Borel Measure Research Articles

Orthogonal Laurent Polynomials of Two Real Variables

ABSTRACTIn this paper, we consider an appropriate ordering of the Laurent monomials , that allows us to study sequences of orthogonal Laurent polynomials of the real variables and with respect to a positive Borel measure defined on such that . This ordering is suitable for considering the multiplication plus inverse multiplication operator on each variable and , and as a result we obtain five‐term recurrence relations, Christoffel–Darboux and confluent formulas for the reproducing kernel and a related Favard's theorem. A connection with the one‐variable case is also presented, along with some applications for future research.

Weighted Dirichlet spaces that are de Branges-Rovnyak spaces with equivalent norms

For a positive Borel measure μ in the unit disc, we examine which weighted Dirichlet spaces Dμ can be identified with de Branges-Rovnyak spaces Hb, with equivalent norms. We prove a necessary condition for the equality Dμ=Hb and we explore its consequences. For Carleson measures μ, we give a necessary and sufficient condition for the equality Dμ=Hbμ, for a certain outer function bμ related with the balayage of μ on the unit circle, and we provide examples of those spaces.

Finite Sum of Integral Operators from the Fractional Cauchy Spaces to Bloch-Type and Zygmund-Type Spaces

The family of fractional Cauchy transforms, defined on the open unit disc in the complex plane, is of classical and modern interest. Membership of an analytic function in the family is determined by the requirement that the function can be expressed as an integral of a certain kernel against a complex Borel measure on the disc. Such an integral representation imposes a growth condition on the function and its derivatives. This exposes a connection between the families of Cauchy transforms and familiar spaces of analytic functions, such as the Bloch spaces and the Zygmund space. The notion of a composition operator has been a fruitful area of study. More generally, many authors have studied weighted composition operators, the differentiation operator, integral-type operators, and various products of such operators, acting from one normed linear space of analytic functions to another such space. A common theme of such works is to characterize the operator-theoretic notions of boundedness and compactness in terms of the inducing symbols of the operator. We extend these studies to a specific linear transformation which will be defined as the sum of finitely many integral operators. Our conclusions include a complete characterization of boundedness and compactness of the integral sum, acting from the fractional Cauchy spaces to the Bloch-type and Zygmund-type spaces.

Amenability of finite energy path and loop groups

It is shown that the groups of finite energy (i.e., Sobolev class H^{1} ) paths and loops with values in a compact Lie group are amenable in the sense of Pierre de la Harpe, that is, every continuous action of such a group on a compact space admits an invariant regular Borel probability measure. To our knowledge, the strongest previously known result concerned the amenability of groups of continuous paths and loops (Malliavin and Malliavin 1992).

The energy operator on a generalized Kato class

In this paper we introduce and study the classes of signed Borel measure Mp(Rn)as well as $\widetilde{\mathcal{M}}_p(Rn) (1 < p < n) which are generalization of the Kato class. For these classes we obtain an upper estimate for the energy operator

On sequences of finitely supported measures related to the Josefson–Nissenzweig theorem

Given a Tychonoff space X, we call a sequence 〈μn:n∈ω〉 of signed Borel measures on X a finitely supported Josefson–Nissenzweig sequence (in short a JN-sequence) if: 1) for every n∈ω the measure μn is a finite combination of one-point measures and ‖μn‖=1, and 2) ∫Xfdμn→0 for every continuous function f∈C(X). Our main result asserts that if a Tychonoff space X admits a JN-sequence, then there exists a JN-sequence 〈μn:n∈ω〉 such that: i) supp(μn)∩supp(μk)=∅ for every n≠k∈ω, and ii) the union ⋃n∈ωsupp(μn) is a discrete subset of X. We also prove that if a Tychonoff space X carries a JN-sequence, then either there is a JN-sequence 〈μn:n∈ω〉 on X such that |supp(μn)|=2 for every n∈ω, or for every JN-sequence 〈μn:n∈ω〉 on X we have limn→∞⁡|supp(μn)|=∞.

