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Given a compact space K, we denote by P(K) the space of all Radon probability measures on K, equipped with the weak* topology inherited from C(K)*. For nonmetrizable compacta K even basic properties of P(K) spaces depend of additional axioms of set theory. We discuss here older and quite recent results on the subject. [ABSTRACT FROM AUTHOR]

There is a long-standing problem, posed by A.T.-M. Lau [ Fixed point theory and its applications , Academic Press, New York-London, 1976, pp. 121–129], whether left amenability is sufficient to ensure the existence of a common fixed point for every jointly weak^{\ast } continuous nonexpansive semigroup action on a nonempty weak^{\ast } compact convex set in a dual Banach space. In this note we discuss the current status of this problem and give a partial solution in the case of weak^{\ast } compact convex sets with the Radon–Nikodým property. [ABSTRACT FROM AUTHOR]

In this paper, we develop the Littman–Stampacchia–Weinberger duality approach to obtain global W 1 , p estimates for a class of elliptic problems involving Leray–Hardy operators and measure sources in a distributional framework associated to a dual formulation with a specific weight function. [ABSTRACT FROM AUTHOR]

Given a positive Borel measure μ on the one-dimensional Euclidean space R , consider the centered Hardy–Littlewood maximal function M μ acting on a finite positive Borel measure ν by M μ ν (x) : = sup r > r 0 (x) ν (B (x , r)) μ (B (x , r)) , x ∈ R , where r 0 (x) = inf { r > 0 : μ (B (x , r)) > 0 } and B(x, r) denotes the closed ball with centre x and radius r > 0 . In this note, we restrict our attention to Radon measures μ on the positive real line [ 0 , + ∞) . We provide a complete characterization of measures having weak-type asymptotic properties for the centered maximal function. Although we don’t know whether this fact can be extended to measures on the entire real line R , we examine some criteria for the existence of the weak-type asymptotic properties for M μ on R . We also discuss further properties, and compute the value of the relevant asymptotic quantity for several examples of measures. [ABSTRACT FROM AUTHOR]

The analogue of polar coordinates in the Euclidean space, a polar decomposition in a metric space, if well-defined, can be very useful in dealing with integrals with respect to a sufficiently regular measure. In this note we handle the technical details associated with such polar decompositions. Supplementary Information: The online version contains the Armenian language version of the article available at https://doi.org/10.1007/s10958-023-06674-w. [ABSTRACT FROM AUTHOR]

In this paper, we compare the Stone–Čech compactification of the space of Radon probability measures on a Tychonoff space , equipped with the weak topology, with the space of Radon probability measures on the Stone–Čech compactification of the space . It is shown that for any noncompact metric space , the compactification does not coincide with . We discuss the case of more general Tychonoff spaces and also the case of the Samuel compactification, for which the coincidence holds. [ABSTRACT FROM AUTHOR]

We develop a global Calderón-Zygmund estimate for the gradients of renormalized solutions to the general nonlinear singular elliptic equations − div A (x , u , D u) = μ on a Reifenberg flat domain with the homogeneous Dirichlet boundary condition, while μ is a finite signed Radon measure. The associated nonlinearity behaves as the elliptic p -Laplacian with respect to Du for the singular case p ∈ (1 , 2 − 1 / n ] , whose discontinuity in the x -variable is measured in terms of small BMO, and the Lipschitz continuity is required with respect to the u -variable. We prove it in two folds: the perturbation technique and the weighted Vitali type covering are first employed to establish the weighted good- λ type inequality, then such inequality is used to prove the desired global gradient estimates in weighted Lorentz spaces and Lorentz-Morrey spaces. As a direct consequence, finally we obtain a global gradient regularity in weighted Orlicz spaces. [ABSTRACT FROM AUTHOR]

We study metrizability of compact sets in spaces of Radon measures with the weak topology. It is shown that if all compacta in a given completely regular topological space are metrizable, then every uniformly tight compact set in the space of Radon measures on this space is also metrizable. It is proved that the property that compact sets of measures on a given space are metrizable is preserved for products of this space with spaces that can be embedded into separable metric spaces. In addition, we construct a Radon probability measure on the space of Radon probability measures on a completely regular space such that its barycenter is not a Radon measure.

