Related papers: Multi-normed spaces, based on non-discrete measure…

In 2011 Lutwak, Yang and Zhang extended the definition of the $L_p$-Minkowski convex combination ($p \geq 1$) introduced by Firey in the 1960s from convex bodies containing the origin in their interiors to all measurable subsets in…

We introduce the notion of a regular mapping on a non-commutative $L_p$-space associated to a hyperfinite von Neumann algebra for $1\le p\le \infty$. This is a non-commutative generalization of the notion of regular or order bounded map on…

Let K \subset L be a field extension. Given K-subspaces A,B of L, we study the subspace spanned by the product set AB = {ab | a \in A, b \in B}. We obtain some lower bounds on the dimension of this subspace and on dim B^n in terms of dim A,…

A new matrix product, called the semi-tensor product (STP), is briefly reviewed. The STP extends the classical matrix product to two arbitrary matrices. Under STP the set of matrices becomes a monoid (semi-group with identity). Some related…

The aim of this paper is to extend the notion of Apq space from its historical context in the work of Herz and to recognise such spaces as preduals of spaces of intertwining operators of induced representations as suggested by the work of…

The notion of $L^p$-distributions is introduced on Riemannian symmetric spaces of noncompact type and their main properties are established. We use a geometric description for the topology of the space of test functions in terms of the…

In a paper published in 2020 in Studia Mathematica, Abrahamsen et al. proved that in the real space $L_1(\mu)$, where $\mu$ is a non-zero $\sigma$-finite (countably additive non-negative) measure, norm-one elements in finite convex…

We provide lower bounds for the norms of embeddings between $\boldsymbol{\gamma}$-weighted Anchored and ANOVA spaces of $s$-variate functions with mixed partial derivatives of order one bounded in $L_p$ norm ($p\in[1,\infty]$). In…

Let $p>n$ and let $L^1_p(R^n)$ be a homogeneous Sobolev space. For an arbitrary Borel measure $\mu$ on $R^n$ we give a constructive characterization of the space $L^1_p(R^n)+L_p(R^n;\mu)$. We express the norm in this space in terms of…

In a previous paper, the authors introduced several vector space norms on the space of algebraic numbers modulo torsion which corresponded to the Mahler measure on a certain class of numbers and allowed the authors to formulate L^p Lehmer…

For $a,b>0$ with $a\neq b$, we define M_{p}=M^{1/p}(a^{p},b^{p})\text{if}p\neq 0 \text{and} M_{0}=\sqrt{ab}, where $M=A,He,L,I,P,T,N,Z$ and $Y$ stand for the arithmetic mean, Heronian mean, logarithmic mean, identric (exponential) mean, the…

Let $p, q \in (0, \infty]$ and $\ell_p^m(\ell_q^n)$ be the mixed-norm sequence space of real matrices $x = (x_{i, j})_{i \leq m, j \leq n}$ endowed with the (quasi-)norm $\Vert x \Vert_{p, q} := \big\Vert \big( \Vert (x_{i, j})_{j \leq n}…

Isomorphic classification of symmetric spaces is an important problem related to the study of symmetric structures in arbitrary Banach spaces. This research was initiated in the seminal work of Johnson, Maurey, Schechtman and Tzafriri…

Let $\vec{p}\in(0,\,\infty)^n$, $A$ be an expansive dilation on $\mathbb{R}^n$,and $H^{\vec{p}}_A({\mathbb {R}}^n)$ be the anisotropic mixed-norm Hardy space defined via the non-tangential grand maximal function studied by \cite{hlyy20}. In…

There are several characterizations of coarse embeddability of a discrete metric space into a Hilbert space. In this note we give such characterizations for general metric spaces. By applying these results to the spaces $L_p(\mu)$, we get…

We find large classes of injective and projective $p$-multinormed spaces. In fact, these classes are universal, in the sense that every $p$-multinormed space embeds into (is a quotient of) an injective (resp. projective) $p$-multinormed…

Lecture notes as per the title. In the first part, the concepts of a measurable space, measurable maps between measurable spaces and that of a measure on a measurable space are introduced, after which the fundamentals of the theory of…

We study mixed unitary channels generated by finite subgroups of the group of all unitary operators in a Hilbert space. Based on the majorization theory we introduce techniques allowing to calculate different characteristics of output…

We study integration and $L_2$-approximation on countable tensor products of function spaces of increasing smoothness. We obtain upper and lower bounds for the minimal errors, which are sharp in many cases including, e.g., Korobov, Walsh,…

Let $1\le p\le q<\infty$ and let $X$ be a $p$-convex Banach function space over a $\sigma$-finite measure $\mu$. We combine the structure of the spaces $L^p(\mu)$ and $L^q(\xi)$ for constructing the new space $S_{X_p}^{\,q}(\xi)$, where…