Related papers: Continuous Rankin Bound for Hilbert and Banach Spa…

Let $(\Omega, \mu)$ be a measure space and $\{\tau_\alpha\}_{\alpha\in \Omega}$ be a normalized continuous Bessel family for a finite dimensional Hilbert space $\mathcal{H}$ of dimension $d$. If the diagonal $\Delta\coloneqq \{(\alpha,…

Let $(\Omega, \mu)$, $(\Delta, \nu)$ be measure spaces and $\{\tau_\alpha\}_{\alpha\in \Omega}$, $\{\omega_\beta\}_{\beta \in \Delta}$ be 1-bounded continuous Parseval frames for a Hilbert space $\mathcal{H}$. Then we show that…

Let $(\Omega, \mu)$, $(\Delta, \nu)$ be measure spaces and $p=1$ or $p=\infty$. Let $(\{f_\alpha\}_{\alpha\in \Omega}, \{\tau_\alpha\}_{\alpha\in \Omega})$ and $(\{g_\beta\}_{\beta\in \Delta}, \{\omega_\beta\}_{\beta\in \Delta})$ be…

Let $(\Omega, \mu)$, $(\Delta, \nu)$ be measure spaces. Let $(\{f_\alpha\}_{\alpha\in \Omega}, \{\tau_\alpha\}_{\alpha\in \Omega})$ and $(\{g_\beta\}_{\beta\in \Delta}, \{\omega_\beta\}_{\beta\in \Delta})$ be continuous p-Schauder frames…

Suppose $\alpha$ is a rotationally symmetric norm on $L^{\infty}\left(\mathbb{T}\right) $ and $\beta$ is a "nice" norm on $L^{\infty}\left(\Omega,\mu \right) $ where $\mu$ is a $\sigma$-finite measure on $\Omega$. We prove a version of…

We prove that among all doubly connected and elastically supported planar membranes $\Omega$ with prescribed values of the area $|\Omega|$ and the lengths of the inner and outer boundaries $|\partial \Omega_{\rm{in}}|_1$, $|\partial…

Wang and Ye conjectured in [22]: Let $\Omega$ be a regular, bounded and convex domain in $\mathbb{R}^{2}$. There exists a finite constant $C({\Omega})>0$ such that \[ \int_{\Omega}e^{\frac{4\pi u^{2}}{H_{d}(u)}}dxdy\le C(\Omega),\;\;\forall…

Let $(\mathbb{X} , d, \mu )$ be a proper metric measure space and let $\Omega \subset \mathbb{X}$ be a bounded domain. For each $x\in \Omega$, we choose a radius $0< \varrho (x) \leq \mathrm{dist}(x, \partial \Omega ) $ and let $B_x$ be the…

Let $\Omega$ be a circular domain, that is, an open disk with finitely many closed disjoint disks removed. Denote by $H^\infty(\Omega)$ the Banach algebra of all bounded holomorphic functions on $\Omega$, with pointwise operations and the…

Motivated by the work of Baronti et al. [J. Math. Anal. Appl. 252(2000) 124-146], where they defined the supremum of an arithmetic mean of the side lengths of a triangle, summing antipodal points on the unit sphere, we introduce a new…

This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert…

A distortion lower bound of $\Omega(\log(h)^{1/p})$ is proven for embedding the complete countably branching hyperbolic tree of height $h$ into a Banach space admitting an equivalent norm satisfying property $(\beta)$ of Rolewicz with…

The aim of this article is to study effective Reifenberg theorems for measures in a Hilbert or Banach space. For Hilbert spaces, we see all the results from $\mathbb{R}^n$ continue to hold with no additional restrictions. For a general…

Let $(E,\|.\|)$ be a Banach space and let $(\Omega,\mu)$ be a Lebesgue measure space. We characterize, for all $p>0$, measurable functions $u:\Omega\rightarrow \mathbb{R}$ for which \begin{equation*} \left\| \int_{\Omega} f\,d\mu…

This paper systematically investigates a new geometric constant associated with isosceles orthogonality in Banach spaces. By establishing the connection between this new constant and a classical function, sharp upper and lower bounds for…

In this work, we revisit the following estimate due to Dahlberg \cite{Dahl}. Let $\textit{\textbf x}_0$ a fixed point in a bounded Lipschitz domain $\Omega$. Then there exists a constant $C > 0$ such that if $u$ is a harmonic function in…

Let $\Omega\subset \mathbb{C}$ be an arbitrary domain in the one-dimensional complex plane equipped with a positive Radon measure $\mu$. For any $1\le p< \infty$, it is shown that the weighted Bergman space $A^p(\Omega, \mu)$ of holomorphic…

We consider H\"older continuous weak solutions $u\in C^\gamma(\Omega)$, $u\cdot n|_{\partial \Omega}=0$, of the incompressible Euler equations on a bounded and simply connected domain $\Omega\subset\mathbb{R}^d$. If $\Omega$ is of class…

Let $X$ be a Banach space, let $(\Omega,\mu)$ be a $\sigma$-finite measure space and let $A,B\colon\Omega\to B(X)$ be strongly measurable $\gamma$-bounded functions. We show that for all $x\in X$ and all $x^*\in X^*$, there exist a Hilbert…

We consider bipartite quantum systems characterized by a continuous angular variable \theta \in [-\pi, \pi[, representing, for instance, the position of a particle on a circle. We show how to reveal non-locality on this type of system using…