Let $(\mathcal{X},d,\mu )$ be a metric measure space, let $L$ be a linear operator that has a bounded $H_{\infty }$-functional calculus and satisfies the Davies–Gaffney estimate, let $\Phi$ be a concave function on $(0,\infty )$ of critical lower type $p_{\Phi }^{-}\in (0,1]$, and let $\rho \left(t\right)\equiv t^{-1}/{\Phi }^{-1}\left(t^{-1}\right)$ for all $t\in (0,\infty )$. In this paper, the authors introduce the generalized VMO space ${VMO}_{\rho ,L}\left(\mathcal{X}\right)$ associated with $L$ and establish its characterization via the tent space. As applications, the authors show that $\left({VMO}_{\rho ,L}\right(\mathcal{X}){)}^{\ast }=B_{\Phi ,L^{\ast }}(\mathcal{X})$, where $L^{\ast }$ denotes the adjoint operator of $L$ in $L^2\left(\mathcal{X}\right)$ and $B_{\Phi ,L^{\ast }}\left(\mathcal{X}\right)$ the Banach completion of the Orlicz– Hardy space $H_{\Phi ,L^{\ast }}\left(\mathcal{X}\right)$.

On a Riemannian manifold, its geometric and analytic properties are crossly related with each other, and to study their relationship is an important subject. This paper studies the absence of eigenvalues, focusing on the curvatures near infinity; we first study rotationally symmetric cases, and, after that, investigate further possibilities, considering perturbations of rotationally symmetric metrics.

In a series of works, Geiss, Leclerc, and Schröer defined the cluster algebra structure on the coordinate ring $\mathbb{C}\left[N\right(w\left)\right]$ of the unipotent subgroup, associated with a Weyl group element $w$. And they proved that cluster monomials are contained in Lusztig’s dual semicanonical basis ${\mathcal{S}}^{\ast }$. We give a setup for the quantization of their results and propose a conjecture that relates the quantum cluster algebras in Berenstein and Zelevinsky’s work to the dual canonical basis ${\mathbf{B}}^{up}$. In particular, we prove that the quantum analogue ${\mathcal{O}}_q\left[N\right(w\left)\right]$ of $\mathbb{C}\left[N\right(w\left)\right]$ has the induced basis from ${\mathbf{B}}^{up}$, which contains quantum flag minors and satisfies a factorization property with respect to the “$q$-center” of ${\mathcal{O}}_q\left[N\right(w\left)\right]$. This generalizes Caldero’s results from finite type to an arbitrary symmetrizable Kac–Moody Lie algebra.

We prove that weak amenability of a locally compact group imposes a strong condition on its amenable closed normal subgroups. This extends non–weak amenability results of Haagerup (1988) and Ozawa and Popa (2010). A von Neumann algebra analogue is also obtained.

We give two types of singularities of maps between $4q$-manifolds whose Thom polynomials with integer coefficients have nonvanishing coefficient of Pontrjagin class $P_q$. We show that an element of the $J$-image of dimension $4q-1$ has a fold map between $S^{4q-1}$ and can be detected by the leading terms of Thom polynomials of those singularities of an extended map between $D^{4q}$ of the fold map.

In a paper published in 1959, Shimura presented an elegant calculation of the critical values of $L$-functions attached to elliptic modular forms using the first cohomology group. We will show that a similar calculation is possible for Hilbert modular forms over real quadratic fields using the second cohomology group. We present explicit numerical examples calculated by this method.