Supports of Weighted Equilibrium Measures: Complete Characterization

In this paper, we prove that a compact set K???? ⁿ is the support of a weighted equilibrium measure if and only it is not pluripolar at each of its points, extending a result of Saff and Totik to higher dimensions. Thus, we characterize the supports of weighted equilibrium measures completely. Our proof is a new proof even in one dimension.     

On the Supports of Vector Equilibrium Measures in the Angelesco Problem with Nested Intervals

A vector logarithmic-potential equilibrium problem with the Angelesco interaction matrix is considered for two nested intervals with a common endpoint. The ratio of the lengths of the intervals is a parameter of the problem, and another parameter is the ratio of the masses of the components of the vector equilibrium measure. Two cases are distinguished, depending on the relations between the parameters. In the first case, the equilibrium measure is described by a meromorphic function on a three-sheeted Riemann surface of genus zero, and the supports of the components do not overlap and are connected. In the second case, a solution to the equilibrium problem is found in terms of a meromorphic function on a six-sheeted surface of genus one, and the supports overlap and are not connected.     

Weakly n-hyponormal Weighted Shifts and Their Examples

The gap between hyponormal and subnormal Hilbert space operators can be studied using the intermediate classes of weakly n-hyponormal and (strongly) n-hyponormal operators. The main examples for these various classes, particularly to distinguish them, have been the weighted shifts. In this paper we first obtain a characterization for a weakly n-hyponormal weighted shift W_α with weight sequence α, from which we extend some known results for quadratically hyponormal (i.e., weakly 2-hyponormal) weighted shifts to weakly n-hyponormal weighted shifts. In addition, we discuss some new examples for weakly n-hyponormal weighted shifts; one illustrates the differences among the classes of 2-hyponormal, quadratically hyponormal, and positively quadratically hyponormal operators.     

Convergence of Weighted Empirical Measures

This article considers a system with infinitely many interacting particles, starting with a system of n interacting particles that is described by a system of n stochastic differential equations for the time-varying locations and weights. Any particle in the system interacts with others through the weighted empirical measure Uⁿ formed by the sum of weighted Dirac measures on the n particles. Weak convergence of the weighted empirical measure is studied under suitable conditions, such as bounded initial values, and linear growth of drift and diffusion coefficients. Thereafter, the limit of the weighted empirical measures is identified to be a martingale solution of the infinite interacting system.     

Two Examples Related to Properties of Discrete Measures

Mathematical Notes - Two examples illustrating properties of discrete measures are given. In the first part of the paper, it is proved that, for any probability measure $$mu$$ with...     

Disintegration of Measures and Contractive 2-Variable Weighted Shifts

In this paper we give a new proof of the existence of disintegration measures using the Hausdorff Moment Problem on a Borel measurable space X × Y, where X ≡ Y is the unit interval. Using this new tool, we can give an abstract solution, moreover, and a concrete necessary condition for the Lifting Problem for contractive 2-variable weighted shifts. In addition, we have a new, computable, and sufficient condition for the Lifting Problem for 2-variable weighted shifts, and an improved version of the Curto-Muhly-Xia conjecture [8] for 2-variable weighted shifts.     

Green Equilibrium Measures and Representations of an External Field

We establish a representation for external fields involving Green potentials. This is the analogue of the representation of Rakhmanov and Buyarov involving logarithmic potentials. We also establish related results and present an example.     

Weighted Logarithmic Sobolev Inequalities for Sub-Gaussian Measures

In this short paper, We establish the weighted Logarithmic Sobolev inequalities for sub-Gaussian measure of high dimension with explicit constants via phase transition and the well-known Bakry-émery criterion.     

Balayage Ping-Pong: A Convexity of Equilibrium Measures

In this paper we establish that the density of the equilibrium measure of finitely many intervals for both logarithmic and Riesz potentials is convex. The main tool is a balayage ping-pong technique. A?similar result is obtained for finitely many arcs on the unit circle. Applications to external field problems and constrained energy problems are presented.     

Some Supports of Fourier Transforms of Singular Measures are not Rajchman

The notion of Riesz sets tells us that a support of Fourier transform of a measure with non-trivial singular part has to be large. The notion of Rajchman sets tells us that if the Fourier transform tends to zero at infinity outside a small set, then it tends to zero even on the small set. Here we present a new angle of an old question: Whether every Rajchman set should be Riesz.     

非双倍测度下的Marcinkiewicz积分的加权估计

Let μ be a nonnegative Radon measure on R d which satisfies the growth condition μ(B(x,r)) ≤ C0rn for all x ∈Rd and r >0,where C0 is a fixed constant and 0相似文献   

加权Bergman空间上的Carleson测度与Toeplitz算子

探讨加权Bergman空间Ap()上的Carleson型测度和具有非负测度符号的Toeplitz算子,给出Carleson测度或消没Carleson测度的若干等价描述并用Carleson测度的方法刻画了Toeplitz算子是有界的或紧致的充要条件.     

