Let $A$ be a compact set in $\Rp$ of Hausdorff dimension $d$. For $s\in(0,d)$, the Riesz $s$-equilibrium measure $μ^{s,A}$ is the unique Borel probability measure with support in $A$ that minimizes $$ \Is(μ):=\iint\Rk{x}{y}{s}dμ(y)dμ(x)$$ over all such probability measures. In this paper we show that if $A$ is a strictly self-similar $d$-fractal, then $μ^{s,A}$ converges in the weak-star topology to normalized $d$-dimensional Hausdorff measure restricted to $A$ as $s$ approaches $d$ from below.
We investigate the asymptotic behavior of minimal $N$-point Riesz $s$-energy on fractal sets of non-integer dimension, with algebraically dependent contraction ratios. For $s$ bigger than the dimension of the set $A$, we prove the asymptotic behavior of the minimal $N$-point Riesz $s$-energy of $A$ along explicit subsequences, but we show that the general asymptotic behavior does not exist.
We study the construction of exponential frames and Riesz sequences for a class of fractal measures on ${\mathbb R}^d$ generated by infinite convolution of discrete measures using the idea of frame towers and Riesz-sequence towers. The exactness and overcompleteness of the constructed exponential frame or Riesz sequence is completely classified in terms of the cardinality at each level of the tower. Using a version of the solution of the Kadison-Singer problem, known as the $R_ε$-conjecture, we show that all these measures contain exponential Riesz sequences of infinite cardinality. Furthermor…
We investigate separation properties of $N$-point configurations that minimize discrete Riesz $s$-energy on a compact set $A\subset \mathbb{R}^p$. When $A$ is a smooth $(p-1)$-dimensional manifold without boundary and $s\in [p-2, p-1)$, we prove that the order of separation (as $N\to \infty$) is the best possible. The same conclusions hold for the points that are a fixed positive distance from the boundary of $A$ whenever $A$ is any $p$-dimensional set. These estimates extend a result of Dahlberg for certain smooth $(p-1)$-dimensional surfaces when $s=p-2$ (the harmonic case). Furthermore, we …
The purpose of this work is twofold. First, we aim to extend for $0<s<1$ the results of one of the authors about equilibrium measures in the real axis in external fields created by point-mass charges for the case of logarithmic potentials ($s=0$). Our second motivation comes from the work of the other two authors on Riesz $s$-equilibrium problems on finitely many intervals on the real line in the presence of external fields. They have shown that when the signed equilibrium measure has concave positive part on every interval, then the $s$-equilibrium support is also a union of finitely many int…
We construct a large class of Riemannian manifolds of arbitrary dimension with Riesz transform unbounded on $L^p(M)$ for all $p > 2$. This extends recent results for Vicsek manifolds, and in particular shows that fractal structure is not necessary for this property.
We consider a special case of Maxwell's problem on the number of equilibrium points of the Riesz potential $1/r^{2β}$ (where $r$ is the Euclidean distance and $β$ is the Riesz parameter) for positive unit point charges placed at the vertices of a regular polygon. We show that the equilibrium points are located on the perpendicular bisectors to the sides of the regular polygon, and study the asymptotic behavior of the equilibrium points with regard to the number of charges $n$ and the Riesz parameter $β$. Finally, we prove that for values of $β$ in a small neighborhood of $β=1$ the Riesz potent…
We investigate properties of minimal $N$-point Riesz $s$-energy on fractal sets of non-integer dimension, as well as asymptotic behavior of $N$-point configurations that minimize this energy. For $s$ bigger than the dimension of the set $A$, we constructively prove a negative result concerning the asymptotic behavior (namely, its nonexistence) of the minimal $N$-point Riesz $s$-energy of $A$, but we show that the asymptotic exists over reasonable sub-sequences of $N$. Furthermore, we give a short proof of a result concerning asymptotic behavior of configurations that minimize the discrete Ries…
In this paper, we investigate Riesz energy problems on unbounded conductors in $\R^d$ in the presence of general external fields $Q$, not necessarily satisfying the growth condition $Q(x)\to\infty$ as $x\to\infty$ assumed in several previous studies. We provide sufficient conditions on $Q$ for the existence of an equilibrium measure and the compactness of its support. Particular attention is paid to the case of the hyperplanar conductor $\R^{d}$, embedded in $\R^{d+1}$, when the external field is created by the potential of a signed measure $ν$ outside of $\R^{d}$. Simple cases where $ν$ is a …
The aim of this note is to provide a full space quadratic external field extension of a classical result of Marcel Riesz for the equilibrium measure on a ball with respect to Riesz s-kernels. We address the case s=d-3 for arbitrary dimension d, in particular the logarithmic kernel in dimension 3. The equilibrium measure for this full space external field problem turns out to be a radial arcsine distribution supported on a ball with a special radius. As a corollary, we obtain new integral identities involving special functions such as elliptic integrals and more generally hypergeometric functio…
We compute the equilibrium measure in dimension d=s+4 associated to a Riesz s-kernel interaction with an external field given by a power of the Euclidean norm. Our study reveals that the equilibrium measure can be a mixture of a continuous part and a singular part. Depending on the value of the power, a threshold phenomenon occurs and consists of a dimension reduction or condensation on the singular part. In particular, in the logarithmic case s=0 (d=4), there is condensation on a sphere of special radius when the power of the external field becomes quadratic. This contrasts with the case d=s+…
