Compact subsets of $\Ps(\N)$ with applications to the embedding of symmetric sequence spaces into $C(α)$

The covering dimension of a distinguished subset of the spectrum $M(H^\infty)$ of $H^\infty$ and the algebra of real-symmetric and continuous functions on $M(H^\infty)$

We show that the covering dimension, $\dim E$, of the closure $E$ of the interval $]-1,1[$ in the spectrum of $H^\infty$ equals one. Using Suárez's result that $\dim M(H^\infty)=2$, we then compute the Bass and topological stable ranks of the algebra $C(M(H^\infty))_{\rm\scriptscriptstyle sym}$ of real-symmetric continuous functions on $M(H^\infty)$.

Measure density and Embeddings of Hajłasz-Besov and Hajłasz-Triebel-Lizorkin spaces

In this paper, we investigate the relation between Sobolev-type embeddings of Hajłasz-Besov spaces (and also Hajłasz-Triebel-Lizorkin spaces) defined on a metric measure space $(X,d,μ)$ and lower bound for the measure $μ.$ We prove that if the measure $μ$ satisfies $μ(B(x,r))\geq cr^Q$ for some $Q>0$ and for any ball $B(x,r)\subset X,$ then the Sobolev-type embeddings hold on balls for both these spaces. On the other hand, if the Sobolev-type embeddings hold in a domain $Ω\subset X,$ then we prove that the domain $Ω$ satisfies the so-called measure density condition, i.e., $μ(B(x,r)\capΩ)\geq …

关于嵌入 $\mathcal{W}^p(K)\subset C(K)$ 的 Kigami 猜想

On Kigami's conjecture of the embedding $\mathcal{W}^p(K)\subset C(K)$

本文证明了在连通紧度量空间中，Kigami 的 Sobolev 空间 $\mathcal{W}^p(K)$ 连续嵌入 $C(K)$ 成立当且仅当 $p$ 大于 Ahlfors 正则维数。

Embeddability of locally finite metric spaces into Banach spaces is finitely determined

The main purpose of the paper is to prove the following results: Let $A$ be a locally finite metric space whose finite subsets admit uniformly bilipschitz embeddings into a Banach space $X$. Then $A$ admits a bilipschitz embedding into $X$. Let $A$ be a locally finite metric space whose finite subsets admit uniformly coarse embeddings into a Banach space $X$. Then $A$ admits a coarse embedding into $X$. These results generalize previously known results of the same type due to Brown-Guentner (2005), Baudier (2007), Baudier-Lancien (2008), and the author (2006, 2009). One of the main steps in th…

Compact embeddings for fractional super and sub harmonic functions with radial symmetry

We prove compactness of the embeddings in Sobolev spaces for fractional super and sub harmonic functions with radial symmetry. The main tool is a pointwise decay for radially symmetric functions belonging to a function space defined by finite homogeneous Sobolev norm together with finite $L^2$ norm of the Riesz potentials. As a byproduct we prove also existence of maximizers for the interpolation inequalities in Sobolev spaces for radially symmetric fractional super and sub harmonic functions.

Lack of isomorphic embeddings of symmetric function spaces into operator ideals

Let $E(0,1)$ be a symmetric space on $(0,1)$ and $C_F$ be a symmetric ideal of compact operators on the Hilbert space $\ell_2$ associated with a symmetric sequence space $F$. We give several criteria for $E(0,1)$ and $ F$ so that $E(0,1)$ does not embed into the ideal $C_F$, extending the result for the case when $E(0,1)=L_p(0,1)$ and $F=\ell_p $, $1\le p<\infty$, due to Arazy and Lindenstrauss.

径向对称函数的紧致Sobolev嵌入

Compact Sobolev embeddings of radially symmetric functions

本文研究了径向对称函数在不同空间中的Sobolev嵌入紧致性，提出了新的技术并给出了加权情况下的最优目标空间。

Some necessary and some sufficient conditions for the compactness of the embedding of weighted Sobolev spaces

We give some necessary conditions and sufficient conditions for the compactness of the embedding of Sobolev spaces $W^{1,p}(Ω,w) \to L^p(Ω,w),$ where $w$ is some weight on a domain $Ω\subset \Real^n$.

欧几里得嵌入、随机聚类和Lipschitz扩展用于有限和双倍集的L_p空间当p>2时

Euclidean embedding, randomized clustering, and Lipschitz extension for finite and doubling subsets of $L_p$ when $p>2$

论文证明了L_p空间中n点集的欧几里得扭曲为O(p^3(logn)^{1/2+o(1)})，并展示了双倍集的分离模和扩展性质。

Embeddings of proper metric spaces into Banach spaces

We show that there exists a strong uniform embedding from any proper metric space into any Banach space without cotype. Then we prove a result concerning the Lipschitz embedding of locally finite subsets of $\mathcal{L}_{p}$-spaces. We use this locally finite result to construct a coarse bi-Lipschitz embedding for proper subsets of any $\mathcal{L}_p$-space into any Banach space $X$ containing the $\ell_p^n$'s. Finally using an argument of G. Schechtman we prove that for general proper metric spaces and for Banach spaces without cotype a converse statement holds.

Almost bi--Lipschitz embeddings using covers of balls centred at the origin

In 2010, Olson \& Robinson [Transactions of the American Mathematical Society, 362(1), 145-168] introduced the notion of an almost homogeneous metric space and showed that if $X$ is a subset of a Hilbert space such that $X-X$ is almost homogeneous, then $X$ admits almost bi--Lipschitz embeddings into Euclidean spaces. In this paper, we extend this result and we show that if $X$ is a subset of a Banach space such that $X-X$ is almost homogeneous at the origin, then $X$ can be embedded in a Euclidean space in an almost bi--Lipschitz way.

An embedding theorem for pseudoconvex domains in Banach spaces

We show that a pseudoconvex open subset of a Banach space with an unconditional basis is biholomorphic to a closed direct submanifold of a Banach space with an unconditional basis.

对称群的$L_1$和$L_2$嵌入

$L_1$ and $L_2$ embeddings of the symmetric group

证明由循环(123...n)和对换(12)生成的对称群Cayley图以O(1)双Lipschitz畸变嵌入L1，首次给出序列群因生成集选择不同而嵌入L1能力不同的例子。

Low Correlation Noise Stability of Symmetric Sets

We study the Gaussian noise stability of subsets A of Euclidean space satisfying A=-A. It is shown that an interval centered at the origin, or its complement, maximizes noise stability for small correlation, among symmetric subsets of the real line of fixed Gaussian measure. On the other hand, in dimension two and higher, the ball or its complement does not always maximize noise stability among symmetric sets of fixed Gaussian measure. In summary, we provide the first known positive and negative results for the Symmetric Gaussian Problem.