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We investigate when the local Lipschitz property of the real-valued function $g(z) = d_Y (f(z),A)$ implies the global Lipschitz property of the mapping $f:X\to Y$ between the metric spaces $(X,d_X)$ and $(Y,d_Y)$. Here, $d_Y(y,A)$ denotes…

Following ideas of Caffarelli and Silvestre in~\cite{CS}, and using recent progress in hyperbolic fillings, we define fractional $p$-Laplacians $(-\Delta_p)^\theta$ with $0<\theta<1$ on any compact, doubling metric measure space…

We prove a new Hamiltonian extension and consequently a fragmentation result in dimension $4$ for the symplectic manifold $\mathbb{D}^{2}\times \mathbb{D}^{2}$. Polterovich and Shelukhin have recently constructed a family of functionals on…

We define functionals generalising the Seiberg-Witten functional on closed $spin^c$ manifolds, involving higher order derivatives of the curvature form and spinor field. We then consider their associated gradient flows and, using a gauge…

This paper deals with generalized Lipschitz spaces $\wedge^k_{\alpha,p,q}(\R)$ in the context of Dunkl harmonic analysis on $\R$, for all real $\alpha$. It also introduces a generalized Dunkl-Lipschitz spaces ${\cal…

Consider a closed manifold $M$ and a time-periodic Tonelli Hamiltonian $H : \mathbb{R}/\mathbb{Z} \times T^*M \to \mathbb{R}$ with flow $\phi_H$. Let $\mathcal{L} \subset T^*M$ be a Lagrangian submanifold Hamiltonianly isotopic to the zero…

Given a family of locally Lipschitz vector fields $X(x)=(X_1(x),\dots,X_m(x))$ on $\mathbb{R}^n$, $m\leq n$, we study functionals depending on $X$. We prove an integral representation for local functionals with respect to $X$ and a result…

We introduce the metric fundamental class for metric spaces that are homeomorphic to compact, non-orientable, smooth manifolds with (possibly empty) boundary. This is an integer rectifiable current that provides an analytic representation…

It is proposed that instead of normal representations one should look at cocycles of group extensions valued in certain groups of unitary operators acting in a Hilbert space (e.g the Fock space of chiral fermions), when dealing with groups…

Let $(W,H,\mu)$ be the classical Wiener space where $H$ is the Cameron-Martin space which consists of the primitives of the elements of $L^2([0,1],\,dt)\otimes \R^d$, we denote by $L^2_a(\mu,H)$ the equivalence classes w.r.t. $dt\times…

We initiate a systematic study of intrinsic dimensional versions of classical functional inequalities which capture refined properties of the underlying objects. We focus on model spaces: Euclidean space, Hamming cube, and manifolds of…

We use sup-convolution to find upper approximations of a bounded $m$-subharmonic function on a compact K\"ahler manifold with nonnegative holomorphic bisectional curvature. As an application, we show the H\"older continuity of solutions to…

In this paper we show that if one writes down the structure equations for the evolution of a curve embedded in an (n)-dimensional Riemannian manifold with constant curvature this leads to a symplectic, a Hamiltonian and an hereditary…

In this paper, we investigate the evolution of certain functionals involving higher powers of a scalar quantity $F$ under Bernard List's extended Ricci flow on a compact Riemannian manifold. By deriving explicit expressions for the time…

For a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function $k\geq 0$ let $\mathbf N=N+\sum_{\alpha \in R} k(\alpha)$. Let $L=-\Delta +V$, $V\geq 0$, be the Dunkl--Schr\"odinger operator on $\mathbb R^N$. Assume that there…

In this paper we study integral estimates of derivatives of conformal mappings $\varphi:\mathbb D\to\Omega$ of the unit disc $\mathbb D\subset\mathbb C$ onto bounded domains $\Omega$ that satisfy the Ahlfors condition. These integral…

The prime number decomposition of a finite dimensional Hilbert space reflects itself in the representations that the space accommodates. The representations appear in conjugate pairs for factorization to two relative prime factors which can…

A classical theorem of Stone and von Neumann says that the Schr\"{o}dinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on…

Let $(M,g)$ be a compact Riemannian surface. Consider a family of $L^2$ normalized Laplace-Beltrami eigenfunctions, written in the semiclassical form $-h_j^2\Delta_g \phi_{h_j} = \phi_{h_j}$, whose eigenvalues satisfy $h h_j^{-1} \in (1, 1…

The main result of this article is that Hitchin representations over real closed field extensions $\mathbb{F}$ of $\mathbb{R}$ correspond precisely to those representations of the fundamental group of a closed surface into…