and hence has continuous selection.

We study Banach spaces X with subspaces Y whose unit ball is densely remotal in X. We show that for several classes of Banach spaces, the unit ball of the space of compact operators is densely remotal in the space of bounded operators. We also show that for several classical Banach spaces, the unit ball is densely remotal in the duals of higher even order. We show that for a separable remotal set E ⊆ X, the set of Bochner integrable functions with values in E is a remotal set in L¹(μ,X).

We provide a structure theorem for Carnot-Carathéodory balls defined by a family of Lipschitz continuous vector fields. From this result a proof of Poincaré inequality follows.

In this article, we describe some basic properties of the Banach algebra which is constructed from all bounded complex-valued functionals.

Let MX,w(ℝ) denote the algebra of the Fourier multipliers on a separable weighted Banach function space X(ℝ,w).We prove that if the Cauchy singular integral operator S is bounded on X(ℝ, w), thenMX,w(ℝ) is continuously embedded into L∞(ℝ). An important consequence of the continuous embedding MX,w(ℝ) ⊂ L∞(ℝ) is that MX,w(ℝ) is a Banach algebra.

For infinite discrete additive semigroups $X\subset [0,\infty )$ we study normed algebras of arithmetic functions $g:X\to ℂ$ endowed with the linear operations and the convolution. In particular, we investigate the problem of scaling the mean deviation of related multiplicative functions for $X=log\mathbb{N}$. This involves an extension of Banach algebras of arithmetic functions by introducing weight functions and proving a weighted inversion theorem of Wiener type in the frame of Gelfand’s theory of commutative Banach algebras.