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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) → + ∞ as | x | → + ∞ 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 supported outside of R d . Simple cases where ν is a discrete measure are analyzed in detail. New theoretic results for Riesz potentials, in particular an extension of a classical theorem by de La Vallée Poussin, are established. These results are of independent interest. [ABSTRACT FROM AUTHOR]

Let be a class of extended numerical functions on a domain of -dimensional Euclidean space , . Given and , we write if there is a function such that on . We consider this special preorder for a pair of subharmonic functions and on in cases when is the space of all harmonic functions on or is the convex cone of all subharmonic functions on . Main results are dual equivalent forms for this preorder in terms of balayage processes for Riesz measures of subharmonic functions and , for Jensen and Arens–Singer (representing) measures, for potentials of these measures, and for special test functions generated by subharmonic functions on complements of non-empty precompact subsets . Applications to holomorphic functions on a domain relate to the distribution of zero sets of functions under upper restrictions on . If a domain is a finitely connected domain with non-empty exterior or a simply connected domain with two different points on the boundary of , then our conditions for the distribution of zeros of with on are both necessary and sufficient. [ABSTRACT FROM AUTHOR]

With the sphere S 2 ⊂ R 3 as a conductor holding a unit charge with logarithmic interactions, we consider the problem of determining the support of the equilibrium measure in the presence of an external field consisting of finitely many point charges on the surface of the sphere. We determine that for any such configuration, the complement of the equilibrium support is the stereographic preimage from the plane of a union of classical quadrature domains, whose orders sum to the number of point charges. [ABSTRACT FROM AUTHOR]

We introduce the space of virtual Markov chains (VMCs) as a pro-jective limit of the spaces of all finite state space Markov chains (MCs), in the same way that the space of virtual permutations is the projective limit of the spaces of all permutations of finite sets. We introduce the notions of virtual initial distribution (VID) and a virtual transition matrix (VTM), and we show that the law of any VMC is uniquely characterized by a pair of a VID and VTM which have to satisfy a certain compatibility condition. Lastly, we study various properties of compact convex sets associated to the theory of VMCs, including that the Birkhoff-von Neumann theorem fails in the virtual setting. [ABSTRACT FROM AUTHOR]

We investigate some properties of balayage, or, sweeping (out), of measures with respect to subclasses of subharmonic functions. The following issues are considered: relationships between balayage of measures with respect to classes of harmonic or subharmonic functions and balayage of measures with respect to significantly smaller classes of specific classes of functions; integration of measures and balayage of measures; sensitivity of balayage of measures to polar sets, etc. [ABSTRACT FROM AUTHOR]

Let X be a Borel right Markov process, let m be an excessive measure for X, and let $\widehat{X}$ be the moderate Markov dual process associated with X and m. The potential theory of co-excessive measures (i.e., measures that are excessive for $\widehat{X}$ ) is developed with special emphasis on the Riesz decomposition. This is then applied to obtain the Riesz decomposition of excessive functions (of X) by exploiting the correspondence between such functions and co-excessive measures. The potential theory of co-excessive measures also enables us to discuss Walsh’s interior réduite under minimal conditions. Many of the tools of the theory of Markov processes are employed in this development. For example, Kuznetsov measures, Ray compactifications, h-transforms, and duality theory for Borel right processes.

The main purpose of the present paper is to establish a link between quadrature surfaces (potential theoretic concept) and sandpile dynamics (Laplacian growth models). For this aim, we introduce a new model of Laplacian growth on the lattice ℤ d (d ≥ 2) which continuously deforms occupied regions of the divisible sandpile model of Levine and Peres (J. Anal. Math. 111(1), 151–219 2010), by redistributing the total mass of the system onto 1 m -sub-level sets of the odometer which is a function counting total emissions of mass from lattice vertices. In free boundary terminology this goes in parallel with singular perturbation, which is known to converge to a Bernoulli type free boundary. We prove that models, generated from a single source, have a scaling limit, if the threshold m is fixed. Moreover, this limit is a ball, and the entire mass of the system is being redistributed onto an annular ring of thickness 1 m . By compactness argument we show that when m tends to infinity sufficiently slowly with respect to the scale of the model, then in this case also there is scaling limit which is a ball, with the mass of the system being uniformly distributed onto the boundary of that ball, and hence we recover a quadrature surface in this case. Depending on the speed of decay of 1/m, the visited set of the sandpile interpolates between spherical and polygonal shapes. Finding a precise characterisation of this shape-transition phenomenon seems to be a considerable challenge, which we cannot address at this moment. [ABSTRACT FROM AUTHOR]