Cesàro operators associated with Borel measures acting on weighted spaces of holomorphic functions with sup-norms

Let μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document} be a positive finite Borel measure on [0, 1). Cesàro-type operators Cμ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\\mu }$$\\end{document} when acting on weighted spaces of holomorphic functions are investigated. In the case of bounded holomorphic functions on the unit disc we prove that Cμ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_\\mu $$\\end{document} is continuous if and only if it is compact. In the case of weighted Banach spaces of holomorphic function defined by general weights, we give sufficient and necessary conditions for the continuity and compactness. For standard weights, we characterize the continuity and compactness on classical growth Banach spaces of holomorphic functions. We also study the point spectrum and the spectrum of Cμ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_\\mu $$\\end{document} on the space of holomorphic functions on the disc, on the space of bounded holomorphic functions on the disc, and on the classical growth Banach spaces of holomorphic functions. All characterizations are given in terms of the sequence of moments (μn)n∈N0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\mu _n)_{n\\in {\\mathbb {N}}_0}$$\\end{document}. The continuity, compactness and spectrum of Cμ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_\\mu $$\\end{document} acting on Fréchet and (LB) Korenblum type spaces are also considered .

Generalized Hilbert operators arising from Hausdorff matrices

For a finite, positive Borel measure μ \mu on ( 0 , 1 ) (0,1) we consider an infinite matrix Γ μ \Gamma _\mu , related to the classical Hausdorff matrix defined by the same measure μ \mu , in the same algebraic way that the Hilbert matrix is related to the Cesáro matrix. When μ \mu is the Lebesgue measure, Γ μ \Gamma _\mu reduces to the classical Hilbert matrix. We prove that the matrices Γ μ \Gamma _\mu are not Hankel, unless μ \mu is a constant multiple of the Lebesgue measure, we give necessary and sufficient conditions for their boundedness on the scale of Hardy spaces H p , 1 ≤ p > ∞ H^p, \, 1 \leq p > \infty , and we study their compactness and complete continuity properties. In the case 2 ≤ p > ∞ 2\leq p>\infty , we are able to compute the exact value of the norm of the operator.

Metric geometry of spaces of persistence diagrams

AbstractPersistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a family of functors $${\mathcal {D}}_p$$ D p , $$1\le p \le \infty $$ 1 ≤ p ≤ ∞ , that assign, to each metric pair (X, A), a pointed metric space $${\mathcal {D}}_p(X,A)$$ D p ( X , A ) . Moreover, we show that $${\mathcal {D}}_{\infty }$$ D ∞ is sequentially continuous with respect to the Gromov–Hausdorff convergence of metric pairs, and we prove that $${\mathcal {D}}_p$$ D p preserves several useful metric properties, such as completeness and separability, for $$p \in [1,\infty )$$ p ∈ [ 1 , ∞ ) , and geodesicity and non-negative curvature in the sense of Alexandrov, for $$p=2$$ p = 2 . For the latter case, we describe the metric of the space of directions at the empty diagram. We also show that the Fréchet mean set of a Borel probability measure on $${\mathcal {D}}_p(X,A)$$ D p ( X , A ) , $$1\le p \le \infty $$ 1 ≤ p ≤ ∞ , with finite second moment and compact support is non-empty. As an application of our geometric framework, we prove that the space of Euclidean persistence diagrams, $${\mathcal {D}}_{{p}}({\mathbb {R}}^{2n},\Delta _n)$$ D p ( R 2 n , Δ n ) , $$1\le n$$ 1 ≤ n and $$1\le p<\infty $$ 1 ≤ p < ∞ , has infinite covering, Hausdorff, asymptotic, Assouad, and Assouad–Nagata dimensions.

Bauer simplices and the small boundary property

We show that, for every minimal action of a countably infinite discrete group on a compact metrizable space, if the extreme boundary of the simplex of invariant Borel probability measures is closed and has finite covering dimension then the action has the small boundary property.

