Finite Positive Borel Measure Research Articles

Hilbert type operators acting from the Bloch space into Bergman spaces

Abstract Let µ be a finite positive Borelmeasure on $[0,1)$ and $\alpha \gt -1$ . The generalized integral operator of Hilbert type $\mathcal {I}_{\mu_{\alpha+1}}$ is defined on the spaces $H(\mathbb{D})$ of analytic functions in the unit disc $\mathbb{D}$ as follows: \begin{equation*}\mathcal {I}_{\mu_{\alpha+1}}(f)(z)=\int_{0}^{1} \frac{f(t)}{(1-tz)^{\alpha+1}}d\mu(t),\ \ f\in H(\mathbb{D}),\ \ z\in \mathbb{D} .\end{equation*} In this paper, we give a unified characterization of the measures µ for which the operator $\mathcal {I}_{\mu_{\alpha+1}}$ is bounded from the Bloch space to a Bergman space for all $\alpha \gt -1$ . Additionally, we also investigate the action of $\mathcal {I}_{\mu_{\alpha+1}}$ from the Bloch space to the Hardy spaces and the Besov spaces.

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.

Approximate spectral cosynthesis in the harmonically weighted Dirichlet spaces

For a finite positive Borel measure $\mu$ on the unit circle, let $\mathcal{D}(\mu)$ be the associated harmonically weighted Dirichlet space. A shift invariant subspace $\mathcal{M}$ recognizes strong approximate spectral cosynthesis if there exists a sequence of shift invariant subspaces $\mathcal{M}_k$, with finite codimension, such that the orthogonal projections onto $\mathcal{M}_k$ converge in the strong operator topology to the orthogonal projection onto $\mathcal{M}$. If $\mu$ is a finite sum of atoms, then we show that shift invariant subspaces of $\mathcal{D}(\mu)$ admit strong approximate spectral cosynthesis.

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.

Toeplitz operators on Bergman spaces with exponential weights

In this paper, we focus on the weighted Bergman spaces in with . We first give characterizations of those finite positive Borel measures µ in such that the embedding is bounded or compact for . Then we describe bounded or compact Toeplitz operators from one Bergman space to another for all possible . Finally, we characterize Schatten class Toeplitz operators on .

Carleson measures on the generalized Hartogs triangles

In this paper, we characterize the Carleson measures of the weighted Bergman spaces on a wide class of non-smooth pseudoconvex Reinhardt domains, which contains some generalizations of the classical Hartogs triangle. In addition, we obtain a necessary condition for a finite positive Borel measure μ to be an vanishing Carleson measure of the weighted Bergman space on the generalized Hartogs triangles.

A two weight inequality for Calderón–Zygmund operators on spaces of homogeneous type with applications

Let (X,d,μ) be a space of homogeneous type in the sense of Coifman and Weiss, i.e. d is a quasi metric on X and μ is a positive measure satisfying the doubling condition. Suppose that u and v are two locally finite positive Borel measures on (X,d,μ). Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calderón–Zygmund operator T from L2(u) to L2(v) in terms of the A2 condition and two testing conditions. For every cube B⊂X, we have the following testing conditions, with 1B taken as the indicator of B‖T(u1B)‖L2(B,v)≤T‖1B‖L2(u),‖T⁎(v1B)‖L2(B,u)≤T‖1B‖L2(v).The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.

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).

Weighted Alpert Wavelets

In this paper we construct a wavelet basis in $$L^{2}({\mathbb {R}}^{n};\mu )$$ possessing vanishing moments of a fixed order for a general locally finite positive Borel measure $$\mu $$ . The approach is based on a clever construction of Alpert in the case of the Lebesgue measure that is appropriately modified to handle the general measures considered here. We then use this new wavelet basis to study a two-weight inequality for a general Calderón–Zygmund operator on $${\mathbb {R}}$$ and conjecture that under suitable natural conditions, including a weaker energy condition, the operator is bounded from $$L^{2}({\mathbb {R}};\sigma )$$ to $$L^{2}({\mathbb {R}};\omega )$$ if certain stronger testing conditions hold on polynomials. An example is provided showing that this conjecture is logically different from existing results in the literature.

