Geometry of Mean Value Sets for General Divergence Form Uniformly Elliptic Operators

In the Fermi Lectures on the obstacle problem in 1998, Caffarelli gave a proof of the mean value theorem which extends to general divergence form uniformly elliptic operators. In the general setting, the result shows that for any such operator L and at any point x₀ in the domain, there exists a nested family of sets {D_r(x₀)} where the average over any of those sets is related to the value of the function at x₀. Although it is known that the {D_r(x₀)} are nested and are comparable to balls in the sense that there exists c,C depending only on L such that B_cr(x₀) ? D_r(x₀) ? B_Cr(x₀) for all r >?0 and x₀ in the domain, otherwise their geometric and topological properties are largely unknown. In this paper we begin the study of these topics and we prove a few results about the geometry of these sets and give a couple of applications of the theorems.

     

Mean Oscillation and Hankel Operators on Fock-type Spaces

For 1 ≤ p ∞ we introduce a notion of “p-mean oscillation” on Cnin terms of the metric induced by reproducing kernel of F_Ψ~2. It is shown that the densely-defined Hankel operators ■ are simultaneously bounded if and only if f is of bounded “p-mean oscillation”.Furthermore, it is also shown that the densely-defined Hankel operators ■ are simultaneously compact if and only if f is of vanishing “p-mean oscillation”. Here the weight Ψ is a positive function of logarithmic growth satisfying certain suitable conditions.     

On Dirichlet Processes Associated with Second Order Divergence Form Operators

Let be a d - dimensional Markov family corresponding to a uniformly elliptic second order divergence form operator. We show that for any quasi continuous in the Sobolev space the process (X) admits under P ^x a decomposition into a martingale additive functional (AF) M and a continuous AF A of zero quadratic variation for almost every starting point x if q=2, for quasi every x if q>2 and for every if is continuous, d=1 and or d>1 and q>d. Our decomposition enables us to show that in the case of symmetric operator the energy of A equals zero if q=2 and that the decomposition of (X) into the martingale AF M and the AF of zero energy A is strict if for some q>d. Moreover, our decomposition provides a probabilistic representation of A .     

Fock Spaces Connected with Spherical Mean Operator and Associated Operators

We define and study the Fock space associated with the spherical mean operator. Next, we establish some results for the Segal-Bergmann transform for this space. Lastly, we prove some properties concerning Toeplitz operators on this space. Received: May 11, 2007. Revised: May 20, 2008. Accepted: May 23, 2008.     

Functions of Vanishing Mean Oscillation

Let X_i, i = 1, 2,…, be i.i.d. symmetric random variables in the domain of attraction of a symmetric stable distribution G_α with 0 < α < 2. Let Y_i, i = 1, 2, …, be i.i.d. symmetric stable random variables with the common distribution G_α. It is known that under certain conditions the sequences {X_i} and {Y_i} can be reconstructed on a new probability space without changing the distribution of each such that \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_{i = 1}^n {(X_i - Y_i) = o(n^{1/\gamma})} $\end{document} a.s. as n → ∞, where α ≦ γ < 2 (see Stout [10]). We will give a second approximation by partial sums of i.i.d. stable (with characteristic exponent α*, α < α* ≦ 2) random variables U_i, i = 1, 2,…, n, and we will obtain strong upperbounds for the differences \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_{i = 1}^n {(X_i - Y_i - U_i)} $\end{document}.     

On Singular Operators in Vanishing Generalized Variable-Exponent Morrey Spaces and Applications to Bergman-Type Spaces

Mathematical Notes - We give a proof of the boundedness of the Bergman projection in generalized variable-exponent vanishing Morrey spaces over the unit disc and the upper half-plane. To this end,...     

散度形式二阶椭圆算子Neumann本征值的上界

设Ω是Rm（m≥2）中的一个有界区域,其边界足够光滑。本文旨在给出散度形式二阶椭圆算子-·（A·）的Neumann本征值的一个上界,该上界除了与维数m及系数张量A的迹trA有关外,仅依赖于区域Ω的体积．     

A Characterization of Vanishing Mean Oscillation

We show that a function has vanishing mean oscillation with respect to a nonatomic measure if and only if it satisfies an asymptotic reverse Jensen inequality.     