On Weighted Toeplitz, Big Hankel Operators and Carleson Measures

We compute the norm of pointwise multiplication operators, Toeplitz and Big Hankel operators with antiholomorphic symbols, defined on Besov spaces. These norms will be given in terms of Carleson measures for Besov spaces related to the symbol.     

非双倍测度的Marcinkiewicz积分交换子的加权估计

设μ是R~d上的非负Radon测度,且满足增长性条件:存在一正常数C_0,使得对任意的x∈R~d和r0,有μ(B(x,r))≤C_0r~n,其中0n≤d.该文研究了相关于非双倍测度μ的Marcinkiewicz积分与RBMO函数生成的交换子,得到了这类交换子的加A_p~p(μ)权的弱型估计.     

Equilibrium Diffusion on the Cone of Discrete Radon Measures

Let \({\mathbb {K}(\mathbb {R}^{d})}\) denote the cone of discrete Radon measures on \(\mathbb {R}^{d}\). There is a natural differentiation on \(\mathbb {K}(\mathbb {R}^{d})\): for a differentiable function \(F:\mathbb {K}(\mathbb {R}^{d})\to \mathbb {R}\), one defines its gradient \(\nabla ^{\mathbb {K}}F\) as a vector field which assigns to each \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) an element of a tangent space \(T_{\eta }(\mathbb {K}(\mathbb {R}^{d}))\) to \(\mathbb {K}(\mathbb {R}^{d})\) at point η. Let \(\phi :\mathbb {R}^{d}\times \mathbb {R}^{d}\to \mathbb {R}\) be a potential of pair interaction, and let μ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on \(\mathbb {R}^{d}\). In particular, μ is a probability measure on \(\mathbb {K}(\mathbb {R}^{d})\) such that the set of atoms of a discrete measure \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) is μ-a.s. dense in \(\mathbb {R}^{d}\). We consider the corresponding Dirichlet form

$$\mathcal{E}^{\mathbb{K}}(F,G)={\int}_{\mathbb K(\mathbb{R}^{d})}\langle\nabla^{\mathbb{K}} F(\eta), \nabla^{\mathbb{K}} G(\eta)\rangle_{T_{\eta}(\mathbb{K})}\,d\mu(\eta). $$

Integrating by parts with respect to the measure μ, we explicitly find the generator of this Dirichlet form. By using the theory of Dirichlet forms, we prove the main result of the paper: If d ≥ 2, there exists a conservative diffusion process on \(\mathbb {K}(\mathbb {R}^{d})\) which is properly associated with the Dirichlet form \(\mathcal {E}^{\mathbb {K}}\).

     

Weighted Bergman Spaces and Carleson Measures on the Unit Ball of Cn

Let be a normal function on [0, 1), B _n the unit ball of C ⁿ , and A ^p (B _n ) the weighted Bergman spaces on B _n with weight . The purpose of this paper is to discuss some relations among A ^p (B _n ), weighted Bergman kernels, and Carleson measures on B _n .     

Dynamics of Supports of Extremal Measures in the Field of a Point Charge

Mathematical Notes -     

Superharmonic Perturbations of a Gaussian Measure, Equilibrium Measures and Orthogonal Polynomials

This work concerns superharmonic perturbations of a Gaussian measure given by a special class of positive weights in the complex plane of the form w(z) = exp(−|z|² + U^μ(z)), where U^μ(z) is the logarithmic potential of a compactly supported positive measure μ. The equilibrium measure of the corresponding weighted energy problem is shown to be supported on subharmonic generalized quadrature domains for a large class of perturbing potentials U^μ(z). It is also shown that the 2 × 2 matrix d-bar problem for orthogonal polynomials with respect to such weights is well-defined and has a unique solution given explicitly by Cauchy transforms. Numerical evidence is presented supporting a conjectured relation between the asymptotic distribution of the zeroes of the orthogonal polynomials in a semi-classical scaling limit and the Schwarz function of the curve bounding the support of the equilibrium measure, extending the previously studied case of harmonic polynomial perturbations with weights w(z) supported on a compact domain. Submitted: July 25, 2008. Accepted: October 1, 2008. Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds de recherche sur la nature et les technologies du Québec (FQRNT).     

Hausdorff Measures versus Equilibrium States of Conformal Infinite Iterated Function Systems

Dealing with infinite iterated function systems we introduce and develop the ergodic theory of Hölder systems of functions similarly as in [HU] and [HMU]. In the context of conformal infinite iterated function systems we prove the volume lemma for the Hausdorff dimension of the projection onto the limit set of a shift invariant measure. This can be considered as a Billingsley type result. Our cenral goal is to demonstrate the appearance of the "singularity-absolute continuity" dichotomy for equilibrium states of Hölder systems of functions which has been observed in [PUZ,I] and [PUZ,II] (see also [DU1] and [DU2]) in the setting of rational functions of the Riemann sphere.