泛函分析

We study the existence and characterization of optimal tensor representations of elements in the space $L_1(\mu,Y)$ of Bochner integrable functions. We completely describe the set of norm-attaining elements in two settings. First, when the…

We give a negative answer to a question of Haller-Langemets-Lima-Nadel-Rueda Zoca asking whether, for all Banach spaces $X$ and $Y$, the Daugavet index of thickness satisfies \[ T(X\oplus_1 Y)=\min\{T(X),T(Y)\}. \] We show that this…

We partially answer two open questions concerning diameter two properties in absolute sums. First, we identify the conditions that a super $\Delta$-point in an absolute sum of Banach spaces imposes on the coordinates. Secondly, we show that…

For a quasinilpotent operator $T$ on a Banach space $X$, Douglas and Yang defined $k_{x}=\limsup\limits_{\lambda\rightarrow 0}\frac{\ln\|(\lambda-T)^{-1}x\|}{\ln\|(\lambda-T)^{-1}\|}$ for each non-zero vector $x$, and called…

We show that there is no real or complex indecomposable Banach space with the primary factorisation property (PFP). We relate the PFP of a Banach space $E$ to ring-theoretic infiniteness of $\mathcal{B}(E)$ and of…

The theory of boundary quadruples and boundary triples is well-studied for symmetric and skew-symmetric operators and in general for dual-pairs. This paper adapts a suitable version for abstract Friedrichs operators and addresses the…

We prove that if an $n\times n\ (n > 3)$ companion matrix $A$ with the spectrum $\sigma(A) = \{ a \}$ has a circular numerical range, then $A$ is the Jordan block. This problem can be described by examining zeros of the Laurent polynomial…

Fix $0<r<1$, and let $X_1,X_2,\dots$ be independent symmetric Weibull$(r)$ random variables, that is, \[ \textsf{P}(|X_i|>t)=e^{-t^r},\qquad t\ge 0. \] We prove that there is no constant $C_r$, depending only on $r$, with the following…

We reformulate the bounded-length-distortion condition for maps between metric spaces in a certain relaxed form that requires the presence of a reference measure on the source space, which makes the new approach more natural from the…

A classical result states that the Hardy--Littlewood maximal operator is bounded on an Orlicz space $L^A(\mathbb{R}^n)$ if and only if its conjugate Young function $\tilde{A}$ satisfies the $\Delta_2$-condition. The same condition also…

The goal of this note is twofold. First, we provide explicit examples of periodic (though not necessarily lattice) sets that give rise to Gabor systems failing to form frames. Our constructions depend only on the parity of the window…

In this article, we study bilinear Calder\'on--Zygmund operators on a Vilenkin group $G$. As a preliminary step, we establish a Grafakos--Torres-type endpoint weak-type result in our setting. Furthermore, we prove that such operators extend…

We provide an explicit construction of a Gabor orthonormal bases for a local field $K$ that provides maximal localization in both time and frequency. Such a localization is not true in case of $\mathbb{R}$ due to the uncertainty principle.…

Let $X$ and $Y$ be locally compact Hausdorff spaces. We study order isomorphisms \[ T:C_0^+(X)\to C_0^+(Y), \] where $C_0(X)$ denotes the Banach space of all real-valued continuous functions on $X$ vanishing at infinity, and \[…

This paper is dedicated to Paul Butzer on the occasion of his 85th birthday. His work and example have strongly influenced not only the first author, but also generations of mathematicians working in approximation theory and Fourier…

Generalized Fofana spaces were recently introduced as generalizations of Fofana spaces and Nakai's generalized Morrey spaces. In this paper, we establish the boundedness properties of the following operators in these spaces: fractional…

We consider the space $C^1[0, 1]$ of continuously differentiable functions on the closed unit interval $[0, 1]$ and the space $\operatorname{Lip}[0, 1]$ of Lipschitz continuous functions on $[0, 1]$, equipped with the norms \begin{align*}…

Given two Banach spaces $X$ and $E$, one can associate a numerical invariant $\mathcal{CR}(X, E)$, called the coarse embeddability ratio, which provides a criterion for coarse and uniform embeddability. We compute the coarse embeddability…

We establish a fundamental extension of Mercer's celebrated theorem by introducing a class of higher-order kernel operators acting on Sobolev spaces $H^k(\Theta)$, where $\Theta \subset \mathbb{R}^d$ is a bounded domain and…

Let $ G $ be a compact metrizable Abelian group, $ L^{1}(G) $ its group algebra and $ M(G) $ its measure algebra. For each proper subset $ E $ of the dual group $ \hat{G} $, let $ L^{1}_{E}(G)=\{f\in L^{1}(G):\hat{f}=0 \text{ on }…