Related papers: Optimality of the rearrangement inequality with ap…

By generalizing a construction of Garling, for each $1\leqslant p<\infty$ and each normalized, nonincreasing sequence of positive numbers $w\in c_0\setminus\ell_1$ we exhibit an $\ell_p$-saturated, complementably homogeneous Banach space…

The aim of this paper is to introduce and investigate a new class of separable Banach spaces modeled after an example of Garling from 1968. For each $1\leqslant p<\infty$ and each nonincreasing weight $\textbf{w}\in c_0\setminus\ell_1$ we…

In this paper we settle in the negative the problem of the superreflexivity of Garling sequence spaces by showing that they contain a complemented subspace isomorphic to a non superreflexive mixed-norm sequence space. As a by-product of our…

Our aim in this article is to contribute to the study of the structure of subsymmetric basic sequences in Banach spaces (even, more generally, in quasi-Banach spaces). For that we introduce the notion of positioning and develop new tools…

Let X_1, X_2,..., X_n be a sequence of independent random variables, let M be a rearrangement invariant space on the underlying probability space, and let N be a symmetric sequence space. This paper gives an approximate formula for the…

{\it We study the class of all rearrangement-invariant (=r.i.) function spaces $E$ on $[0,1]$ such that there exists $0<q<1$ for which $ \Vert \sum_{_{k=1}}^n\xi_k\Vert_{E}\leq Cn^{q}$, where $\{\xi_k\}_{k\ge 1}\subset E$ is an arbitrary…

Let (f_n) and (g_n) be two sequences of random variables adapted to an increasing sequence of $\sigma$-algebras $({\cal F}_n)$ such that the conditional distributions of f_n and g_n given ${\cal F}_{n-1}$ coincide, and such that the…

We define and study asymptotically symmetric Banach spaces (a.s.) and its variations: weakly a.s. (w.a.s.) and weakly normalized a.s. (w.n.a.s.). If X is a.s. then all spreading models of X are uniformly symmetric. We show that the converse…

Garling sequence spaces admit a renorming with respect to which their standard unit vector basis is 1-greedy. We also discuss some additional properties of these Banach spaces related to uniform convexity and superreflexivity. In…

We prove optimality of the Gagliardo-Nirenberg inequality $$ \|\nabla u\|_{X}\lesssim\|\nabla^2 u\|_Y^{1/2}\|u\|_Z^{1/2}, $$ where $Y, Z$ are rearrangement invariant Banach function spaces and $X=Y^{1/2}Z^{1/2}$ is the…

This article is dedicated to geometric structure of the Lorentz and Marcinkiewicz spaces in case of the pure atomic measure. We study complete criteria for order continuity, the Fatou property, strict monotonicity and strict convexity in…

We consider the problem of sequencing a set of positive numbers. We try to find the optimal sequence to maximize the variance of its partial sums. The optimal sequence is shown to have a beautiful structure. It is interesting to note that…

In this paper, we study the existence of infinite dimensional closed linear subspaces of a rearrangement invariant space on [0,1] every nonzero element of which does not belong to any included rearrangement invariant space of the same class…

Let $G$ be an abelian group, let $S$ be a sequence of terms $s_1,s_2,...,s_{n}\in G$ not all contained in a coset of a proper subgroup of $G$, and let $W$ be a sequence of $n$ consecutive integers. Let $$W\odot S=\{w_1s_1+...+w_ns_n:\;w_i…

Let $m$ be a positive integer larger than $1$, let $w$ be a finite word over $\left\{0,1,...,m-1\right\}$ and let $a_{m;w}(n)$ be the number of occurrences of the word $w$ in the $m$-expansion of $n$ mod $p$ for any non-negative integer…

In this article, we show that Orlicz-Lorentz spaces $\ell^n_{M,a}$, $n\in\mathbb N$ with Orlicz function $M$ and weight sequence $a$ are uniformly isomorphic to subspaces of $L_1$ if the norm $\|\cdot\|_{M,a}$ satisfies certain Hardy-type…

Let $r$ and $n$ be positive integers such that $r<n$, and $\mathbb{K}$ be an arbitrary field. In a recent work, we have determined the maximal dimension for a linear subspace of $n$ by $n$ symmetric matrices with rank less than or equal to…

Let $H$ be a Hilbert space. Using Ball's solution of the "complex plank problem" we prove that the following properties of a sequence $a_n>0$ are equivalent: (1) There is a sequence $x_n \in H$ with $\|x_n\|=a_n$, having 0 as a weak cluster…

We study the subleading structure of asymptotically-flat spacetimes and its relationship to the $w_{1+\infty}$ loop algebra of higher spin charges. We do so using both the Bondi-Sachs and the Newman-Penrose formalism, via a dictionary built…

Orlicz-Lorentz spaces provide a common generalization of Orlicz spaces and Lorentz spaces. They have been studied by many authors, including Masty\l o, Maligranda, and Kami\'nska. In this paper, we consider the problem of comparing the…