Boundedness in Lebesgue spaces for commutators of multilinear singular integrals and RBMO functions with non-doubling measures

The boundedness in Lebesgue spaces for commutators generated by multilinear singular integrals and RBMO(μ) functions of Tolsa with non-doubling measures is obtained, provided that‖μ‖=∞and multilinear singular integrals are bounded from L1(μ)×L1(μ)to L1/2,∞(μ).     

Carleson measures on planar sets

In this paper, we investigate what are Carleson measures on open subsets in the complex plane. A circular domain is a connected open subset whose boundary consists of finitely many disjoint circles. We call a domain G multi-nicely connected if there exists a circular domain W and a conformal map ψ from W onto G such that ψ is almost univalent with respect the arclength on δW. We characterize all Carleson measures for those open subsets so that each of their components is multinicely connected and harmonic measures of the components are mutually singular. Our results suggest the extension of Carleson measures probably is up to this class of open subsets     

Weighted estimates for commutators of singular integral operators with non-doubling measures

Letμbe a nonnegative Radon measure on R~d which only satisfiesμ(B(x,r))≤C_0r~n for all x∈R~d,r>0,and some fixed constants C_0>0 and n∈(0,d].In this paper,some weighted weak type estimates with A_(p,(log L)~σ)~ρ(μ) weights are established for the commutators generated by Calder■n-Zygmund singular integral operators with RBMO(μ) functions.     

The structure on invariant measures of <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> generic diffeomorphisms

Let Λ be an isolated non-trivial transitive set of a C 1 generic diffeomorphism f ∈ Diff (M ). We show that the space of invariant measures supported on Λ coincides with the space of accumulation measures of time averages on one orbit. Moreover, the set of points having this property is residual in Λ (which implies that the set of irregular+ points is also residual in Λ). As an application, we show that the non-uniform hyperbolicity of irregular+ points in Λ with totally 0 measure (resp., the non-uniform hyperbolicity of a generic subset in Λ) determines the uniform hyperbolicity of Λ.     

Hausdorff-type Measures of the Sample Path of Gaussian Random Fields

Let φ be a Hausdorff measure function and A be an infinite increasing sequence of positive integers. The Hausdorff-type measure φ - mA associated to φ and A is studied. Let X（t）（t ∈ R^N） be certain Gaussian random fields in R^d. We give the exact Hausdorff measure of the graph set GrX（[0, 1]N）, and evaluate the exact φ - mA measure of the image and graph set of X（t）. A necessary and sufficient condition on the sequence A is given so that the usual Hausdorff measure function for X（[0, 1] ^N） and GrX（[0, 1]^N） are still the correct measure functions. If the sequence A increases faster, then some smaller measure functions will give positive and finite （ φ A）-Hausdorff measure for X（[0, 1]^N） and GrX（[0, 1]N）.     

Constructing Restricted Patterson Measures for Geometrically Infinite Kleinian Groups

In this paper, we study exhaustions, referred to as p-restrictions, of arbitrary nonelementary Kleinian groups with at most finitely many bounded parabolic elements. Special emphasis is put on the geometrically infinite case, where we obtain that the limit set of each of these Kleinian groups contains an infinite family of closed subsets, referred to as p-restricted limit sets, such that there is a Poincaré series and hence an exponent of convergence δp, canonically associated with every element in this family. Generalizing concepts which are well known in the geometrically finite case, we then introduce the notion of p-restricted Patterson measure, and show that these measures are non-atomic, δp-harmonic, δp-subconformal on special sets and δp-conformal on very special sets. Furthermore, we obtain the results that each p-restriction of our Kleinian group is of δp-divergence type and that the Hausdorff dimension of the p-restricted limit set is equal to δp.     

The Exact Hausdorff Measure Function of the Level Sets of Multi-parameter Symmetric Stable Process

In this paper, we investigate the Hausdorff measure for level sets of N-parameter Rd-valued stable processes, and develop a means of seeking the exact Hausdorff measure function for level sets of N-parameter Rd-valued stable processes. We show that the exact Hausdorff measure function of level sets of N-parameter Rd-valued symmetric stable processes of index α is Ф（r） = r^N-d/α （log log l/r）d/α when Nα 〉 d. In addition, we obtain a sharp lower bound for the Hausdorff measure of level sets of general （N, d, α） strictly stable processes.     

Invariant Measure for the Markov Process Corresponding to a PDE System

In this paper, we consider the Markov process （X^∈（t）, Z^∈（t）） corresponding to a weakly coupled elliptic PDE system with a small parameter ∈ 〉 0. We first prove that （X^∈（t）, Z^∈（t）） has the Feller continuity by the coupling method, and then prove that （X^∈（t）, Z^∈（t）） has an invariant measure μ^∈（·） by the Foster-Lyapunov inequality. Finally, we establish a large deviations principle for μ^∈（·） as the small parameter e tends to zero.     

Differentiation in the branges spaces and embedding theorems

The boundedness conditions for the differentiation operator in Hilbert spaces of entire functions (Branges spaces) and conditions under which the embedding K_и⊂L₂(μ) holds in spaces K_и associated with the Branges spacesH(E) are studied. Measure μ such that the above embedding is isometric are of special interest. It turns out that the condition E'/E∈H^∞(C⁺) is sufficient for the boundedness of the differentiation operator inH(E). Under certain restrictions on E, this condition is also necessary. However, this fact fails in the general case, which is demonstrated by the counterexamples constructed in this paper. The convex structure of the set of measures μ such that the embedding K_E ^* _/E⊂L²(μ) is isometric (the set of such measures was described by de Brages) is considered. Some classes of measures that are extreme points in the set of Branges measures are distinguished. Examples of measures that are not extreme points are also given. Bibliography: 7 titles. Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 27–68.     