We are concerned with the minimal entropy conditions for one-dimensional scalar conservation laws with general convex flux functions. For such scalar conservation laws, we prove that a single entropy-entropy flux pair (\eta (u),q(u)) with \eta (u) of strict convexity is sufficient to single out an entropy solution from a broad class of weak solutions in L^\infty _{\mathrm { loc}} that satisfy the inequality: \eta (u)_t+q(u)_x\leq \mu in the distributional sense for some non-negative Radon measure \mu. Furthermore, we extend this result to the class of weak solutions in L^p_{\mathrm {loc}}, based on the asymptotic behavior of the flux function f(u) and the entropy function \eta (u) at infinity. The proofs are based on the equivalence between the entropy solutions of one-dimensional scalar conservation laws and the viscosity solutions of the corresponding Hamilton-Jacobi equations, as well as the bilinear form and commutator estimates as employed similarly in the theory of compensated compactness. [ABSTRACT FROM AUTHOR]

This paper is devoted to the invariance of time dependent sets under dynamical systems defined on a metric space. In the absence of vector structure, evolutions are described by the so-called mutational inclusions, that extend differential inclusions of the classical Euclidean framework to the one of general metric spaces. The main difficulty we have to overcome is the absence of local compactness of the constraints implying that the distance between a point and a closed set, in general, is not realized. The fairy technical proof we propose has a potential to be applicable also to problems of invariance of closed sets under various evolution laws. The reason to seek such generality lies in the desire to have universal results that can be applied to various settings, as for instance to classical differential inclusions, to continuity inclusions in the Wasserstein spaces, the controlled transport equation in the space of Radon measures or morphological systems in the space of closed subsets of a Banach space. The obtained results are illustrated by the example of dynamics described by a non-homogeneous controlled continuity equation on the space of Radon measures. [ABSTRACT FROM AUTHOR]

We study two topologies $ \tau_{KR} $ and $ \tau_K $ on the space of measures on a completely regular space generated by Kantorovich–Rubinshtein and Kantorovich seminorms analogous to their classical norms in the case of a metric space. The Kantorovich–Rubinshtein topology $ \tau_{KR} $ coincides with the weak topology on nonnegative measures and on bounded uniformly tight sets of measures. A sufficient condition is given for the compactness in the Kantorovich topology. We show that for logarithmically concave measures, some $ s $-convex measures and stable measures weak convergence implies convergence in the Kantorovich topology. We also obtain an efficiently verified condition for convergence of the barycenters of Radon measures from a sequence or net weakly converging on a locally convex space. As an application it is shown that for weakly convergent logarithmically concave measures and stable measures convergence of their barycenters holds without additional conditions. The same is true for measures given by polynomial densities of a fixed degree with respect to logarithmically concave measures. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of at least one renormalized solution for the p(x)-Laplacian equation associated with a maximal monotone operator and Radon measure data. The functional setting involves Sobolev spaces with variable exponent W1,p(·)(Ω).

In this article, a simple linear method is established for designing optimal revolving objects with desired physical properties. First, the problem is converted to an optimal control problem by defining an artificial control related to an unknown generator curve. Next, considering a variational presentation and applying an embedding procedure, the problem is transferred into one whose unknown is an optimal Radon measure. Using two stages of approximation determines the optimal control, and the optimal generator rotating curve is obtained from the results of a finite linear programming problem. The accuracy and applications of the method are shown by presenting some classical numerical simulations and comparing the method with the level set method in those examples. [ABSTRACT FROM AUTHOR]

The extension of lattice linear space of continuous bounded functions on a completely regular space, generated by -Riemann-integrable functions on this space, is considered in the paper. To characterize this -Riemann extension some new functionally-analytical category of -latlineals with refinements ( -latlineals) is used. On its base the notion of -completion of some definite type is introduced. A functionally-analytical characterization of the -Riemann extension as some -completion of some definite type of the -latlineal of continuous bounded functions is given. [ABSTRACT FROM AUTHOR]

