Related papers: Riesz $s$-equilibrium measures on $d$-rectifiable …

We construct an example of a purely unrectifiable measure $\mu$ in $\mathbb{R}^d$ for which the singular integrals associated to the kernels $\displaystyle{K(x)=\frac{P_{2k+1}(x)}{|x|^{2k+d}}}$, with $k\geq 1$ and $P_{2k+1}$ a homogeneous…

In this paper we establish a Besicovitch-Federer type projection theorem for general measures. Specifically, let $\mu$ be a finite Borel measure on $\mathbb{R}^n$ and let $0 < m < n$ be an integer. We show that, under the sole assumption…

A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures $\mu$ in $n$-dimensional Euclidean space for all $n\geq 2$ in terms of…

We construct a probability measure $\mu$ supported on a set of zero $2d/p$-Hausdorff measure such that $\hat{\mu}\in L_{p}(\mathbb{R}^d)$.

Given a probability measure space $(X,\Sigma,\mu)$, it is well known that the Riesz space $L^0(\mu)$ of equivalence classes of measurable functions $f: X \to \mathbf{R}$ is universally complete and the constant function $\mathbf{1}$ is a…

In view of a recent example of a positive Radon measure $\mu$ on a domain $D\subset\mathbb R^n$, $n\geqslant3$, such that $\mu$ is of finite energy $E_g(\mu)$ relative to the $\alpha$-Green kernel $g$ on $D$, though the energy of…

The halfspace depth of a $d$-dimensional point $x$ with respect to a finite (or probability) Borel measure $\mu$ in $\mathbb{R}^d$ is defined as the infimum of the $\mu$-masses of all closed halfspaces containing $x$. A natural question is…

Given $N$ i.i.d. samples from a probability measure $\mu$ on $\mathbf{R}^d$, we study the rate of convergence of the empirical measure $\mu_N \to \mu$ in the negative Sobolev space $W^{-\alpha, p}$. When $W^{-\alpha, p}$ contains point…

For the Riesz kernel $\kappa_\alpha(x,y):=|x-y|^{\alpha-n}$ on $\mathbb R^n$, where $n\geqslant2$, $\alpha\in(0,2]$, and $\alpha<n$, we consider the problem of minimizing the Gauss functional…

Given a Borel measure $\mu$ on ${\mathbb R}^{n}$, we define a convex set by \[ M({\mu})=\bigcup_{\substack{0\le f\le1,\\ \int_{{\mathbb R}^{n}}f\,{\rm d}{\mu}=1 } }\left\{ \int_{{\mathbb R}^{n}}yf\left(y\right)\,{\rm…

Let ${\mathbb X}$ be a compact, connected, Riemannian manifold (without boundary), $\rho$ be the geodesic distance on ${\mathbb X}$, $\mu$ be a probability measure on ${\mathbb X}$, and $\{\phi_k\}$ be an orthonormal system of continuous…

In this work we obtain a geometric characterization of the measures $\mu$ in $\mathbb{R}^{n+1}$ with polynomial upper growth of degree $n$ such that the $n$-dimensional Riesz transform $\mathcal{R}\mu (x) = \int…

Let $A(\cdot)$ be an $(n+1)\times (n+1)$ uniformly elliptic matrix with H\"older continuous real coefficients and let $\mathcal E_A(x,y)$ be the fundamental solution of the PDE $\mathrm{div} A(\cdot) \nabla u =0$ in $\mathbb R^{n+1}$. Let…

We present a Riesz integral representation theory in which functions, operators and measures take values in uniform commutative monoids (a commutative monoid with a uniformity making the binary operation of the monoid uniformly continuous).…

We give a simple criterion on the set of probability tangent measures $\mathrm{Tan}(\mu,x)$ of a positive Radon measure $\mu$, which yields lower bounds on the Hausdorff dimension of $\mu$. As an application, we give an elementary and…

We prove a structure theorem for any $n$-rectifiable set $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, satisfying a weak version of the lower ADR condition, and having locally finite $H^n$ ($n$-dimensional Hausdorff) measure. Namely, that…

Riesz Theorem establishes a correspondence between the set of $\sigma$-additive regular Borel measures and the set of linear positively defined functionals. We consider an idempotent analogue of this correspondence between possibility…

We generalize various notions of stability of invariant sets of dynamical systems to invariant measures, by defining a topology on the set of measures. The defined topology is similar, but not topologically equivalent to weak* topology, and…

We quantify Prokhorov's Theorem by establishing an explicit formula for the Hausdorff measure of non-compactness (HMNC) for the parametrized Prokhorov metric on the set of Borel probability measures on a Polish space. Furthermore, we…

Let \(\mu\) be a finite Borel measure on \((-\pi,\pi)\). Consider the one-dimensional Poisson equation \(-u''=\mu\), where equality holds in the sense of distributions, with Dirichlet boundary conditions \(u(\pm\pi)=0\). In this paper, we…