Fractional boundary value problems and elastic sticky brownian motions

AbstractWe extend the results obtained in [14] by introducing a new class of boundary value problems involving non-local dynamic boundary conditions. We focus on the problem to find a solution to a local problem on a domain $$\varOmega $$ Ω with non-local dynamic conditions on the boundary $$\partial \varOmega $$ ∂ Ω . Due to the pioneering nature of the present research, we propose here the apparently simple case of $$\varOmega =(0, \infty )$$ Ω = ( 0 , ∞ ) with boundary $$\{0\}$$ { 0 } of zero Lebesgue measure. Our results turn out to be instructive for the general case of boundary with positive (finite) Borel measures. Moreover, in our view, we bring new light to dynamic boundary value problems and the probabilistic description of the associated models.

Regularity with Respect to the Parameter of Lyapunov Exponents for Diffeomorphisms with Dominated Splitting

We consider families of diffeomorphisms with dominated splittings and preserving a Borel probability measure, and we study the regularity of the Lyapunov exponents associated to the invariant bundles with respect to the parameter. We obtain that the regularity is at least the sum of the regularities of the two invariant bundles (for regularities in [ 0 , 1 ] [0,1] ), and under suitable conditions we obtain formulas for the derivatives. Similar results are obtained for families of flows, and for the case when the invariant measure depends on the map. We also obtain several applications. Near the time one map of a geodesic flow of a surface of negative curvature the metric entropy of the volume is Lipschitz with respect to the parameter. At the time one map of a geodesic flow on a manifold of constant negative curvature the topological entropy is differentiable with respect to the parameter, and we give a formula for the derivative. Under some regularity conditions, the critical points of the Lyapunov exponent function are nonflat (the second derivative is nonzero for some families). Also, again under some regularity conditions, the criticality of the Lyapunov exponent function implies some rigidity of the map, in the sense that the volume decomposes as a product along the two complementary foliations. In particular for area preserving Anosov diffeomorphisms, the only critical points are the maps smoothly conjugated to the linear map, corresponding to the global extrema.

A moment approach for entropy solutions of parameter-dependent hyperbolic conservation laws

We propose a numerical method to solve parameter-dependent scalar hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work from Marx et al. (2020). This approach relies on a very weak notion of solution of nonlinear equations, namely parametric entropy measure-valued (MV) solutions, satisfying linear equations in the space of Borel measures. The infinite-dimensional linear problem is approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems, called Lasserre’s hierarchy. This gives us a sequence of approximations of the moments of the occupation measure associated with the parametric entropy MV solution, which is proved to converge. In the end, several post-treatments can be performed from this approximate moments sequence. In particular, the graph of the solution can be reconstructed from an optimization of the Christoffel–Darboux kernel associated with the approximate measure, that is a powerful approximation tool able to capture a large class of irregular functions. Also, for uncertainty quantification problems, several quantities of interest can be estimated, sometimes directly such as the expectation of smooth functionals of the solutions. The performance of our approach is evaluated through numerical experiments on the inviscid Burgers equation with parametrised initial conditions or parametrised flux function.

Bilevel Optimization of the Kantorovich Problem and Its Quadratic Regularization

This paper is concerned with an optimization problem which is governed by the Kantorovich problem of optimal transport. More precisely, we consider a bilevel optimization problem with the underlying problem being the Kantorovich problem. This task can be reformulated as a mathematical problem with complementarity constraints in the space of regular Borel measures. Because of the non-smoothness that is induced by the complementarity constraints, problems of this type are often regularized, e.g., by an entropic regularization. However, in this paper we apply a quadratic regularization to the Kantorovich problem. By doing so, we are able to drastically reduce its dimension while preserving the sparsity structure of the optimal transportation plan as much as possible. As the title indicates, this is the second part in a series of three papers. While the existence of optimal solutions to both the bilevel Kantorovich problem and its regularized counterpart were shown in the first part, this paper deals with the (weak-∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}) convergence of solutions to the regularized bilevel problem to solutions of the original bilevel Kantorovich problem for vanishing regularization parameters.

Point-set games and functions with the hereditary small oscillation property

Given a metric space X, we consider certain families of functions f:X→R having the hereditary oscillation property HSOP and the hereditary continuous restriction property HCRP on large sets. When X is Polish, among them there are families of Baire measurable functions, μ‾-measurable functions (for a finite nonatomic Borel measure μ on X) and Marczewski measurable functions. We obtain their characterizations using a class of equivalent point-set games. In similar aspects, we study cliquish functions, SZ-functions and countably continuous functions.