We study existence and uniqueness of solutions of ( E 1 ) − Δ u + μ | x | 2 u + g (u) = ν in Ω , u = λ on ∂ Ω , where Ω ⊂ ℝ + N is a bounded smooth domain such that 0 ∈ ∂ Ω , μ ≥ − N 2 4 is a constant, g a continuous nondecreasing function satisfying some integral growth condition and ν and λ two Radon measures, respectively, in Ω and on ∂ Ω. We show that the situation differs considerably according the measure is concentrated at 0 or not. When g is a power we introduce a capacity framework which provides necessary and sufficient conditions for the solvability of problem ( E 1 ). [ABSTRACT FROM AUTHOR]

This study focuses on semilinear fractional elliptic problems with concave-convex type nonlinearities and measures as data. Suitable iteration techniques and embedding results are employed to ensure the existence and multiplicity of solutions. [ABSTRACT FROM AUTHOR]

The existence of a nontrivial solution in some 'weaker' sense of the system of equations: L s u + l (x) ϕ u + w (x) u k = μ i n Ω , (− Δ) s ϕ = l (x) u 2 i n Ω , u > 0 i n Ω , u = ϕ = 0 i n R N ∖ Ω is proved. Here, L s u = (− Δ) s u − λ u − β , λ > 0 , 0 < β ≤ s < 1 , l, w are bounded, nonnegative real-valued functions in Ω, μ is a nonnegative Radon measure and k>1 belongs to a certain range. [ABSTRACT FROM AUTHOR]

The purpose of this work is to study a class of periodic parabolic equations having a critical growth nonlinearity with respect to the gradient and bounded Radon measure. By the main of the sub- and super-solution method, we employ some new technics to prove the existence of a nonnegative weak periodic solution to the studied problems. [ABSTRACT FROM AUTHOR]

In this paper we are going to prove that the Hardy-Litllewood maximal operators on variable Lebesgue spaces Lp(·)(µ) with respect to a probability Borel measure µ, are weak type and strong type for two conditions of regularity on the exponent function p(·); following [2] and [3]. Also following [2] we extend some important properties of this operator to a probability Borel measure µ, the key to extend these results is using the Besicovitch covering lemma instead of the Calderón-Zygmund decomposition. [ABSTRACT FROM AUTHOR]

We construct an example of a Hilbert -module which shows that Troitsky's theorem on the geometric essence of -compact operators between Hilbert -modules cannot be extended to modules which are not countably generated case (even in the case of a stronger uniform structure, which is also introduced). In addition, the constructed module admits no frames. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we first prove the local L s $L^{s}$ norm estimate of composite operators △ k G m ( u ) $\triangle^{k}G^{m}(u)$ by use of the L s $L^{s}$ norm of u. Then we establish the local and global higher norm inequalities of △ k G m ( u ) $\triangle^{k}G^{m}(u)$ . Simultaneously, we also give a global higher norm estimate with Radon measure. Finally, as applications of these results, we give two examples to estimate the higher norm of △ k G m ( u ) $\triangle^{k}G^{m}(u)$ .

We show from a categorical point of view that probability measures on certain measurable or topological spaces arise canonically as the extension of probability distributions on countable sets. We do this by constructing probability monads as the codensity monads of functors that send a countable set to the space of probability distributions on that set. On (pre)measurable spaces we discuss monads of probability (pre)measures and their finitely additive analogues. We also give codensity constructions for monads of Radon measures on compact Hausdorff spaces and compact metric spaces and for the monad of Baire measures on Hausdorff spaces. A crucial role in these constructions is given by integral representation theorems, which we derive from a generalized Daniell-Stone theorem. [ABSTRACT FROM AUTHOR]

We establish local continuity of solutions to the $G$-Laplace equation involving measures, i.e., \begin{align*} -\text{div}\ \bigg(\frac{g(|\nabla u|)}{|\nabla u|}\nabla u\bigg)=\mu,\end{align*} where $\mu$ is a nonnegative Radon measure satisfying $\mu (B_{r}(x_{0}))\leq Cr^{m}$ for any ball $B_{r}(x_{0})\subset\subset \Omega$ with $r\leq 1$ and $m>n-1-\delta\geq 0$. The function $g$ is supposed to be nonnegative and $C^{1}$-continuous on $[0,+\infty)$, satisfying $g(0)=0$ and \begin{align*}\delta\leq \frac{tg'(t)}{g(t)}\leq g_{0}, \forall t>0\end{align*} with positive constants $\delta$ and $g_{0}$, which generalizes the structural conditions of Ladyzhenskaya–Ural'tseva for an elliptic operator.

