We build a locally convex algebra of real analytic functions defined in a strip of the Poincaré half-plane in which a class of periodic hyperfunctions on the real line is topologically embedded. This is accomplished via a harmonic regularization method. In this algebra, we can give a sense to differential problems involving products of hyperfunctions which are a priori not defined in the classical setting. Some examples and an application are given.

For an infinite-dimensional Lie group $G$ modeled on a locally convex Lie algebra $\mathfrak{g}$, we prove that every smooth projective unitary representation of $G$ corresponds to a smooth linear unitary representation of a Lie group extension $G^{♯}$ of $G$. (The main point is the smooth structure on $G^{♯}$.) For infinite-dimensional Lie groups $G$ which are $1$-connected, regular, and modeled on a barreled Lie algebra $\mathfrak{g}$, we characterize the unitary $\mathfrak{g}$-representations which integrate to $G$. Combining these results, we give a precise formulation of the correspondence between smooth projective unitary representations of $G$, smooth linear unitary representations of $G^{♯}$, and the appropriate unitary representations of its Lie algebra ${\mathfrak{g}}^{♯}$.

Consider a finitely generated group $G$ that is relatively hyperbolic with respect to a family of subgroups $H_1,\dots ,H_n$. We present an axiomatic approach to the problem of extending metric properties from the subgroups $H_i$ to the full group $G$. We use this to show that both (weak) finite decomposition complexity and straight finite decomposition complexity are extendable properties. We also discuss the equivalence of two notions of straight finite decomposition complexity.

We present two analytic applications of the fact that a hyperbolic group can be endowed with a strongly hyperbolic metric. The first application concerns the crossed product $C^{\ast }$-algebra defined by the action of a hyperbolic group on its boundary. We construct a natural time flow, involving the Busemann cocycle on the boundary. This flow has a natural KMS state, coming from the Hausdorff measure on the boundary, which is furthermore unique when the group is torsion-free. The second application is a short new proof of the fact that a hyperbolic group admits a proper isometric action on an ${\ell }^p$-space for large enough $p$.

We consider an iterative procedure for constructing maximal abelian ${}^{\ast }$-subalgebras (MASAs) satisfying prescribed properties in II${}_1$ factors. This method pairs well with the intertwining by bimodules technique and with properties of the MASA and of the ambient factor that can be described locally. We obtain such a local characterization for II${}_1$ factors $M$ that have an s-MASA, $A\subset M$ (i.e., for which $A\vee JAJ$ is maximal abelian in $\mathcal{B}\left(L^{2}M\right)$), and use this strategy to prove that any factor in this class has uncountably many nonintertwinable singular (resp., semiregular) s-MASAs.

We show relationships between uniform K-stability and plt blowups of log Fano pairs. We see that it is enough to evaluate certain invariants defined by volume functions for all plt blowups in order to test the uniform K-stability of log Fano pairs. We also discuss the uniform K-stability of two log Fano pairs under crepant finite covers. Moreover, we give another proof of the K-semistability of the projective plane.

Based on the theory of variable exponents and atomic decomposition, we study the boundedness of $n$-dimensional fractional Hardy operators on variable Herz and Herz-type Hardy spaces, where the three main indices are variable exponents. The corresponding boundedness for the $m$th order commutators generated by the $n$-dimensional fractional Hardy operators and bounded mean oscillation (BMO) function are also considered. We note that, even in the special case of $m=1$, the obtained results are also new.

We study necessary conditions for compactness of the weighted $\overline{\partial }$-Neumann operator on the space $L^2({\mathbb{C}}^n,e^{-\phi })$ for a plurisubharmonic function $\phi$. Under the assumption that the corresponding weighted Bergman space of entire functions has infinite dimension, a weaker result is obtained by simpler methods. Moreover, we investigate (non)compactness of the $\overline{\partial }$-Neumann operator for decoupled weights, which are of the form $\phi (z)={\phi }_1(z_1)+\cdots +{\phi }_n(z_n)$. More can be said if every $\Delta {\phi }_j$ defines a nontrivial doubling measure.

We prove Angehrn–Siu type effective freeness and effective point separation for quasi-log canonical pairs. As a natural consequence, we obtain that these two results hold for semi-log canonical pairs. One of the main ingredients of our proof is inversion of adjunction for quasi-log canonical pairs, which is established in this paper.

We study a notion of cusp forms for the symmetric spaces $G/H$ with $G=\mathrm{SL}(n,\mathbb{R})$ and $H=\mathrm{S}(\mathrm{GL}(n-1,\mathbb{R})\times \mathrm{GL}(1,\mathbb{R}))$. We classify all minimal parabolic subgroups of $G$ for which the associated cuspidal integrals are convergent and discuss the possible definitions of cusp forms. Finally, we show that the closure of the direct sum of the discrete series representations of $G/H$ coincides with the space of cusp forms.