We prove theorems on boundedness of operators weak type $(\varphi_0, \psi_0, \varphi_1, \psi_1)$ from Lorentz space $\Lambda_{\varphi,a}(\mathbb{R}^n)$ to $\Lambda_{\varphi,b}(\mathbb{R}^n)$ in “limit” cases, when one functions $\varphi(t) / \varphi_0(t)$, \varphi_1(t)$ slowly changes at zero and infinity.

Abstract The Jánossy density for a determinantal point process is the probability that an interval $I$ contains exactly $p$ points except those at $k$ designated loci. associated with integrable kernel $\mathbf{K}\doteq (\varphi(x)\psi(y)-\psi(x)\varphi(y))/(x-y)$ shown to be expressed as Fredholm determinant $\mathrm{Det}(\mathbb{I}-\tilde{\mathbf{K}}|_I)$ of transformed $\tilde{\mathbf{K}}\do...

In this paper, we will define and study amenability of Hopf C-algebras as well as Fourier duality. 1991 AMS Mathematics Classification number: 46L55, 46L05

By studying the so-called traveling salesman groups, we obtain a new metric criterion for non-amenability. As an application, we give a new and very short proof of non-amenability of free Burnside groups with sufficiently big odd exponent. Amenable groups have been introduced by John von Neumann in 1929 in connection with Banach-Tarski Paradox, although earlier, Banach himself had understood th...