Finite Positive Measure Research Articles

Nodal sets and continuity of eigenfunctions of Krein-Feller operators

Let \(\mu\) be a compactly supported positive finite Borel measure on \(\mathbb{R}^d\). Let \(0<\lambda_1\leq\lambda_2\leq\cdots\) be eigenvalues of the Krein-Feller operator \(\Delta_{\mu}\). We prove that, on a bounded domain, the nodal set of a continuous \(\lambda_n\)-eigenfunction of a Krein-Feller operator divides the domain into at least 2 and at most \(n+r_n-1\) subdomains, where \(r_n\) is the multiplicity of \(\lambda_n\). This work generalizes the nodal set theorem of the classical Laplace operator to Krein-Feller operators on bounded domains. We also prove that on bounded domains on which the classical Green function exists, the eigenfunctions of a Krein-Feller operator are continuous. For more information see https://ejde.math.txstate.edu/Volumes/2025/12/abstr.html

Operators of Hilbert and Cesàro type acting on Dirichlet spaces

If α>-1-1$$\\end{document}]]> the space of Dirichlet type Dα2 consists of those functions f which are analytic in the unit disc D such that f′ belongs to the weighted Bergman space Aα2. The space D02 is the classical Dirichlet space D. If g is an analytic function in D, we study the generalized Hilbert operator Hg defined by Hg(f)(z)=∫01f(t)g′(tz)dtacting on the spaces Dα2 (0≤α≤1). We obtain a characterization of those g for which Hg is bounded, compact, or Hilbert-Schmidt on the Dirichlet space D. In addition to this, we use our results concerning the operators Hg to study certain Cesàro-type operators C(η) acting on the spaces Dα2 (0≤α≤1). We give also a characterization of the positive finite Borel measures μ in [0, 1) for which a certain Cesàro type operator Cμ associated to μ is bounded on the Bergman space Aα1 (α>-1-1$$\\end{document}]]>). This is an extension of the previously known results for the spaces Aαp with p>11$$\\end{document}]]> and α>-1-1$$\\end{document}]]>.

Transcendental Julia sets of positive finite Lebesgue measure

We show that there exists a transcendental entire function whose Julia set has positive finite Lebesgue measure.

On spaces of integrable functions associated to vector measures and limiting real interpolation

We investigate which spaces are obtained when considering the limiting class of real interpolation spaces (0,q;J) for ordered Banach couples of spaces of (scalar) integrable functions with respect to a vector measure m, defined on a σ-algebra, with values in a Banach space. If m is in particular a finite positive scalar measure, previous known results are derived from ours. Furthermore, we study the interpolation of p-th power factorable operators by the extreme real interpolation method (1,q;K). We also deduce interpolation results for the (1,q;K)-method that apply to other related classes of operators to p-th power factorable operators, such as bidual (p,q)-power-concave operators and q-concave operators.

SUBSETS OF POSITIVE AND FINITE MULTIFRACTAL MEASURES

Sets of infinite multifractal measures are awkward to work with, and reducing them to sets of positive finite multifractal measures is a very useful simplification. The aim of this paper is to show that the multifractal Hausdorff measures satisfy the “subset of positive and finite measure” property. We apply our main result to prove that the multifractal function dimension is defined as the supremum over the multifractal dimension of all Borel probability measures.

Invertibility in weak-star closed algebras of analytic functions

For K⊂C a compact subset and μ a positive finite Borel measure supported on K, let R∞(K,μ) be the weak-star closure in L∞(μ) of rational functions with poles off K. We show that if R∞(K,μ) has no non-trivial L∞ summands and f∈R∞(K,μ), then f is invertible in R∞(K,μ) if and only if Chaumat's map for K and μ applied to f is bounded away from zero on the envelope with respect to K and μ. The result proves the conjecture ⋄ posed by J. Dudziak [6] in 1984.

Explicit expressions and computational methods for the Fortet–Mourier distance of positive measures to finite weighted sums of Dirac measures

Explicit expressions and computational approaches are given for the Fortet–Mourier distance between a positively weighted sum of Dirac measures on a metric space and a positive finite Borel measure. Explicit expressions are given for the distance to a single Dirac measure. For the case of a sum of several Dirac measures one needs to resort to a computational approach. In particular, two algorithms are given to compute the Fortet–Mourier norm of a molecular measure, i.e. a finite weighted sum of Dirac measures. It is discussed how one of these can be modified to allow computation of the dual bounded Lipschitz (or Dudley) norm of such measures.

