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,...     

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.     

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.     

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.     

Pseudodifferential Operators on Spaces of Distributions Associated with Non-negative Self-Adjoint Operators

We consider Hörmander type symbols on a family of spaces associated with non-negative self-adjoint operators, and we prove boundedness of the corresponding pseudodifferential operators on both classical and non-classical Besov and Triebel–Lizorkin spaces. Consequently, this also covers the case of Sobolev spaces. As an application, we obtain boundedness of spectral multipliers on the mentioned spaces.     

伴随Herz空间的Hardy空间上的向量值Calderón-Zygmund算子(英文)

本文证明了一类具有向量值核的Calderon－Zygmund算子是Herz型Hard,空间HKp到向量值Herz空间KE,p有界的,应用这一结果,得到了粗糙核Calderon－Zygmund算子,极大型Calderon－Zygmund算子,极大算子等是HKp到Kp有界的．     

Predual Spaces of Banach Completions of Orlicz-Hardy Spaces Associated with Operators

Let L be a linear operator in L ²(? ⁿ ) and generate an analytic semigroup {e ^?tL } _t??0 with kernels satisfying an upper bound of Poisson type, whose decay is measured by ??(L)??(0,??]. Let ?? on (0,??) be of upper type 1 and of critical lower type $\widetilde{p}_{0}(\omega)\in(n/(n+\theta(L)),1]$ and ??(t)=t ^?1/?? ^?1(t ^?1) for t??(0,??). In this paper, the authors first introduce the VMO-type space VMO _??,L (? ⁿ ) and the tent space $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ and characterize the space VMO _??,L (? ⁿ ) via the space $T^{\infty}_{\omega,\mathrm{v}}({{\mathbb{R}}}^{n+1}_{+})$ . Let $\widetilde{T}_{\omega}({{\mathbb{R}}}^{n+1}_{+})$ be the Banach completion of the tent space $T_{\omega}({\mathbb{R}}^{n+1}_{+})$ . The authors then prove that $\widetilde{T}_{\omega}({\mathbb{R}}^{n+1}_{+})$ is the dual space of $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ . As an application of this, the authors finally show that the dual space of $\mathrm{VMO}_{\rho,L^{\ast}}({\mathbb{R}}^{n})$ is the space B _??,L (? ⁿ ), where L ^* denotes the adjoint operator of L in L ²(? ⁿ ) and B _??,L (? ⁿ ) the Banach completion of the Orlicz-Hardy space H _??,L (? ⁿ ). These results generalize the known recent results by particularly taking ??(t)=t for t??(0,??).     

Random Walks Associated with Non-Divergence Form Elliptic Equations

This paper is concerned with the study of the diffusion process associated with a nondivergence form elliptic operator in d dimensions, d2. The authors introduce a new technique for studying the diffusion, based on the observation that the probability of escape from a d–1 dimensional hyperplane can be explicitly calculated. They use the method to estimate the probability of escape from d–1 dimensional manifolds which are C ^1, , and also d–1 dimensional Lipschitz manifolds. To implement their method the authors study various random walks induced by the diffusion process, and compare them to the corresponding walks induced by Brownian motion.     

On Elliptic Homogeneous Differential Operators in Grand Spaces

We give an application of so-called grand Lebesgue and grand Sobolev spaces, intensively studied during last decades, to partial differential equations. In the case of unbounded domains such spaces are defined using so-called grandizers. Under some natural assumptions on the choice of grandizers, we prove the existence, in some grand Sobolev space, of a solution to the equation Pm(D)u(x) = f(x), x ∈ ℝn, m < n, with the right-hand side in the corresponding grand Lebesgue space, where Pm(D) is an arbitrary elliptic homogeneous in the general case we improve some known facts for the fundamental solution of the operator Pm(D): we construct it in the closed form either in terms of spherical hypersingular integrals or in terms of some averages along plane sections of the unit sphere.