The two weight T1 theorem for fractional Riesz transforms when one measure is supported on a curve

Let σ and ω be locally finite positive Borel measures on ℝn. We assume that at least one of the two measures σ and ω is supported on a regular C1,δ curve in ℝn. Let Rα,n be the α-fractional Riesz transform vector on ℝn. We prove the T1 theorem for Rα,n: namely that Rα,n is bounded from L2(σ) to L2(σ) if and only if the \({\cal A}_2^\alpha \) conditions with holes hold, the punctured \(A_2^\alpha \) conditions hold, and the cube testing condition for Rα,n and its dual both hold. The special case of the Cauchy transform, n = 2 and α = 1, when the curve is a line or circle, was established by Lacey, Sawyer, Shen, Uriarte-Tuero and Wick in [LaSaShUrWi]. This T1 theorem represents essentially the most general T1 theorem obtainable by methods of energy reversal. More precisely, for the pushforwards of the measures σ and ω, under a change of variable to straighten out the curve to a line, we use reversal of energy to prove that the quasienergy conditions in [SaShUr7] are implied by the \({\cal A}_2^\alpha \) with holes, punctured \(A_2^\alpha \), and quasicube testing conditions for Rα,n. Then we apply the main theorem in [SaShUr7] to deduce the T1 theorem above.

Rational approximation and Sobolev-type orthogonality

In this paper, we study the sequence of orthogonal polynomials {Sn}n=0∞ with respect to the Sobolev-type inner product 〈f,g〉=∫−11f(x)g(x)dμ(x)+∑j=1Nηjf(dj)(cj)g(dj)(cj)where μ is a finite positive Borel measure whose support suppμ⊂[−1,1] contains an infinite set of points, ηj>0, N,dj∈Z+ and {c1,…,cN}⊂R∖[−1,1]. Under some restriction of order in the discrete part of 〈⋅,⋅〉, we prove that for sufficiently large n the zeros of Sn are real, simple, n−N of them lie on (−1,1) and each of the mass points cj “attracts” one of the remaining N zeros.The sequences of associated polynomials {Sn[k]}n=0∞ are defined for each k∈Z+. If μ is in the Nevai class M(0,1), we prove an analogue of Markov’s Theorem on rational approximation to Markov type functions and prove that convergence takes place with geometric speed.

A two weight local $Tb$ theorem for the Hilbert transform

We obtain a two weight local Tb theorem for any elliptic and gradient elliptic fractional singular integral operator T^{\alpha} on the real line \mathbb{R} , and any pair of locally finite positive Borel measures (\sigma,\omega) on \mathbb{R} . The Hilbert transform is included in the case \alpha = 0 , and is bounded from L^{2}(\sigma) to L^{2}(\omega) if and only if the Muckenhoupt and energy conditions hold, as well as b_{Q} and b_{Q}^{\ast} testing conditions over intervals Q , where the families \{b_{Q}\} and \{b_{Q}^{\ast}\} are p -weakly accretive for some p > 2 . A number of new ideas are needed to accommodate weak goodness, including a new method for handling the stubborn nearby form, and an additional corona construction to deal with the stopping form. In a sense, this theorem improves the T1 theorem obtained by the authors and M. Lacey.

Characterizations of Positive Operator-Monotone Functions and Monotone Riemannian Metrics via Borel Measures

We show that there is a one-to-one correspondence between positive operator-monotone functions on the positive reals, monotone Riemannian metrics, and finite positive Borel measures on the unit interval. This correspondence appears as an integral representation of weighted harmonic means with respect to that measure on the unit interval. We also investigate the normalized/symmetric conditions for operator-monotone functions. These conditions turn out to characterize monotone metrics and Morozowa–Chentsov functions as well. Concrete integral representations of such functions related to well-known monotone metrics are also provided. Moreover, we use this integral representation to decompose positive operator-monotone functions. Such decomposition gives rise to a decomposition of the associated monotone metric.