We study differentiability properties of Riesz potentials of finite Borel measures in dimension d larger than 2. The Riesz kernel has homogeneity 2-d. In dimension 2 we consider logarithmic potentials. We introduce a notion of differentiability in the capacity sense, capacity being Newtonian capacity in dimension larger than 2 and Wiener capacity in the plane. It turns out that differentiability in the capacity sense at a point is related to the existence of principal values of the measure with respect to the vector valued Riesz potential x/|x|^d of homogeneity 1-d. This leads to Calderón-Zygm…
Focusing first on the inner $α$-harmonic measure $\varepsilon_y^A$ ($\varepsilon_y$ being the unit Dirac measure, and $μ^A$ the inner $α$-Riesz balayage of a Radon measure $μ$ to $A\subset\mathbb R^n$ arbitrary), we describe its Euclidean support, provide a formula for evaluation of its total mass, establish the vague continuity of the map $y\mapsto\varepsilon_y^A$ outside the inner $α$-irregular points for $A$, and obtain necessary and sufficient conditions for $\varepsilon_y^A$ to be of finite energy (more generally, for $\varepsilon_y^A$ to be absolutely continuous with respect to inner cap…
First introduced by J. Deny, the classical principle of positivity of mass states that if $κ_αμ\leqslantκ_αν$ everywhere on $\mathbb{R}^n$, then $μ(\mathbb{R}^n)\leqslantν(\mathbb{R}^n)$. Here $μ,ν$ are positive Radon measures on $\mathbb{R}^n$, $n\geqslant2$, and $κ_αμ$ is the potential of $μ$ with respect to the Riesz kernel $|x-y|^{α-n}$ of order $α\in(0,2]$, $α<n$. We strengthen Deny's principle by showing that $μ(\mathbb{R}^n)\leqslantν(\mathbb{R}^n)$ still holds even if $κ_αμ\leqslantκ_αν$ is fulfilled only on a proper subset $A$ of $\mathbb{R}^n$ that is not inner $α$-thin at infinity; …
We study the constrained minimum energy problem with an external field relative to the $α$-Riesz kernel $|x-y|^{α-n}$ of order $α\in(0,n)$ for a generalized condenser $\mathbf A=(A_i)_{i\in I}$ in $\mathbb R^n$, $n\geqslant 3$, whose oppositely charged plates intersect each other over a set of zero capacity. Conditions sufficient for the existence of minimizers are found, and their uniqueness and vague compactness are studied. Conditions obtained are shown to be sharp. We also analyze continuity of the minimizers in the vague and strong topologies when the condenser and the constraint both var…
We study the properties and asymptotics of the Jacobi matrices associated with equilibrium measures of the weakly equilibrium Cantor sets. These family of Cantor sets were defined and different aspects of orthogonal polynomials on them were studied recently. Our main aim is numerically examine some conjectures concerning orthogonal polynomials which do not directly follow from previous results. We also compare our results with more general conjectures made for recurrence coefficients associated with fractal measures supported on $\mathbb{R}$.
We investigate minimum weak $α$-Riesz energy problems with external fields in both the unconstrained and constrained settings for generalized condensers $(A_1,A_2)$ such that the closures of $A_1$ and $A_2$ in $\mathbb R^n$ are allowed to intersect one another. (Such problems with the standard $α$-Riesz energy in place of the weak one would be unsolvable, which justifies the need for the concept of weak energy when dealing with condenser problems.) We obtain sufficient and/or necessary conditions for the existence of minimizers, provide descriptions of their supports and potentials, and single…
We study the asymptotic equidistribution of points near arbitrary compact sets of positive capacity in $\R^d,\ d\ge 2$. Our main tools are the energy estimates for Riesz potentials. We also consider the quantitative aspects of this equidistribution in the classical Newtonian case. In particular, we quantify the weak convergence of discrete measures to the equilibrium measure, and give the estimates of convergence rates for discrete potentials to the equilibrium potential.
We study properties of the $α$-Green kernel $g_D^α$ of order $0<α\leqslant2$ for a domain $D\subset\mathbb R^n$, $n\geqslant3$. This kernel is associated with the $α$-Riesz kernel $|x-y|^{α-n}$, $x,y\in\mathbb R^n$, in a manner particularly well known in the case $α=2$. Besides the usual principles of potential theory, we establish for the $α$-Green kernel the property of consistency. This allows us to prove the completeness of the cone of positive measures $μ$ on $D$ with finite energy $g_D^α(μ,μ):=\iint g_D^α(x,y)\,dμ(x)\,dμ(y)$ in the topology defined by the energy norm $\|μ\|_{g_D^α}=\sqrt…
We establish the theory of balayage for the Riesz kernel $|x-y|^{α-n}$, $α\in(0,2]$, on $\mathbb R^n$, $n\geqslant3$, alternative to that suggested in the book by Landkof. A need for that is caused by the fact that the balayage in that book is defined by means of the integral representation, which, however, so far is not completely justified. Our alternative approach is mainly based on Cartan's ideas concerning inner balayage, formulated by him for the Newtonian kernel. Applying the theory of inner Riesz balayage thereby developed, we obtain a number of criteria for the existence of an inner e…
We study a physical system of $N$ interacting particles in $\mathbb{R}^d$, $d\geq1$, subject to pair repulsion and confined by an external field. We establish a large deviations principle for their empirical distribution as $N$ tends to infinity. In the case of Riesz interaction, including Coulomb interaction in arbitrary dimension $d>2$, the rate function is strictly convex and admits a unique minimum, the equilibrium measure, characterized via its potential. It follows that almost surely, the empirical distribution of the particles tends to this equilibrium measure as $N$ tends to infinity. …
