Riesz capacities of a set due to Dobi\'nski

We study a capacity theory based on a definition of a Riesz potential in metric spaces with a doubling measure. In this general setting, we study the basic properties of the Riesz capacity, including monotonicity, countable subadditivity and several convergence results. We define a modified version of the Hausdorff measure and provide lower bound and upper bound estimates for the capacity in terms of the modified Hausdorff content. We also study isocapacitary inequalities and boundedness of the Riesz potential.

Indiana University Mathematics Journal, 2011

Analytic capacity is associated with the Cauchy kernel 1/z and the space L ∞ . One has likewise capacities associated with the real and imaginary parts of the Cauchy kernel and L ∞ . Striking results of Tolsa and a simple remark show that these three capacities are comparable. We present an extension of this fact to R n , n ≥ 3, involving the vector valued Riesz kernel of homogeneity −1 and n − 1 of its components.

Integral Equations and Operator Theory, 2002

We discuss the potential theory related to the variational capacity and the Sobolev capacity on metric measure spaces. We prove our results in the axiomatic framework of [16].

SUMMARY. A definition of "monotone integral" is given, similarly as in [5], for real-valued maps and with respect to Dedekind complete Riesz space-valued "capacities". Some representation theorems are proved; in particular, we give here a version of Riesz representation theorem. Moreover, a concept of weak convergence is introduced, and some Portmanteau-type theorems, Vitali convergence and Fatou theorems are proved. Finally, a version of both strong and weak laws of large numbers is demonstrated.

Contemporary Mathematics, 2003

It has been recently proved that analytic capacity, γ, is semiadditive. This result is a consequence of the comparability between γ and γ + , a version of γ originated by bounded Cauchy potentials of positive measures. In this paper we describe the main ideas involved in the proof of this result and we give a complete proof of it in the particular case of the N -th approximation of the corner quarters Cantor set.

Mathematica Slovaca, 2009

A definition of concave integral is given for real-valued maps and with respect to Dedekind complete Riesz space-valued “capacities”. Some comparison results with other integrals are given and some convergence theorems are proved.

Journal Fur Die Reine Und Angewandte Mathematik, 2005

There is a natural capacity associated to any vector valued Riesz kernel of a given homogeneity. If we are in the plane and the kernel is the Cauchy kernel, then this capacity is analytic capacity. Our main result states that if the homogeneity of the kernel is negative and larger than minus one, then the capacity is comparable to one

International Journal of Approximate Reasoning, 2003

The characterization of the extreme points constitutes a crucial issue in the investigation of convex sets of probabilities, not only from a purely theoretical point of view, but also as a tool in the management of imprecise information. In this respect, different authors have found an interesting relation between the extreme points of the class of probability measures dominated by a second order alternating Choquet capacity and the permutations of the elements in the referential. However, they have all restricted their work to the case of a finite referential space. In an infinite setting, some technical complications arise and they have to be carefully treated. In this paper, we extend the mentioned result to the more general case of separable metric spaces. Furthermore, we derive some interesting topological properties about the convex sets of probabilities here investigated. Finally, a closer look to the case of possibility measures is given: for them, we prove that the number of extreme points can be reduced even in the finite case. alternating capacities [5] or submodular set functions[11]. They play an important role as upper bounds of sets of probability measures (see, for instance [12]). This means that they are coherent [25] upper probabilities. A Choquet capacity does not need to be 2-alternating to satisfy the property of coherence. Nevertheless, this property leads to important mathematical simplifications, e.g. in the calculus of the formulae of upper expectations and conditional upper probabilities. Thus we can find detailed studies in the literature concerning this type of Choquet capacities. On the other hand, several authors have studied the properties of convex sets of probability measures, or credal sets [15]. They are well suited to provide robustness in certain situations [25, 28]. Moreover, they have been successfully applied in different contexts, such as information theory [1] or classification [30]. In particular, Walley [25] has showed that a coherent upper probability µ contains the same information as the credal set M (µ), the set of probability measures dominated by µ.

Proceedings of The London Mathematical Society, 2010

Our aim is to give sharp upper bounds for the size of the set of points where the Riesz transform of a linear combination of N point masses is large. This size will be measured by the Hausdorff content with various gauge functions. Among other things, we shall characterize all gauge functions for which the estimates do not blow up as N tends to infinity (in this case a routine limiting argument will allow us to extend our bounds to all finite Borel measures). We also show how our techniques can be applied to estimates for certain capacities.

Cardinal functions of partially ordered sets, topological spaces and Boolean algebras; precalibers; ideals of sets.