Boundedness of the dyadic maximal function on graded Lie groups

Abstract Let $1 \lt p\leq \infty$ and let $n\geq 2.$ It was proved independently by Calderón, Coifman and Weiss that the dyadic maximal function $$ \mathcal{M}^{d\sigma}_Df(x)=\sup_{j\in\mathbb{Z}}\left|\smallint\limits_{\mathbb{S}^{n-1}}f(x-2^jy)d\sigma(y)\right| \\[4pt] $$ is a bounded operator on $L^p(\mathbb{R}^n)$, where $d\sigma(y)$ is the surface measure on $\mathbb{S}^{n-1}.$ In this paper we prove an analogue of this result on arbitrary graded Lie groups. More precisely, to any finite Borel measure $d\sigma$ with compact support on a graded Lie group $G,$ we associate the corresponding dyadic maximal function $\mathcal{M}_D^{d\sigma}$ using the homogeneous structure of the group. Then, we prove a criterion in terms of the order (at zero and at infinity) of the group Fourier transform $\widehat{d\sigma}$ of $d\sigma$ with respect to a fixed Rockland operator $\mathcal{R}$ on G that assures the boundedness of $\mathcal{M}_D^{d\sigma}$ on $L^p(G)$ for all $1 \lt p\leq \infty.$

Generalized Cesàro operators on weighted Dirichlet spaces

Given a complex Borel measure η∈M(D), we study the boundedness of the Cesàro-type operator Cη given byCη(f)(z)=∑n=0∞(∫Dwndη(w))(∑k=0nak)zn where f(z)=∑n=0∞anzn, acting on the weighted Dirichlet spaces Dρ consisting in functions f∈H(D) with ∑n=0∞|an|2(n+1)ρn<∞ where (ρn) is a sequence of positive real numbers. We get new estimates on Taylor coefficients of functions in the Dirichlet space D0 and extend the results known for Dα and positive measures ν supported in [0,1) to more general weights and general measures.

General fractal dimensions of typical sets and measures

Consider (Y,ρ) as a complete metric space and S as the space of probability Borel measures on Y. Let dim‾BΨ,Φ(E) be the general upper box dimension of the set E⊂Y. We begin by proving that the general packing dimension of the typical compact set, in the sense of the Baire category, is at least inf⁡{dim‾BΨ,Φ(B(x,r))|x∈Y,r>0} where B(x,r) is the closed ball in Y with center at x and radii r>0. Next, we obtain some estimates of the general upper and lower box dimensions of typical measures in the sense of the Baire category. Finally, we demonstrate that if S is equipped with the weak topology and under some assumptions then the set of measures possessing the general upper and lower correlation dimension zero are residual. Furthermore, the general upper correlation dimension of typical measures (in the sense of the Baire category) is approximated through the general local lower and upper entropy dimensions of Y.

On points avoiding measures

We say that an element x of a topological space X avoids measures if for every Borel measure μ on X if μ({x})=0, then there is an open U∋x such that μ(U)=0. The negation of this property can viewed as a local version of the property of supporting a strictly positive measure. We study points avoiding measures in the general setting as well as in the context of ω⁎, the remainder of Stone-Čech compactification of ω.

On the support of measures with fixed marginals with applications in optimal mass transportation

Abstract Let $\mu $ and $\nu $ be Borel probability measures on complete separable metric spaces X and Y, respectively. Each Borel probability measure $\gamma $ on $X\times Y$ with marginals $\mu $ and $\nu $ can be described through its disintegration $\big ( \gamma _{x}\big )_{x \in X}$ with respect to the initial distribution $\mu .$ Assume that $\mu $ is continuous, i.e., $\mu \big (\{x\}\big )=0$ for all $x \in X.$ We shall analyze the structure of the support of the measure $\gamma $ provided $\text {card } \big (\mathrm{spt} (\gamma _{x}) \big )$ is finitely countable for $\mu $ -a.e. $x\in X.$ We shall also provide an application to optimal mass transportation.