A new definition of random set

A new definition of random sets is proposed in the presented paper. It is based on a special distance in a measurable space and uses negative definite kernels for continuation from the initial space to the one of the random sets. Motivation for introducing the new definition is that the classical approach deals with Hausdorff distance between realisations of the random sets, which is not satisfactory for statistical analysis in many cases. We place the realisations of the random sets in a complete Boolean algebra (B.A.) endowed with a positive finite measure intended to capture important characteristics of the realisations. A distance on B.A. is introduced as a square root of measure of symmetric difference between its two elements. The distance is then used to define a class of Borel subsets of B.A. Consequently, random sets are defined as measurable mappings taking values in the B.A. This approach enables us to use more general family of distances between realisations of random sets which allows us to make new statistical tests concerning equality of some characteristics of random set distributions. As an extra result, the notion of stability of newly defined random sets with respect to intersections is proposed and limit theorems are obtained.

Differentiation of measures on an arbitrary measurable space

Let μ,ν be positive finite measures on an arbitrary measurable space (Ω,F), ν=νa+νs be the Lebesgue decomposition of ν with respect to μ, P be the family of all finite partitions π⊆F of Ω, andfπ(μ):=∑A∈π:μ(A)>01Aν(A)μ(A),π∈P. We recall that (fπ(μ))π∈P→dνadμ in L1(μ), as is (essentially) known. Here we identify(πn)n∈N⊆P such thatfπn(μ)→dνadμμ a.s. as n→∞; in the setting of separable F, a (rather trivial) way to do this was already known. To do all this, we characterise the case of equality in Jensen's conditional inequality (generalising the known case of the standard Jensen inequality), and use this to determine how, given a p-uniformly integrable martingale (fi)i∈I, one can identify a sequence(in)n∈N⊆I such that (fin)n converges in Lp to some f which closes the whole net(fi)i. We also give a new proof of the (already-known) characterisation of p-uniformly integrable martingales, without relying on the martingale a.s. convergence theorem.

Cauchy transform and uniform approximation by polynomial modules

For a compact subset K of the complex plane C, let C(K) denote the algebra of continuous functions on K. For an open subset U⊂K, let A(K,U)⊂C(K) be the algebra of functions that are analytic in U. We show that there exists ϕ∈A(K,U) so that each f∈A(K,U) can uniformly be approximated by {pn+qnϕ} on K, where pn and qn are analytic polynomials in z. In particular, ϕ can be chosen as a Cauchy transform of a finite positive measure η compactly supported in C∖U. Recent developments of analytic capacity and Cauchy transform provide us useful tools in our proofs.

A Sweeping Process Control Problem Subject To Mixed Constraints

In this study, we investigate optimal control problems that involve sweeping processes with a drift term and mixed inequality constraints. Our goal is to establish necessary optimality conditions for these problems. We address the challenges that arise due to the combination of sweeping processes and inequality mixed constraints in two contexts: regular and non-regular. This requires working with different types of multipliers, such as finite positive Radon measures for the sweeping term and integrable functions for regular mixed constraints. For non-regular mixed constraints, the multipliers correspond to purely finitely additive set functions.

Calderón–Zygmund operators on RBMO

Let μ \mu be an n n -dimensional finite positive measure on R m \mathbb {R}^m . We obtain a T 1 T1 condition sufficient for the boundedness of Calderón–Zygmund operators on R B M O ( μ ) \mathrm {RBMO}(\mu ) , the regular BMO space of Tolsa [Math. Ann. 319 (2001), pp. 89–149].

Extremal functions and invariant subspaces in Dirichlet spaces

Let μ be a positive finite Borel measure on the unit circle of the complex plane, and let D(μ) be the associated Dirichlet space. The Beurling-Deny capacity associated with D(μ) is denoted by cμ. Brown-Shields conjecture for D(μ) says that a function f∈D(μ) is cyclic for D(μ), meaning that {pf:pis a polynomial} is dense in D(μ), if and only if f is outer and the boundary zeros set of f is of cμ− capacity zero. It was recently proved, using Banach algebras techniques, that Brown-Shields conjecture is true for D(μ) whenever the support of μ is countable. In this paper, we produce a new class of measures for which Brown-Shields conjecture holds. This class includes the canonical measures on Cantor sets. For these measures, we also give an explicit description of all invariant subspaces for the shift operator, i.e. the operator of multiplication by the independent variable. Our method is based on a study of the behavior of extremal functions for Dirichlet spaces. We prove that if the support of μ is a Carleson set, then every outer extremal function φ can be explicitly recovered from the restriction of its boundary values |φ⁎| on the support of μ. This allows us to give estimates of φ for a large class of measures.