Let M be a subharmonic function in a domain D ⊂ ℂn with Riesz measure νM, and let Z ⊂ D. As was shown in the first of the preceding papers, if there exists a holomorphic function f ≠ 0 in D such that f(Z) = 0 and |f| ⩽ exp M on D, then one has a scale of integral uniform upper bounds for the distribution of the set Z via νM. The present paper shows that for n = 1 this result "almost has a converse." Namely, it follows from such a scale of estimates for the distribution of points of the sequence Z ≔ {zk}k=1,2,... ⊂ D ⊂ ℂ via νM that there exists a nonzero holomorphic function f in D such that f(Z) = 0 and |f| ⩽ exp M↑r on D, where the function M↑r ⩾ M on D is constructed from the averages of M over circles rapidly narrowing when approaching the boundary of D with a possible additive logarithmic term associated with the rate of narrowing of these circles. [ABSTRACT FROM AUTHOR]

By BMOo(R) we denote the space consisting of all those odd and bounded mean oscillation functions on R. In this paper we characterize the functions in BMOo(R) with bounded support as those ones that can be written as a sum of a bounded function on (0,∞) plus the balayage of a Carleson measure on (0,∞)×(0,∞) with respect to the Poisson semigroup associated with the Bessel operator Bλ:=-x-λddxx2λddxx-λ,λ>0.This result can be seen as an extension to Bessel setting of a classical result due to Carleson. [ABSTRACT FROM AUTHOR]

Let L be a Schrödinger operator of the form L = −Δ+ V acting on L (R), n ≥ 3, where the nonnegative potential V belongs to the reverse Hölder class B for some q ≥ n: Let BMO(R) denote the BMO space associated to the Schrödinger operator L on R. In this article, we show that for every f ∈ BMO(R) with compact support, then there exist g ∈ L∞(R) and a finite Carleson measure μ such that with $${\left\| g \right\|_\infty } + |||\mu ||{|_c} \leqslant C{\left\| f \right\|_{BM{O_L}\left( {{\mathbb{R}^n}} \right)}}$$ ; where , and P ( x; y) is the kernel of the Poisson semigroup $$\left\{ {{e^{ - t\sqrt L }}} \right\}t > 0$$ on L (R). Conversely, if μ is a Carleson measure, then S belongs to the space BMO(R). This extends the result for the classical John-Nirenberg BMO space by Carleson (1976) (see also Garnett and Jones (1982), Uchiyama (1980) and Wilson (1988)) to the BMO setting associated to Schrödinger operators. [ABSTRACT FROM AUTHOR]

We establish a general uniqueness theorem for subharmonic functions of several variables on a domain. A corollary from this uniqueness theorem for holomorphic functions is formulated in terms of the zero subset of holomorphic functions and restrictions on the growth of functions near the boundary of domain. [ABSTRACT FROM AUTHOR]

Given a bounded strongly pseudoconvex domain D in ℂn with smooth boundary, we characterize (p, q, α)-Bergman Carleson measures for 0 < p < ∞, 0 < q < ∞, and α > -1. As an application, we show that the Bergman space version of the balayage of a Bergman Carleson measure on D belongs to BMO in the Kobayashi metric. [ABSTRACT FROM AUTHOR]

Let φ be a plurisubharmonic function on a pseudoconvex domain D in an n-dimensional complex space. We show that there exists a nonzero holomorphic function f on D such that some local mean value of φ with logarithmic additional terms majorizes log| f|. A similar problem is discussed for a locally integrable function on D in terms of balayage of positive measures. [ABSTRACT FROM AUTHOR]

For various applications (including: regularity study of free boundaries, and approximation in the mean of harmonic functions on unbounded domains by rapidly decreasing ones) it is desirable to have an estimate, valid for an arbitrary smooth function φ of compact support in IRn, bounding sup |φ| in the unit ball B in terms of sup |∆φ| on IR^n. Such an estimate cannot exist with no further assumptions on φ, but in the predecessor of this paper it was shown that if φ vanishes on a subset E of B with volume |E| > 0, such a bound holds , and uniformly with respect to all sets E with |E| not less than any prescribed positive constant. In the present paper an analogous estimate is obtained for φ which |φ(0)| ≤ 1, and grad(φ) vanishes on a subset E of B whose Newtonian capacity exceeds a positive constant. Various applications are given. The derivation of the basic estimate involves ideas which may have some independent interest, in particular “quasi-balayage”, the sweeping-out of a measure from its support to some other compact set such that the potentials of the two measures, while not equal on a neighborhood of infinity as required by (true) balayage, are asymptotically equal to some prescribed degree.

