有非负Ricci曲率的度量测度空间上调和函数的边界问题

设(X,d,μ)是满足非负Ricci曲率条件的度量测度空间.本文研究了(开)上半空间X×R+上调和函数的边界问题.我们得到了:若u(x,t)是定义在上半空间X×R+上的调和函数,且满足Carleson测度条件supxB,rB∫rB0fB(xB,rB)|t▽u(x,t)|2dμ(x)dt/t≤C＜∞,其中▽=(▽x,?)...     

Ricci curvature of Markov chains on metric spaces

We define the coarse Ricci curvature of metric spaces in terms of how much small balls are closer (in Wasserstein transportation distance) than their centers are. This definition naturally extends to any Markov chain on a metric space. For a Riemannian manifold this gives back, after scaling, the value of Ricci curvature of a tangent vector. Examples of positively curved spaces for this definition include the discrete cube and discrete versions of the Ornstein-Uhlenbeck process. Moreover this generalization is consistent with the Bakry-Émery Ricci curvature for Brownian motion with a drift on a Riemannian manifold.Positive Ricci curvature is shown to imply a spectral gap, a Lévy-Gromov-like Gaussian concentration theorem and a kind of modified logarithmic Sobolev inequality. The bounds obtained are sharp in a variety of examples.     

Carleson measure characterization of Bloch functions

We give several equivalences of Bloch functions and little Bloch functions. Using these results we obtain the generalized Carleson measure characterization of Bloch functions and the generalized vanishing Carleson measure characterization of little Bloch functions, that is,f B if and only if |D f(z)| ^p (1-|z|²)^p-1 dm(z) is a generalized Carleson measure;f B ₀ if and only if |D f(z)| ^p (1-|z|²)^p-1 dm(z) is a generalized vanishing Carleson measure, whereD f( > 0) is the fractional derivative of analytic functionf of order, m denotes the normalised Lebesgue measure.Supported partly by the Young Teacher Natural Science Foundation of Shandong Province.     

Sobolev embeddings in metric measure spaces with variable dimension

In this article we study metric measure spaces with variable dimension. We consider Lebesgue spaces on these sets, and embeddings of the Riesz potential in these spaces. We also investigate Hajłasz-type Sobolev spaces, and prove Sobolev and Trudinger inequalities with optimal exponents. All of these questions lead naturally to function spaces with variable exponents. Supported the Research Council of Norway, Project 160192/V30.     

De Lellis-Topping type inequalities on smooth metric measure spaces

We obtain some De Lellis-Topping type inequalities on the smooth metric measure spaces, some of them are as generalization of De Lellis-Topping type inequality that was proved by X. Cheng [Ann. Global Anal. Geom., 2013, 43: 153-160].     

Analytic approaches and harmonic functions on Alexandrov spaces with nonnegative Ricci curvature

In this paper, the author gets a sharp dimension estimate of the space of harmonic functions with polynomial growth of a fixed order on Alexandrov spaces, which extends the result of Colding and Minicozzi from Riemannian manifolds to Alexandrov spaces.     

Some new characterizations on spaces of functions with bounded mean oscillation

Let X be a space of homogeneous type. The authors introduce some generalized approximations to the identity (for short, GAI) with optimal decay conditions in the sense that these conditions are the sufficient and necessary conditions for these GAI's to characterize BMO(X), the space of functions with bounded mean oscillation on X. The authors also obtain a new John‐Nirenberg‐type inequality associated with GAI's, which leads some new characterizations of BMO(X) in terms of rearrangement functions, and certain maximal functions related to GAI's. Some variants of these characterizations for BMO‐type spaces on X of Duong and Yan are also established. Moreover, the equivalence between BMO and the space of BMO type on Ahlfors n ‐regular quasimetric measure spaces is obtained, which confirms an assertion of Duong and Yan (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Luzin approximation of functions from classes Wαp on metric spaces with measure

In this paper we prove an analog of the Luzin theorem on correction for spaces of the Sobolev type on an arbitrary metric space with a measure, satisfying the doubling condition. The correcting function belongs to the H?lder class and approximates a given function in the metrics of the initial space. Dimensions of exceptional sets are evaluated in terms of Hausdorff capacities and volumes. Original Russian Text ? V.G. Krotov and M.A. Prokhorovich, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 5, pp. 55–66. Dedicated to the memory of Petr Lavrent’evich Ul’yanov     

Hausdorff dimension and doubling measures on metric spaces

Volberg and Konyagin have proved that a compact metric space carries a nontrivial doubling measure if and only if it has finite uniform metric dimension. Their construction of doubling measures requires infinitely many adjustments. We give a simpler and more direct construction, and also prove that for any , the doubling measure may be chosen to have full measure on a set of Hausdorff dimension at most .

