Related papers: On the Convergence of Optimal Measures

We establish a new global endpoint Sobolev inequality for measures that extends the classical theorem of Meyers-Ziemer by placing a maximal function on the right-hand side. This result has several significant consequences. It extends…

We extend almost everywhere convergence in Wiener-Wintner ergodic theorem for $\sigma$-finite measure to a generally stronger almost uniform convergence and present a larger, universal, space for which this convergence holds. We then extend…

For a large class of irreducible shift spaces $X\subset\tA^{\Z^d}$, with $\tA$ a finite alphabet, and for absolutely summable potentials $\Phi$, we prove that equilibrium measures for $\Phi$ are weak Gibbs measures. In particular, for…

Let $(X, \omega)$ be a compact K\"ahler manifold of complex dimension n and $\theta$ be a smooth closed real $(1,1)$-form on $X$ such that its cohomology class $\{ \theta \}\in H^{1,1}(X, \mathbb{R})$ is pseudoeffective. Let $\varphi$ be a…

We provide a new proof of a result of X.X.Chen and G.Tian : for a polarized extremal K\"ahler manifold, an extremal metric attains the minimum of the modified K-energy. The proof uses an idea of C.Li adapted to the extremal metrics using…

We discuss smooth metric measure spaces admitting two weighted Einstein representatives of the same weighted conformal class. First, we describe the local geometries of such manifolds in terms of certain Einstein and quasi-Einstein warped…

We prove that a metric measure space $(X,d,m)$ satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space $W^{1,2}$ is Hilbert is rectifiable. That is, a $RCD^*(K,N)$-space is rectifiable, and in particular for…

In this article, we establish weighted strong and weak type inequalities for non-commutative square functions that naturally arise in the analysis of differences between ball averages and martingale sequences within the framework of group…

This paper starts with a brief review of the topic of strong and weak pre- and post-selected (PPS) quantum measurements, as well as weak values, and afterwards presents original work. In particular, we develop a nonperturbative theory of…

Consider a homeomorphism $f$ defined on a compact metric space $X$ and a continuous map $\phi\colon X \to \mathbb{R}$. We provide an abstract criterion, called \emph{control at any scale with a long sparse tail} for a point $x\in X$ and the…

This work will incorporate a few related tools for addressing the conceptual difficulties arising from sewing together classical and quantum mechanics: deterministic operators, weak measurements and post-selection. Weak Measurement, based…

One says that a Riemannian four-manifold is \emph{weakly Einstein} if the three-index contraction of its curvature tensor against itself equals a function times the metric. Since this includes all four-manifolds that are Einstein, or…

Let $G$ be a connected compact group equipped with the normalised Haar measure $\mu$. Our first result shows that given $\alpha, \beta>0$, there is a constant $c = c(\alpha,\beta)>0$ such that for any compact sets $A,B\subseteq G$ with $…

We establish the existence and fundamental properties of the equilibrium measure in uniformly quasiregular dynamics. We show that a uniformly quasiregular endomorphism $f$ of degree at least 2 on a closed Riemannian manifold admits an…

A property $\Pi$ on a finite set $U$ is \emph{monotone} if for every $X \subseteq U$ satisfying $\Pi$, every superset $Y \subseteq U$ of $X$ also satisfies $\Pi$. Many combinatorial properties can be seen as monotone properties. The problem…

The study deals with a minimal energy problem in the presence of an external field over noncompact classes of vector measures of infinite dimension in a locally compact space. The components are positive measures (charges) satisfying…

We show that the set of max-plus measures representing a given max-plus harmonic vector has a least element. This may be viewed as an analogue of the uniqueness of the integral representation of harmonic functions in Potential Theory. As an…

For $0<p<\infty$, $\Psi:[0,\infty)\to(0,\infty)$ and a finite positive Borel measure $\mu$ on the unit disc $\mathbb{D}$, the Lebesgue--Zygmund space $L^p_{\mu,\Psi}$ consists of all measurable functions $f$ such that $\lVert f…

An approach to complex interpolation of compact subsets of $\Bbb C^n$, including Brunn-Minkowski type inequalities for the capacities of the interpolating sets, was developed recently by means of plurisubharmonic geodesics between relative…

We prove upper pointwise estimates for the Bergman kernel of the weighted Fock space of entire functions in $L^2(e^{-2\phi})$ where $\phi$ is a subharmonic function with $\Delta \phi$ a doubling measure. We derive estimates for the…