Related papers: Isometric classification of norms in rearrangement…

Using a nonlinear version of the well known Hardy-Littlewood inequalities, we derive new formulas for decreasing rearrangements of functions and sequences in the context of convex functions. We use these formulas for deducing several…

Let M and N be Orlicz functions. We establish some combinatorial inequalities and show that the product spaces l^n_M(l^n_N) are uniformly isomorphic to subspaces of L_1 if M and N are "separated" by a function t^r, 1<r<2.

In this article, we investigate the existence of closed vector subspaces (i.e.spaceability) in various nonlinear subsets of Orlicz-Lorentz spaces $\Lambda_{\varphi,w}$, equipped with the Luxemburg norm. If a family of Orlicz functions…

In this paper, we study the noncommutative Orlicz space $L_{\varphi}(\widetilde{\mathcal{M}},\tau)$, which generalizes the concept of noncommutative $L^{p}$ space, where $\mathcal{M}$ is a von Neumann algebra, and $\varphi$ is an Orlicz…

The rearrangement inequalities of Hardy-Littlewood and Riesz say that certain integrals involving products of two or three functions increase under symmetric decreasing rearrangement. It is known that these inequalities extend to integrands…

We generalize the celebrated isoperimetric inequality of Khot, Minzer, and Safra~(SICOMP 2018) for Boolean functions to the case of real-valued functions $f \colon \{0,1\}^d\to\mathbb{R}$. Our main tool in the proof of the generalized…

Motivated by the result of Dantas et. al. (2023) that there exist metric spaces for which the set of strongly norm-attaining Lipschitz functions does not contain an isometric copy of $c_0$, we introduce and study a weaker notion of…

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…

In the paper, it is given isomorphic classification of $F$-spaces of $log$-integrable measurable functions constructed using different measure spaces. At the same time, it is proved that such spaces are non-isometric.

We provide necessary and sufficient conditions for the coincidence, up to equivalence of the norms, between strong and weak Orlicz spaces. Roughly speaking, this coincidence holds true only for the so-called exponential spaces. We find also…

Poincar\'{e}-Sobolev-type inequalities involving rearrangement-invariant norms on the entire $\mathbb{R}^n$ are provided. Namely, inequalities of the type $\|u-P\|_{Y(\mathbb{R}^n)}\leq C\|\nabla^m u\|_{X(\mathbb{R}^n)}$, where $X$ and $Y$…

Let $X$ be a rearrangement-invariant space. An operator $T: X\to X$ is called narrow if for each measurable set $A$ and each $\epsilon > 0$ there exists $x \in X$ with $x^2= \chi_A, \int x d \mu = 0$ and $\| Tx \| < \epsilon$. In particular…

In this note we prove that $\frac{1}{n!} \sum_{\pi} (\sum_{i=1}^n |x_i a_{i,\pi(i)} |^2)^{1/2}$ is equivalent to a Musielak-Orlicz norm $\norm{x}_{\sum M_i}$. We also obtain the inverse result, i.e., given the Orlicz functions, we provide a…

A unified approach to embedding theorems for Sobolev type spaces of vector-valued functions, defined via their symmetric gradient, is proposed. The Sobolev spaces in question are built upon general rearrangement-invariant norms. Optimal…

Rearrangement-invariance in function spaces can be viewed as a kind of generalization of 1-symmetry for Schauder bases. We define subrearrangement-invariance in function spaces as an analogous generalization of 1-subsymmetry. It is then…

Let $\msp$ be a measure space and let $1 < p < \infty$. The {\em weak $L^p$}\/ space $\wlp$ consists of all measurable functions $f$ such that \[ \|f\| = \sup_{t>0}t^{\frac{1}{p}}f^*(t) < \infty,\] where $f^*$ is the decreasing…

Let $X$ be a possibly non-reduced space of pure dimension. We introduce an essentially intrinsic pointwise Hermitian norm on smooth $(0,*)$-forms, in particular on holomorphic functions, on $X$. We prove that the space of holomorphic…

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 $X$…

In this paper, we present a characterization of support functionals and smooth points in $L_{0}^{\Phi}$, the Musielak-Orlicz space equipped with the Orlicz norm. As a result, criterion for the smoothness of $L_{0}^{\Phi}$ is also obtained.…

{\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…