关于消失平均振动的一个注记

本文得到了极大算子的几个不等式,并从这些结果推广了函数f∈L_(loc)~1(R,dx)属于VMO(R,dx)的一个必要条件。     

A Characterization of Vanishing Mean Oscillation

We show that a function has vanishing mean oscillation with respect to a nonatomic measure if and only if it satisfies an asymptotic reverse Jensen inequality.     

Riesz Transform Characterizations of Hardy Spaces Associated to Degenerate Elliptic Operators

Let \({L_{w}}{:=-w^{-1}{\rm div}(A\nabla)}\) be the degenerate elliptic operator on the Euclidean space \({{\mathbb{R}^{n}}}\), where w is a Muckenhoupt \({A_{2}({\mathbb{R}^{n}})}\) weight. In this article, the authors establish the Riesz transform characterization of the Hardy space \({H^{p}_{L_{w}}({\mathbb{R}}^{n})}\) associated with L_w, for \({w \in A_{q}({\mathbb{R}}^{n}) \cap RH_{\frac{n}{n-2}}({\mathbb{R}^{n}})}\) with \({n \geq 3}\), \({q \in [1,2]}\) and \({p \in (q(\frac{1}{r}+\frac{q-1}{2}+\frac{1}{n})^{-1},1]}\) if, for some \({r \in (1,\,2]}\), \({{\{tL_w e^{-tL_w}\}}_{t\geq 0}}\) satisfies the weighted \({L^{r}-L^{2}}\) full off-diagonal estimates.     

带权散度型椭圆算子的低阶特征值估计

本文首先给出紧致带边(边界可以为空集)光滑度量测度空间上带权散度型算子的低阶特征值的一个一般不等式,通过使用这个一般不等式,可以得到光滑度量测度空间中有界连通区域上带权散度型算子的低阶特征值的一些万有不等式.     

相伴于非散度型椭圆算子的Riesz变换的L~p有界性

本文利用小波方法在一般阶的非散度椭圆算子的系数BMO模非常小的情形下,证明了广义 Riesz变换的 L~p(2≤p<+∞)有界性。     

完备黎曼流形上四阶散度型椭圆算子的特征值估计

研究了完备黎曼流形上四阶散度型椭圆算子的特征值问题,得到了特征值的一个基本不等式.由这个基本不等式,得到具有特殊函数的完备黎曼流形上四阶散度型椭圆算子的特征值估计的万有不等式,同时给出具有这些特殊函数的完备黎曼流形的例子.在此基础上,证明了Chen-Zheng-Yang的一个猜想是成立的.     

散度型椭圆方程的解在Morrey空间上的细正则性

对具有不连续系数的散度型椭圆方程-(aijuxi)xj=(fj)xj的解在Morrey空间中的细正则性进行了研究,即如果aij∈VMO ∩ L∞(Ω),fj∈Lp,λ(Ω),u∈W1,q(Ω)(1＜q≤p)是方程的解,则 u∈W1,ploc(Ω)且uxj∈Lp,λloc(Ω).     

Mean Value Operators, Differential Operators and D'Atri Spaces

We study commutativity relations between differential operators and spherical means on Riemannian manifolds. The results show that all D'Atri spaces are ball-homogeneous. A further consequence characterizes complete nonpositively curved, simply connected D'Atri spaces by commuting mean value operators.     

Toeplitz Operators on Generalized Bergman Spaces

We consider the weighted Bergman spaces HL²(\mathbb B^d, m_l){\mathcal {H}L^{2}(\mathbb {B}^{d}, \mu_{\lambda})}, where we set dm_l(z) = c_l(1-|z|²)^l dt(z){d\mu_{\lambda}(z) = c_{\lambda}(1-|z|^2)^{\lambda} d\tau(z)}, with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert–Schmidt operators on the generalized Bergman spaces.     

广义Fock空间上的Hankel算子

利用有界(消失)平均振荡函数的性质,本文刻画了一类广义Fock空间上的Hankel算子的有界性(紧性),同时,还刻画了换位子[M_f,P]的有界性和紧性,其中P是一个Toeplitz投影算子,而M_f表示符号为f的乘子.最后,应用Berezin变换来研究了BMO空间和VMO空间的几何性质.