Type alternative for Frostman measures

For a finite positive Borel measure μ on R its exponential type, Tμ, is defined as the infimum of a>0 such that finite linear combinations of complex exponentials with frequencies between 0 and a are dense in L2(μ). The definition can be easily extended from finite to broader classes of measures. In this paper we prove a new formula for Tμ and use it to study growth and additivity properties of measures with finite positive type. As one of the applications, we show that Frostman measures on R may only have type zero or infinity.

Cyclicity in Dirichlet Spaces

AbstractLet $\unicode[STIX]{x1D707}$ be a positive finite Borel measure on the unit circle and ${\mathcal{D}}(\unicode[STIX]{x1D707})$ the associated harmonically weighted Dirichlet space. In this paper we show that for each closed subset $E$ of the unit circle with zero $c_{\unicode[STIX]{x1D707}}$ -capacity, there exists a function $f\in {\mathcal{D}}(\unicode[STIX]{x1D707})$ such that $f$ is cyclic (i.e., $\{pf:p\text{ is a polynomial}\}$ is dense in ${\mathcal{D}}(\unicode[STIX]{x1D707})$ ), $f$ vanishes on $E$ , and $f$ is uniformly continuous. Next, we provide a sufficient condition for a continuous function on the closed unit disk to be cyclic in ${\mathcal{D}}(\unicode[STIX]{x1D707})$ .

Kernel and capacity estimates in Dirichlet spaces

We study the capacity in the sense of Beurling–Deny associated with the Dirichlet space D(μ) where μ is a finite positive Borel measure on the unit circle. First, we obtain a sharp asymptotic estimate of the norm of the reproducing kernel of D(μ). It allows us to give an estimate of the capacity of points and arcs of the unit circle. We also provide new conditions on closed sets to be polar. Our method is based on sharp estimates of norms of some outer functions, which allow us to transfer these problems to an estimate of the reproducing kernel of an appropriate weighted Sobolev space.

QpSpaces and Dirichlet Type Spaces

AbstractIn this paper, we show that the Möbius invariant function space Qpcan be generated by variant Dirichlet type spaces 𝒟μ,pinduced by finite positive Borel measures μ on the open unit disk. A criterion for the equality between the space 𝒟μ,pand the usual Dirichlet type space 𝒟pis given. We obtain a sufficient condition to construct different 𝒟μ,pspaces and provide examples. We establish decomposition theorems for 𝒟μ,pspaces and prove that the non-Hilbert space Qpis equal to the intersection of Hilbert spaces 𝒟μ,p. As an application of the relation between Qpand 𝒟μ,pspaces, we also obtain that there exist different 𝒟μ,pspaces; this is a trick to prove the existence without constructing examples.

Asymptotics for varying discrete Sobolev orthogonal polynomials

We consider a varying discrete Sobolev inner product such as(f,g)S=∫f(x)g(x)dμ+Mnf(j)(c)g(j)(c),where μ is a finite positive Borel measure supported on an infinite subset of the real line, c is adequately located on the real axis, j≥0, and {Mn}n≥0 is a sequence of nonnegative real numbers satisfying a very general condition. Our aim is to study asymptotic properties of the sequence of orthonormal polynomials with respect to this Sobolev inner product. In this way, we focus our attention on Mehler-Heine type formulae as they describe in detail the asymptotic behavior of these polynomials around c, just the point where we have located the perturbation of the standard inner product. Moreover, we pay attention to the asymptotic behavior of the (scaled) zeros of these varying Sobolev polynomials and some numerical experiments are shown. Finally, we provide other asymptotic results which strengthen the idea that Mehler–Heine asymptotics describe in a precise way the differences between Sobolev orthogonal polynomials and standard ones.

Existence of Continuous Functions That Are One-to-One Almost Everywhere

It is shown that given a metric space $X$ and a $\sigma$-finite positive regular Borel measure $\mu$ on $X$, there exists a bounded continuous real-valued function on $X$ that is one-to-one on the complement of a set of $\mu$ measure zero.