We develop a theory of non-uniform sampling in the context of the theory of frames for the settings of the short time fourier transform and pseudo-differential operators. Our theory is based on profound historical precedents including Beurling's theory of balayage, emanating from the nineteenth century work of Christoffel and Poincaré, the theory and results from spectral synthesis due to Wiener and Beurling and a host of the major harmonic analysts of the twentieth century, and the theory of sets of multiplicity, going back to Riemann and emerging fundamentally from the Russian school of harmonic analysis in the early twentieth century. Our results are meant to serve as the underpinnings for both theoretical and practical results in the realm of non-uniform sampling. They can also be compared with several other distinct forays into non-uniform sampling, including the settings of quasi-crystals and modulation spaces, where proofs for the latter setting require the analysis of convolution operators on the Heisenberg group. Our theory herein is the first step in which the ultimate goal is computational implementation for non-uniform sampling and its myriad applications, where balayage, spectral synthesis, and sets of multiplicity are computationally quantified. A critical component is to resurrect the formulation of balayage in terms of covering criteria. [ABSTRACT FROM AUTHOR]

We introduce the Bergman space version of the balayage on the unit disk, which we call B-balayage. We show that the B-balayage of a Bergman-Carleson measure satisfies a Lipschitz condition in the Bergman metric. Consequently, the B-balayage of a Bergman-Carleson measure belongs to BMO in the Bergman metric. This extends a classical result about the balayage to the case of Bergman spaces. [ABSTRACT FROM PUBLISHER]

We consider the minimal energy problem on the unit sphere 핊 in the Euclidean space ℝ in the presence of an external field Q, where the energy arises from the Riesz potential 1/ r (where r is the Euclidean distance and s is the Riesz parameter) or the logarithmic potential log(1/ r). Characterization theorems of Frostman-type for the associated extremal measure, previously obtained by the last two authors, are extended to the range d − 2 ≤ s < d − 1. The proof uses a maximum principle for measures supported on 핊. When Q is the Riesz s-potential of a signed measure and d − 2 ≤ s < d, our results lead to explicit point-separation estimates for ( Q, s)-Fekete points, which are n-point configurations minimizing the Riesz s-energy on 핊 with external field Q. In the hyper-singular case s > d, the short-range pair-interaction enforces well-separation even in the presence of more general external fields. As a further application, we determine the extremal and signed equilibria when the external field is due to a negative point charge outside a positively charged isolated sphere. Moreover, we provide a rigorous analysis of the three point external field problem and numerical results for the four point problem. [ABSTRACT FROM AUTHOR]

In this paper we establish that the density of the equilibrium measure of finitely many intervals for both logarithmic and Riesz potentials is convex. The main tool is a balayage ping-pong technique. A similar result is obtained for finitely many arcs on the unit circle. Applications to external field problems and constrained energy problems are presented. [ABSTRACT FROM AUTHOR]

It is shown that, for open sets in classical potential theory and-more generally-for elliptic harmonic spaces Y, the set J( Y) of Jensen measures (representing measures with respect to superharmonic functions on Y) for a point x ∈ Y is a simple union of closed faces of the compact convex set $M_x(\mathcal P(Y))$ of representing measures with respect to potentials on Y, a set which has been thoroughly studied a long time ago. In particular, the set of extreme Jensen measures can be immediately identified. The results hold even without ellipticity (thus capturing also many examples for the heat equation) provided a rather weak approximation property for superharmonic functions holds. Equally sufficient are a certain transience property and a weak regularity property. More important, each of these properties turns out to be necessary and sufficient for obtaining (in the classical case) that J( Y) coincides with the set of all compactly supported probability measures in $M_x(\mathcal P(Y))$. [ABSTRACT FROM AUTHOR]

Abstract: Given a probability measure supported on some compact set and with orthonormal polynomials , define the measures and let be the normalized zero counting measure for the polynomial . If is supported on a compact subset of the real line or on the unit circle, we provide a new proof of a 2009 theorem due to Simon that for any fixed the th moment of and differ by at most as . [Copyright &y& Elsevier]

