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A function space, $L^{\theta,\infty)}(\Omega)$, $0 \leq \theta <\infty$, is defined. It is proved that $L^{\theta,\infty)}(\Omega)$ is a Banach space which is a generalization of exponential class. An alternative definition of…

In this work the authors use their contour integral method to derive a double integral connected to the modified Bessel function of the second kind and express it in terms of the Lerch function. There are some useful results relating double…

We introduce and characterize, on the Banach lattice valued continuous function space, multiplication operators generating strongly continuous multiplication operator semigroups. Our characterization is the generalization of known results…

An algebra of bounded linear operators on a Banach space is said to be {\em strongly compact} if its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Banach space is said to be {\em strongly…

Let $\mu$ be a positive finite Borel measure on $[0,1).$ Ces\`aro-type operators $C_{\mu}$ when acting on weighted spaces of holomorphic functions are investigated. In the case of bounded holomorphic functions on the unit disc we prove that…

Given two metric measure spaces $X$ and $Y$. Let $f:X\to Y$ be a measurable mapping and $A\subset X$. The Banach indicatrix (multiplicity function) is defined as $N(y,f,A) = \#\{x\in A \mid f(x) = y\}$. We prove measurability of this…

The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency domain. This description preserves more explicitly the symmetry between time and frequency…

In contrast to the usual Lipschitz seminorms associated to ordinary metrics on compact spaces, we show by examples that Lipschitz seminorms on possibly non-commutative compact spaces are usually not determined by the restriction of the…

We study Birkhoff-James orthogonality of compact (bounded) linear operators between Hilbert spaces and Banach spaces. Applying the notion of semi-inner-products in normed linear spaces and some related geometric ideas, we generalize and…

The classical Korenblum-Roberts Theorem characterizes the cyclic singular inner functions in the Bergman spaces of the unit disk $\mathbb{D}$ as those for which the corresponding singular measure vanishes on Beurling-Carleson sets of…

We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For $n\ge 2$ and $1<p<\infty$, it is shown that $\ell_\infty^n$ is representable in a Banach space $X$ if and only if it…

We obtain an explicit characterization of the $K$-functional of a pair of weighted classical Lorentz spaces of type $S$. We develop a method for obtaining such characterization based on a relation between the desired quantity and the…

We consider integration of functions with values in a partially ordered vector space, and two notions of extension of the space of integrable functions. Applying both extensions to the space of real valued simple functions on a measure…

Uniform integer-valued Lipschitz functions on a domain of size $N$ of the triangular lattice are shown to have variations of order $\sqrt{\log N}$. The level lines of such functions form a loop $O(2)$ model on the edges of the hexagonal…

In this paper we introduce and study a new class of operators related to norm bounded sets on Banach Lattice and which brings together several classical classes of operators (as o-weakly compact operators, b-weakly compact operators,…

For a fixed $K\gg 1$ and $n\in\mathbb{N}$, $n\gg 1$, we study metric spaces which admit embeddings with distortion $\le K$ into each $n$-dimensional Banach space. Classical examples include spaces embeddable into $\log n$-dimensional…

For a metric space $(K,d)$ the Banach space $\Lip(K)$ consists of all scalar-valued bounded Lipschitz functions on $K$ with the norm $\|f\|_{L}=\max(\|f\|_{\infty},L(f))$, where $L(f)$ is the Lipschitz constant of $f$. The closed subspace…

One of the goals of this article is to define a an unified setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure. We first remark that some…

Let $X$ be a rearrangement-invariant space over a non-atomic $\sigma$-finite measure space $(\mathscr{R},\mu)$ and let $\alpha\in(0,\infty)$. We define the functional \begin{equation*} \|f\|_{X^{\langle \alpha \rangle}} =…

A Lipschitz space is defined in the Ornstein-Uhlenbeck setting, by means of a bound for the gradient of the Ornstein-Uhlenbeck Poisson integral. This space is then characterized with a Lipschitz-type continuity condition. These functions…