Let $\mathsf{G}$ be a locally compact group satisfying some technical requirements and $\widehat{\sf{G}}$ its unitary dual. Using the theory of twisted crossed product $C^{*}$-algebras, we develop a twisted global quantization for symbols defined on $\mathsf{G}\times\widehat{\sf{G}}$ and taking operator values. The emphasis is on the representation-theoretic aspect. For nilpotent Lie groups, the connection is made with a scalar quantization of the cotangent bundle $T^{*}(\mathsf{G})$ and with a Quantum Mechanical theory of observables in the presence of variable magnetic fields.

The main purpose of this paper is to provide new characterizations of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type with the “reverse” doubling property. To achieve our goal, the key idea is to prove a $T1$ theorem with only half the usual smoothness and cancellation conditions.

The fixed-point index of a homeomorphism of Jordan curves measures the number of fixed-points, with multiplicity, of the extension of the homeomorphism to the full Jordan domains in question. The now-classical Circle Index Lemma says that the fixed-point index of a positive-orientation-preserving homeomorphism of round circles is always non-negative. We begin by proving a generalization of this lemma, to accommodate Jordan curves bounding domains which do not disconnect each other. We then apply this generalization to give a new proof of Schramm’s Incompatibility Theorem, which was used by Schramm to give the first proof of the rigidity of circle packings filling the complex and hyperbolic planes. As an example application, we include outlines of proofs of these circle packing theorems.

We then introduce a new tool, the so-called torus parametrization, for working with fixed-point index, which allows some problems concerning this quantity to be approached combinatorially. We apply torus parametrization to give the first purely topological proof of the following lemma: given two positively oriented Jordan curves, one may essentially prescribe the images of three points of one of the curves in the other, and obtain an orientation-preserving homeomorphism between the curves, having non-negative fixed-point index, which respects this prescription. This lemma is essential to our proof of the Incompatibility Theorem.

For transversely homogeneous foliations on compact manifolds whose global holonomy group has connected closure, it is shown that either all holonomy covers of the leaves have polynomial growth with degree bounded by a common constant, or all holonomy covers of the leaves have exponential growth. This is an extension of a recent answer given by Breuillard and Gelander to a question of Carrière. Examples of transversely projective foliations satisfying the above condition were constructed by Chihi and ben Ramdane.

On a smooth, non-compact, complete, boundaryless, connected Riemannian manifold $(M,g)$, there are three kinds of objects that have been studied extensively:

$\bullet $ Viscosity solutions to the Hamilton–Jacobi equation determined by the Riemannian metric;

$\bullet $ Ends introduced by Freudenthal and more general other remainders from compactification theory;

$\bullet $ Various kinds of ideal boundaries introduced by Gromov.

In this paper, we will present some initial relationship among these three kinds of objects and some related topics are also considered.

In this article, by following the strategies in dealing with supercritical cubic and quintic wave equations in (J. Eur. Math. Soc. (JEMS) 16 (2014) 1–30) and (J. Math. Pures Appl. (9) 105 (2016) 342–366), we obtain that, the equation \begin{equation*}(\partial^{2}_{t}-\Delta)u+|u|^{p-1}u=0,\quad3<p<5\end{equation*} is almost surely global well-posed with initial data $(u(0),\partial_{t}u(0))\in H^{s}(\mathbb{T}^{3})\times H^{s-1}(\mathbb{T}^{3})$ for any $s\in(\frac{p-3}{p-1},1)$. The key point here is that $\frac{p-3}{p-1}$ is much smaller than the critical index $\frac{3}{2}-\frac{2}{p-1}$ for $3<p<5$.

We let the central Fourier algebra, $\operatorname{ZA}(G)$, be the subalgebra of functions $u$ in the Fourier algebra $\mathrm{A}(G)$ of a compact group, for which $u(xyx^{-1})=u(y)$ for all $x$, $y$ in $G$. We show that this algebra admits bounded point derivations whenever $G$ contains a non-Abelian closed connected subgroup. Conversely when $G$ is virtually Abelian, then $\operatorname{ZA}(G)$ is amenable. Furthermore, for virtually Abelian $G$, we establish which closed ideals admit bounded approximate identities. We also show that $\operatorname{ZA}(G)$ is weakly amenable, in fact hyper-Tauberian, exactly when $G$ admits no non-Abelian connected subgroup. We also study the amenability constant of $\operatorname{ZA}(G)$ for finite $G$ and exhibit totally disconnected groups $G$ for which $\operatorname{ZA}(G)$ is non-amenable. In passing, we establish some properties related to spectral synthesis of subsets of the spectrum of $\operatorname{ZA}(G)$.

The aim of this paper is to prove that a compact almost Ricci soliton with null Cotton tensor is isometric to a standard sphere provided one of the following conditions associated to the Schouten tensor holds: the second symmetric function is constant and positive; two consecutive symmetric functions are non null multiple or some symmetric function is constant and the quoted tensor is positive.

Take $(R,\mathfrak{m})$ any normal Noetherian domain, either local or $\mathbb{N}$-graded over a field. We study the question of when $R$ satisfies the uniform symbolic topology property (USTP) of Huneke, Katz, and Validashti: namely, that there exists an integer $D>0$ such that for all prime ideals $P\subseteq R$, the symbolic power $P^{(Da)}\subseteq P^{a}$ for all $a>0$. Reinterpreting results of Lipman, we deduce that when $R$ is a two-dimensional rational singularity, then it satisfies the USTP. Emphasizing the non-regular setting, we produce explicit, effective multipliers $D$, working in two classes of surface singularities in equal characteristic over an algebraically closed field, using: (1) the volume of a parallelogram in $\mathbb{R}^{2}$ when $R$ is the coordinate ring of a simplicial toric surface; or (2) known invariants of du Val isolated singularities in characteristic zero due to Lipman.

We find necessary and sufficient conditions for a function to be equal almost everywhere to a function of bounded variation. These results can be applied to broaden the class of general monotone functions.

We establish a connection between compactness of Hankel operators and geometry of the underlying domain through compactness multipliers for the $\overline{\partial}$-Neumann operator. In particular, we prove that any compactness multiplier induces a compact Hankel operator. We also generalize the notion of compactness multipliers to vector fields and matrices and then we use this generalization to generate compact Hankel operators.

In this paper, we generalize the Malliavin calculus for jump processes in the infinite-dimensional setting and obtain an integration by parts formula for jump processes on Hilbert spaces. By using this formula, we investigate derivative formula and exponential convergence for SPDEs driven by purely jump processes.

We study the Wiener–Ikehara theorem under the so-called log-linearly slowly decreasing condition. Moreover, we clarify the connection between two different hypotheses on the Laplace transform occurring in exact forms of the Wiener–Ikehara theorem, that is, in “if and only if” versions of this theorem.