     

Every complete doubling metric space carries a doubling measure

We prove that a complete metric space carries a doubling measure if and only if is doubling and that more precisely the infima of the homogeneity exponents of the doubling measures on and of the homogeneity exponents of are equal. We also show that a closed subset of carries a measure of homogeneity exponent . These results are based on the case of compact due to Volberg and Konyagin.

     

On extensions of Sobolev functions defined on regular subsets of metric measure spaces

We characterize the restrictions of first-order Sobolev functions to regular subsets of a homogeneous metric space and prove the existence of the corresponding linear extension operator.     

Cheeger-harmonic functions in metric measure spaces revisited

Let (X,d,μ)$(X,d,\mu )$ be a complete metric measure space, with μ   a locally doubling measure, that supports a local weak L²$L^2$-Poincaré inequality. By assuming a heat semigroup type curvature condition, we prove that Cheeger-harmonic functions are Lipschitz continuous on (X,d,μ)$(X,d,\mu )$. Gradient estimates for Cheeger-harmonic functions and solutions to a class of non-linear Poisson type equations are presented.     

Lower bound estimates for the first eigenvalue of the weighted p-Laplacian on smooth metric measure spaces

New lower bounds of the first nonzero eigenvalue of the weighted p-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the m-Bakry–Émery Ricci curvature, the Escobar–Lichnerowicz–Reilly type estimates are proved; under the assumption of nonnegative ∞-Bakry–Émery Ricci curvature and the m-Bakry–Émery Ricci curvature bounded from below by a non-positive constant, the Li–Yau type lower bound estimates are given. The weighted p-Bochner formula and the weighted p-Reilly formula are derived as the key tools for the establishment of the above results.     

Monotonicity of quotients of theta functions related to an extremal problem on harmonic measure

We prove that for fixed u and v such that u,v∈[0,1/2), the quotients θj(u|iπt)/θj(v|iπt), j=1,2,3,4, of the theta functions are monotone on 0<t<∞. The case v=0 has been used by the second author to study a generalization of Gonchar's problem on harmonic measure of radial slits.     

On an equivalence of topological vector space valued cone metric spaces and metric spaces

Scalarization method is an important tool in the study of vector optimization as corresponding solutions of vector optimization problems can be found by solving scalar optimization problems. Recently this has been applied by Du (2010) [14] to investigate the equivalence of vectorial versions of fixed point theorems of contractive mappings in generalized cone metric spaces and scalar versions of fixed point theorems in general metric spaces in usual sense. In this paper, we find out that the topology induced by topological vector space valued cone metric coincides with the topology induced by the metric obtained via a nonlinear scalarization function, i.e any topological vector space valued cone metric space is metrizable, prove a completion theorem, and also obtain some more results in topological vector space valued cone normed spaces.     

Dirac-Lu space with pseudo-Riemannian metric of constant curvature

In this paper, we will discuss the geometries of the Dirac-Lu space whose boundary is the third conformal space N defined by Dirac and Lu. We firstly give the SO(3, 3) invariant pseudo-Riemannian metric with constant curvature on this space, and then discuss the timelike geodesics. Finally we get a solution of the Yang-Mills equation on it by using the reduction theorem of connections.     

Complete noncompact manifolds with harmonic curvature

Let (M ⁿ , g) be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper, by employing an elliptic estimation method, we show that (M ⁿ , g) is a space form if it has sufficiently small L ^n/2-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for (M ⁿ , g) with positive scalar curvature.     

Carleson measures and balayage for Bergman spaces of strongly pseudoconvex domains

Given a bounded strongly pseudoconvex domain D in with smooth boundary, we characterize ‐Bergman Carleson measures for , , and . As an application, we show that the Bergman space version of the balayage of a Bergman Carleson measure on D belongs to BMO in the Kobayashi metric.