On quasi-similarity of subnormal operators

For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K,v) denote the closure of Rat(K) in L2(v) and let Sv denote the operator of multiplication by the independent variable z on R2(K, v), that is, Svf = zf for every f∈R2(K, v). SupposeΩis a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H2(Ω). Letσdenote a harmonic measure forΩ. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R2(Ω,σ). We show that for a given v supported onΩ, Sv is quasi-similar to Sσif and only if v/■Ω■σ and log(dv/dσ)∈L1(σ). Our result extends a well-known result of Clary on the unit disk.     

Definition of Measure-theoretic Pressure Using Spanning Sets

We introduce a new definition of measure–theoretic pressure for ergodic measures of continuous maps on a compact metric space. This definition is similar to those of topological pressure involving spanning sets. As an application, for C ^{1+ α} (α > 0) diffeomorphisms of a compact manifold, we study the relationship between the measure–theoretic pressure and the periodic points. Project Supported by National Natural Science Foundation of China     

Modified Logarithmic Sobolev Inequalities and Transportation Cost Inequalities in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, the modified logarithmic Sobolev inequalities and transportation cost inequalities for measures with density e ^− V in ℝ ⁿ are established. It is proved by using Prékopa–Leindler inequalities following the idea of Bobkov–Ledoux, but a different type of condition is used which recovers Bakry–Emery criterion. As an application, we establish the modified logarithmic Sobolev and transportation cost inequalities for probability measures with p > 1 in ℝ ⁿ , and give out explicit estimates for their constants. This work is supported by NSFC (No. 10721091), 973-Project (No.2006CB805901) and DFMEC (NO. 20070027007).     

On convex risk measures on <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-spaces

Much of the recent literature on risk measures is concerned with essentially bounded risks in L ^∞. In this paper we investigate in detail continuity and representation properties of convex risk measures on L ^p spaces. This frame for risks is natural from the point of view of applications since risks are typically modelled by unbounded random variables. The various continuity properties of risk measures can be interpreted as robustness properties and are useful tools for approximations. As particular examples of risk measures on L ^p we discuss the expected shortfall and the shortfall risk. In the final part of the paper we consider the optimal risk allocation problem for L ^p risks.     

Ordered representations of spaces of integrable functions

Let X be a Banach space, (Ω,Σ) a measurable space and let m : Σ → X be a (countably additive) vector measure. Consider the corresponding space of integrable functions L¹(m). In this paper we analyze the set of (countably additive) vector measures n satisfying that L¹(n) = L¹(m). In order to do this we define a (quasi) order relation on this set to obtain under adequate requirements the simplest representation of the space L¹(m) associated to downward directed subsets of the set of all the representations. This research has been partially supported by La Junta de Andalucía. The support of D.G.I. under project MTM2006–11690–C02 (M.E.C. Spain) and FEDER is gratefully acknowledged.     

Gibbs measures for the SOS model with four states on a Cayley tree

We analyze the SOS (solid-on-solid) model with spins 0, 1, 2, 3 on a Cayley tree of order k ≥ 1. We consider translation-invariant and periodic splitting Gibbs measures for this model. The majority of the constructed Gibbs measures are mirror symmetric. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 18–31, October, 2006.     

Decrease rate of the probabilities ofε-deviations for the means of stationary processes

The asymptotic behavior asn → ∞ of the normed sumsσn =n ⁻¹ Σ _k ⁼⁰ⁿ⁻¹ Xk for a stationary processX = (X _n ,n ∈ ℤ) is studied. For a fixedε > 0, upper estimates for P(sup _k≥n |σ _k | ≥ε) asn → ∞ are obtained. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 366–372, September, 1998.     

Yan Theorem in L∞ with Applications to Asset Pricing

We prove an L~∞ version of the Yan theorem and deduce from it a necessary condition for theabsence of free lunches in a model of financial markets,in which asset prices are a continuous R~d valued processand only simple investment strategies are admissible.Our proof is based on a new separation theorem for convexsets of finitely additive measures.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-singular dichotomy for orbital measures of classical simple Lie algebras

We prove that the G-invariant orbital measures supported on adjoint orbits in the Lie algebra of a classical, compact, connected, simple Lie group satisfy a smoothness dichotomy: Either μ ^k is singular to Lebesgue measure or μ ^k ∈ L ². The minimum k for which μ ^k ∈ L ² is specified and is also the minimum k such that the k-fold sum of the orbit has positive measure. S. K. Gupta appreciates the hospitality of the Department of Pure Mathematics at the University of Waterloo where some of this research was done. K. E. Hare was supported in part by NSERC.     

The Dichotomy Problem for Orbital Measures of <Emphasis Type="Italic">SU</Emphasis>(<Emphasis Type="Italic">n</Emphasis>)

For many orbital measures μ, on SU(n), we show that either μ^k ∈ L² or μ^k is singular to L¹. The least k for which μ^k ∈ L² is determined and is shown to be the minimum k for which the k-fold product of the conjugacy class supporting the measure has positive measure. It would be interesting to know if all orbital measures satisfy this dichotomy.     

The Dichotomy Problem for Orbital Measures of SU(n)

For many orbital measures μ, on SU(n), we show that either μ^k ∈ L² or μ^k is singular to L¹. The least k for which μ^k ∈ L² is determined and is shown to be the minimum k for which the k-fold product of the conjugacy class supporting the measure has positive measure. It would be interesting to know if all orbital measures satisfy this dichotomy.