In this article, we prove the existence of multiple nontrivial solutions to the equation $$\displaylines{ -\Delta_{p}u = \frac{\lambda}{u^{\gamma}}+g(u)+\mu\quad \text{in }\Omega,\cr u = 0\quad \text{on } \partial\Omega,\cr u>0 \quad \text{in }\Omega, }$$ where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain with $N \geq 3$, $1 < p-1 < q$, $ \lambda>0$, $\gamma>0$, g satisfies certain conditions, $\mu\geq 0$ is a bounded Radon measure.

We study existence and stability of solutions of $$ -\Delta u +\frac{\mu}{|x|^{2 }}u+ g(u)= ν \text{ in }\Omega,\ \ \ u=0\text{ on } ∂Ω,$$ where $\Omega$ is a bounded, smooth domain of $\mathbb{ R}^N$, $N\geq 2$, containing the origin, $\mu\geq-\frac{(N-2)^2}{4}$ is a constant, $g$ is a nondecreasing function satisfying some integral growth assumption and the weak ∆2-condition, and ν is a Radon measure in Ω. We show that the situation differs depending on whether the measure is diffuse or concentrated at the origin. When $g$ is a power function, we introduce a capacity framework to find necessary and sufficient conditions for solvability.

A labelled Markov process (LMP) consists of a measurable space S together with an indexed family of Markov kernels from S to itself. This structure has been used to model probabilistic computations in Computer Science, and one of the main problems in the area is to define and decide whether two LMP S and S′ "behave the same". There are two natural categorical definitions of sameness of behavior: S and S′ are bisimilar if there exist an LMP T and measure preserving maps forming a diagram of the shape S←T→S′; and they are behaviorally equivalent if there exist some U and maps forming a dual diagram S→U←S′. These two notions differ for general measurable spaces but Doberkat (extending a result by Edalat) proved that they coincide for analytic Borel spaces, showing that from every diagram S→U←S′ one can obtain a bisimilarity diagram as above. Moreover, the resulting square of measure preserving maps is commutative (a semipullback). In this paper, we extend the previous result to measurable spaces S isomorphic to a universally measurable subset of a Polish space with the trace of the Borel σ-algebra, using a version of Strassen's theorem on common extensions of finitely additive measures. [ABSTRACT FROM AUTHOR]

We study a class of nonlinear elliptic equations involving measure data −divA(x,Du)=μinΩ,where μ is a Radon measure. Under the main assumption of A(x,ξ) that there exists a constant Λ>0 such that |A(x,ξ)−A(x0,ξ)|≤Λ(a(x)+a(x0))|x−x0|α(|ξ|2+s2)p−12,α∈(0,1],where 0≤a(x)∈Lm(Ω) for some integrable index m>1, we obtain the Calderón–Zygmund estimates in the Sobolev–Morrey spaces for refined fractional‐order derivatives of distributional solutions depending on α. [ABSTRACT FROM AUTHOR]

This article concerns about the existence of a positive SOLA (Solutions Obtained as Limits of Approximations) for the following singular critical Choquard problem involving fractional power of Laplacian and a critical Hardy potential. 0.1 (- Δ) s u - α u | x | 2 s = λ u + u - γ + β ∫ Ω u 2 b ∗ (y) | x - y | b d y u 2 b ∗ - 1 + μ in Ω , u > 0 in Ω , u = 0 in R N \ Ω. Here, Ω is a bounded domain of R N , s ∈ (0 , 1) , α , λ and β are positive real parameters, N > 2 s , γ ∈ (0 , 1) , 0 < b < min { N , 4 s } , 2 b ∗ = 2 N - b N - 2 s is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality and μ is a bounded Radon measure in Ω . [ABSTRACT FROM AUTHOR]