Discrete analytic functions, structured matrices and a new family of moment problems

Using Zeilberger generating function formula for the values of a discrete analytic function in a quadrant we make connections with the theory of structured reproducing kernel spaces, structured matrices and a generalized moment problem. An important role is played by a Krein space realization result of Dijksma, Langer and de Snoo for functions analytic in a neighborhood of the origin. A key observation is that, using a simple Moebius transform, one can reduce the study of discrete analytic functions in the upper right quadrant to problems of function theory in the open unit disk. As an example, we associate to each finite positive measure on [0,2π] a discrete analytic function on the right-upper quarter plane with values on the lattice defining a positive definite function. Emphasis is put on the rational case, both when an underlying Carathéodory function is rational and when, in the positive case, the spectral function is rational. The rational case and the general case are linked via the existence of a unitary dilation, possibly in a Krein space.

Новые свойства рекуррентных движений и предельных множеств динамических систем

In the earlier article by the authors [A.P. Afanas’ev, S. M. Dzyuba “About new properties of recurrent motions and minimal sets of dynamical systems”, Russian Universities Reports. Mathematics, 26:133 (2021), 5–14] a connection between general motions and recurrent motions in a compact metric space is established, and a very simple behavior of recurrent motions is proved. Based on these results, we introduce here a new definition of recurrent motion which, in contrast to the one widely used in modern literature, provides fairly complete information about the structure of a recurrent motion as a function of time and, therefore, is more illustrative. At the same time, we show that in an abstract metric space, the proposed definition is equivalent to Birkhoff’s definition and is equivalent to the generally accepted modern definition in a complete metric space. Necessary and sufficient conditions for recurrence (in the sense of the definition proposed in the article) of a motion in a compact metric space are obtained. It is proved that α- and ω-limit sets of any motion are minimal in a compact metric space (this assertion was announced in an earlier paper by the authors). From the minimality of α- and ω-limit sets, it is deduced that in a compact metric space, each positively (negatively) Poisson-stable point lies on the trajectory of a recurrent motion, i.e. is a point of a minimal set, and thus, in a compact metric space with a finite positive invariant measure almost all points are points of minimal sets.

Steffensen Type Inequalites for Convex Functions on Borel σ-Algebra

In the paper, we prove Steffensen type inequalities for positive finite measures by using functions which are convex in point. Further, we prove Steffensen type inequalities on Borel σ-algebra for the function of the form f/h which is convex in point. We conclude the paper by showing that these results also hold for convex functions.

English

Let μ be a finite positive measure on the unit disk and let j ⩾ 1 be an integer. D. Suarez (2015) gave some conditions for a generalized Toeplitz operator $$T_\mu ^{(j)}$$ to be bounded or compact. We first give a necessary and sufficient condition for $$T_\mu ^{(j)}$$ to be in the Schatten p-class for 1 ⩽ p < ∞ on the Bergman space A2, and then give a sufficient condition for $$T_\mu ^{(j)}$$ to be in the Schatten p-class (0 < p < 1) on A2. We also discuss the generalized Toeplitz operators with general bounded symbols. If ϕ ∈ L∞ (D, dA) and 1 < p < ∞, we define the generalized Toeplitz operator $$T_\varphi ^{(j)}$$ on the Bergman space Ap and characterize the compactness of the finite sum of operators of the form $$T_{{\varphi _1}}^{(j)} \ldots T_{{\varphi _n}}^{(j)}$$ .

Besicovitch Pseudodistances with Respect to Non-Følner Sequences

The Besicovitch pseudodistance defined in [1] for biinfinite sequences is invariant by translations. We generalize the definition to arbitrary locally compact second-countable groups and study how properties of the pseudodistance, including invariance by translations, are determined by those of the sequence of sets of finite positive measure used to define it. In particular, we restate from [2] that if the Besicovitch pseudodistance comes from an exhaustive Følner sequence, then every shift is an isometry. For non-Følner sequences, it is proved that some shifts are not isometries, and the Besicovitch pseudodistance with respect to some subsequences even makes them discontinuous.

Havin-Mazya type uniqueness theorem for Dirichlet spaces

Let μ be a positive finite Borel measure on the unit circle. The associated Dirichlet space D(μ) consists of holomorphic functions on the unit disc whose derivatives are square integrable when weighted against the Poisson integral of μ. We give a sufficient condition on a Borel subset E of the unit circle which ensures that E is a uniqueness set for D(μ). We also give some examples of positive Borel measures μ and uniqueness sets for D(μ).

When and where the Orlicz and Luxemburg (quasi-) norms are equivalent?

We study the equivalence between the Orlicz and Luxemburg (quasi-) norms in the context of the generalized Orlicz spaces associated to an N-function Φ and a (quasi-) Banach function space X over a positive finite measure μ. We show that the Orlicz and the Luxemburg spaces do not coincide in general, and also that under mild requirements (σ-Fatou property, strictly monotone renorming) the coincidence holds. We use as a technical tool the classes LwΦ(m), LΦ(m) and LΦ(‖m‖) of Orlicz spaces of scalar integrable functions with respect to a Banach-space-valued countably additive vector measure m, providing also some new results on these spaces.