Abstract: Let X be a Green domain in , , , and let denote the compact convex set of all representing measures for x. Recently it has been proven that the set of harmonic measures , U open in X, , which is contained in the set of extreme points of , is dense in . In this paper, it is shown that is not a simplex (and hence not a Poulsen simplex). This is achieved by constructing open neighborhoods , , , of x such that the harmonic measures are pairwise different and . In fact, these measures form a square with respect to a natural -structure. Since the construction is mainly based on having certain symmetries, it can be carried out just as well for Riesz potentials, the Heisenberg group (or any stratified Lie algebra), and the heat equation (or more general parabolic situations). [ABSTRACT FROM AUTHOR]

Abstract: We develop a theory of balayage on complete doubling metric measure spaces supporting a Poincaré inequality. In particular, we are interested in continuity and -harmonicity of the balayage. We also study connections to the obstacle problem. As applications, we characterize regular boundary points and polar sets in terms of balayage. [Copyright &y& Elsevier]

Abstract: We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz potentials with . For a given axis-supported external field, the support and the density of the corresponding extremal measure on is determined. The special case yields interesting phenomena, which we investigate in detail. A asymptotic analysis is provided as . [Copyright &y& Elsevier]

We discuss quadrature domains for subharmonic functions and prove the existence of core quadrature domains for certain positive measures. The core quadrature domains are the smallest quadrature domains as measures and inherit good properties from quadrature domains with finite volume. We next discuss new balayage for the class of harmonic functions integrable in a neighborhood of ∞. We give several estimates of balayage measures. The new balayage is introduced to construct quadrature domains for harmonic functions. [ABSTRACT FROM AUTHOR]

Abstract: It is shown that any convex combination of harmonic measures , where are relatively compact open neighborhoods of a given point , , can be approximated by a sequence of harmonic measures such that each is an open neighborhood of x in . This answers a question raised in connection with Jensen measures. Moreover, it implies that, for every Green domain X containing x, the extremal representing measures for x with respect to the convex cone of potentials on X (these measures are obtained by balayage of the Dirac measure at x on Borel subsets of X) are dense in the compact convex set of all representing measures. This is achieved approximating balayage on open sets by balayage on unions of balls which are pairwise disjoint and very small with respect to their mutual distances and then reducing the size of these balls in a suitable manner. These results, which are presented simultaneously for the classical potential theory and for the theory of Riesz potentials, can be sharpened if the complements or the boundaries of the open sets have a capacity doubling property. The methods developed for this purpose (continuous balayage on increasing families of compact sets, approximation using scattered sets with small capacity) finally lead to answers even in a very general potential-theoretic setting covering a wide class of second order partial differential operators (uniformly elliptic or in divergence form, or sums of squares of vector fields satisfying Hörmander''s condition, for example, sub-Laplacians on stratified Lie algebras). [Copyright &y& Elsevier]

Blackwell (1951), in his seminal work on comparison of experiments, ordered two experiments using a dilation ordering: one experiment, Y, is ‘more spread Out’ in the sense of dilation than another one, X, if E(c(Y)) ⩾ E(c(X)) for all convex functions c. He showed that this ordering is equivalent to two other orderings, namely (i) a total time on test ordering and (ii) a martingale relationship E(Y′ ∣ X′) = X′, where (X′, Y′) has a joint distribution with the same marginals as X and Y. These comparisons are generalized to balayage orderings that are defined in terms of generalized convex functions. These balayage orderings are equivalent to (i) iterated total integral of survival orderings and (ii) martingale-type orderings which we refer to as k-mart orderings. These comparisons can arise naturally in model fitting and data confidentiality contexts. [ABSTRACT FROM AUTHOR]

A minimal element, mother body, for a family of bodies producing the same external gravitational field is the body in the family, whose support has Lebesgue measure zero and satisfies some additional requirements. The finite algorithm of constructing mother bodies in is suggested. The local structure of mother bodies near singular points of continued logarithmic potential is investigated in generic positions. [ABSTRACT FROM AUTHOR]

We consider coherent risk measures satisfying the Fatou property which are monotonous with respect to balayage or dilatation. An equivalent condition ensuring balayage-monotonicity is given and a representation result is derived. [ABSTRACT FROM AUTHOR]