We give a systematic account of iterated function systems (IFS) of weak contractions of different types (Browder, Rakotch, topological). We show that the existence of attractors and asymptotically stable invariant measures, and the validity of the random iteration algorithm ('chaos game'), can be obtained rather easily for weakly contractive systems. We show that the class of attractors of weakly contractive IFSs is essentially wider than the class of classical IFSs' fractals. On the other hand, we show that, in reasonable spaces, a typical compact set is not an attractor of any weakly contractive IFS. We explore the possibilities and restrictions to break the contractivity barrier by employing several tools from fixed point theory: geometry of balls, average contractions, remetrization technique, ordered sets, and measures of noncompactness. From these considerations it follows that while the existence of invariant sets and invariant measures can be assured rather easily for general iterated function systems under mild conditions, to establish the existence of attractors and unique invariant measures is a substantially more difficult problem. This explains the central role of contractive systems in the theory of IFSs. [ABSTRACT FROM AUTHOR]

We study the kernel function of the operator u ↦ L μ u = − Δ u + μ | x | 2 u in a bounded smooth domain Ω ⊂ R + N such that 0 ∈ ∂ Ω , where μ ≥ − N 2 4 is a constant. We show the existence of a Poisson kernel vanishing at 0 and a singular kernel with a singularity at 0. We prove the existence and uniqueness of weak solutions of L μ u = 0 in Ω with boundary data ν + k δ 0 , where ν is a Radon measure on ∂ Ω ∖ { 0 } , k ∈ R and show that this boundary data corresponds in a unique way to the boundary trace of positive solution of L μ u = 0 in Ω. [ABSTRACT FROM AUTHOR]

We study the existence of a solution to the mixed boundary value problem for Helmholtz and Poisson type equations in a bounded Lipschitz domain Ω ⊂ R N and in R N \ Ω for N ≥ 3 . The boundary ∂ Ω of Ω is the decomposition of Γ 1 , Γ 2 ⊂ ∂ Ω such that ∂ Ω = Γ = Γ ¯ 1 ∪ Γ 2 = Γ 1 ∪ Γ ¯ 2 and Γ 1 ∩ Γ 2 = ∅ . We have shown that if the Neumann data f 2 ∈ H - 1 2 (Γ 2) and the Dirichlet data f 1 ∈ H 1 2 (Γ 1) then the Helmholtz problem with mixed boundary data admits a unique solution. We have also shown the existence of a weak solution to a mixed boundary value problem governed by the Poisson equation with a measure data and the Dirichlet, Neumann data belongs to f 1 ∈ H 1 2 (Γ 1) , f 2 ∈ H - 1 2 (Γ 2) , respectively. [ABSTRACT FROM AUTHOR]

In this paper, we first prove the local L s norm estimate of composite operators △ k G m (u) by use of the L s norm of u. Then we establish the local and global higher norm inequalities of △ k G m (u) . Simultaneously, we also give a global higher norm estimate with Radon measure. Finally, as applications of these results, we give two examples to estimate the higher norm of △ k G m (u) . [ABSTRACT FROM AUTHOR]

This paper deals with properties of the ultraproducts for various structures. We introduce and study the concept of the ergodic action of a group with respect to a normal state on an abelian von Neumann algebra. In particular, we provide an example showing that the ultraproduct of ergodic states, generally speaking, is not ergodic. The ultraproduct of the Radon measures on a compact convex subset of a locally convex space is also investigated in the paper. As is well-known, the study of the extreme points in the state set for a algebra is a very interesting problem in itself. Considering the ultraproducts of -algebras and the states on these algebras, we get quite nontrivial results. [ABSTRACT FROM AUTHOR]

This paper offers a new perspective to look at the Riesz potential. On the one hand, it is shown that not only 𝔏q,q⁢p-1⁢(n-α⁢p)∩𝔏p,κ-α⁢p\mathfrak{L}^{q,qp^{-1}(n-\alpha p)}\cap\mathfrak{L}^{p,\kappa-\alpha p} contains Iα⁢Lp,κ{I_{\alpha}L^{p,\kappa}} under the conditions 1