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Register a new userAn Alternative Definition of the Completion of Metric Spaces
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Authors
Cheng Hao
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In this article, the author proposes another way to define the completion of a metric space, which is different from the classical one via the dense property, and prove the equivalence between two definitions. This definition is based on considerations from category theory, and can be generalized to arbitrary categories.
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Motivated by the local theory of Banach spaces we introduce a notion of finite representability for metric spaces. This allows us to develop a new technique for comparing the generalized roundness of metric spaces. We illustrate this technique in two different ways by applying it to Banach spaces and metric trees. In the realm of Banach spaces we obtain results such as the following: (1) if $mathcal{U}$ is any ultrafilter and $X$ is any Banach space, then the second dual $X^{astast}$ and the ultrapower $(X)_{mathcal{U}}$ have the same generalized roundness as $X$, and (2) no Banach space of positive generalized roundness is uniformly homeomorphic to $c_{0}$ or $ell_{p}$, $2 < p < infty$. Our technique also leads to the identification of new classes of metric trees of generalized roundness one. In particular, we give the first examples of metric trees of generalized roundness one that have finite diameter. These results on metric trees provide a natural sequel to a paper of Caffarelli, Doust and Weston. In addition, we show that metric trees of generalized roundness one possess special Euclidean embedding properties that distinguish them from all other metric trees.
Enflo constructed a countable metric space that may not be uniformly embedded into any metric space of positive generalized roundness. Dranishnikov, Gong, Lafforgue and Yu modified Enflos example to construct a locally finite metric space that may not be coarsely embedded into any Hilbert space. In this paper we meld these two examples into one simpler construction. The outcome is a locally finite metric space $(mathfrak{Z}, zeta)$ which is strongly non embeddable in the sense that it may not be embedded uniformly or coarsely into any metric space of non zero generalized roundness. Moreover, we show that both types of embedding may be obstructed by a common recursive principle. It follows from our construction that any metric space which is Lipschitz universal for all locally finite metric spaces may not be embedded uniformly or coarsely into any metric space of non zero generalized roundness. Our construction is then adapted to show that the group $mathbb{Z}_omega=bigoplus_{aleph_0}mathbb{Z}$ admits a Cayley graph which may not be coarsely embedded into any metric space of non zero generalized roundness. Finally, for each $p geq 0$ and each locally finite metric space $(Z,d)$, we prove the existence of a Lipschitz injection $f : Z to ell_{p}$.
In this note we give several characterisations of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardys original inequality. We give examples obtaining new weighted Hardy inequalities on $mathbb R^n$, on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds.
Let $Gamma(E)$ be the family of all paths which meet a set $E$ in the metric measure space $X$. The set function $E mapsto AM(Gamma(E))$ defines the $AM$--modulus measure in $X$ where $AM$ refers to the approximation modulus. We compare $AM(Gamma(E))$ to the Hausdorff measure $comathcal H^1(E)$ of codimension one in $X$ and show that $$comathcal H^1(E) approx AM(Gamma(E))$$ for Suslin sets $E$ in $X$. This leads to a new characterization of sets of finite perimeter in $X$ in terms of the $AM$--modulus. We also study the level sets of $BV$ functions and show that for a.e. $t$ these sets have finite $comathcal H^1$--measure. Most of the results are new also in $mathbb R^n$.
We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present ageneral positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysisand metric geometry and provide a number of examples.
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