Banach January 2017

banach@mathdept.okstate.edu

2 participants
17 discussions

Abstract of a paper by Rita Ferreira, Peter Hästö, Ana Margarida Ribeiro
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “Characterization of generalized Orlicz spaces” by Rita Ferreira<https://arxiv.org/find/math/1/au:+Ferreira_R/0/1/0/all/0/1>, Peter Hästö<https://arxiv.org/find/math/1/au:+Hasto_P/0/1/0/all/0/1>, Ana Margarida Ribeiro<https://arxiv.org/find/math/1/au:+Ribeiro_A/0/1/0/all/0/1>. Abstract: The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Because the difference quotient is based on shifting the function, it cannot be used in generalized Orlicz spaces. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the generalized Orlicz-Sobolev space. Our results are new even in Orlicz spaces and variable exponent spaces. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.04566

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Abstract of a paper by Han Huang, Feng Wei
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “Upper bound for the Dvoretzky dimension in Milman-Schechtman theorem” by Han Huang<https://arxiv.org/find/math/1/au:+Huang_H/0/1/0/all/0/1>, Feng Wei<https://arxiv.org/find/math/1/au:+Wei_F/0/1/0/all/0/1>. Abstract: For a symmetric convex body $K\subset R_n$, the Dvoretzky dimension $k(K)$ is the largest dimension for which a random central section of $K$ is almost spherical. A Dvoretzky-type theorem proved by V.~D.~Milman in 1971 provides a lower bound for $k(K)$ in terms of the average $M(K)$ and the maximum $b(K)$ of the norm generated by $K$ over the Euclidean unit sphere. Later, V.~D.~Milman and G. Schechtman obtained a matching upper bound for $k(K)$ in the case when $M(K)b(K)$>c(\log (n)/n)^{1/2}$. In this paper, we will give an elementary proof of the upper bound in Milman-Schechtman theorem which does not require any restriction on $M(K)$ and $b(K)$ The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.03572

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Abstract of a paper by Florent P. Baudier, Robert Deville
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “Lipschitz Embeddings of Metric Spaces into $c_0$” by Florent P. Baudier<https://arxiv.org/find/math/1/au:+Baudier_F/0/1/0/all/0/1>, Robert Deville<https://arxiv.org/find/math/1/au:+Deville_R/0/1/0/all/0/1>. Abstract: Let M be a separable metric space. We say that $f=(f_n): M\rightarrow c_0$ is a good-$\lambda$-embedding if, whenever $x, y\in M, x\neq y$ implies $d(x, y)\leq \|f(x)-f(y)\|$ and, for each $n, Lip(f_n)<\lambfda$, where $Lip(f_n)$ denotes the Lipschitz constant of $f_n$. We prove that there exists a good-$\lambda$-embedding from $M$ into $c_0$ if and only if $M$ satisfies an internal property called $\pi(\lambda)$. As a consequence, we obtain that for any separable metric space $M$, there exists a good-$2$-embedding from $M$ into $c_0$. These statements slightly extend former results obtained by N. Kalton and G. Lancien, with simplified proofs. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.02025

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Abstract of a paper by Atanu Manna
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “Certain geometric structure of Λ-sequence spaces” by Atanu Manna<https://arxiv.org/find/math/1/au:+Manna_A/0/1/0/all/0/1>. Abstract: The $Lambda$-sequence spaces $\Lambda_p$ for $1<p\leq\infty$ and its generalization $\Lambda_{\hat{p}}$ for $1<\hat{p}<\infty, \hat{p}=(p_n)$ is introduced. The James constants and strong $n$-th James constants of $\Lambda_p$ for $1<p\leq\infty$ is determined. It is proved that generalized $\Lambda$-sequence space $\Lambda_{\hat{p}}$ is embedded isometrically in the Nakano sequence space $\ell_{\hat{p}}(R_{n+1})$ of finite dimensional Euclidean space $R_{n+1}$. Hence it follows that sequence spaces $\Lambda_{p}$ and $\Lambda_{\hat{p}}$ possesses the uniform Opial property, property ($\beta$) of Rolewicz and weak uniform normal structure. Moreover, it is established that $\Lambda_{\hat{p}}$ possesses the coordinate wise uniform Kadec-Klee property. Further necessary and sufficient conditions for element $x\in S(\Lambda_{\hat{p}})$ to be an extreme point of $B(\Lambda_{\hat{p}})$ are derived. Finally, estimation of von Neumann-Jordan and James constants of two dimensional $\Lambda$-sequence space $\Lambda_2^{(2)}$ is being carried out. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.01519

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Abstract of a paper by Aude Dalet, Gilles Lancien
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “Some properties of coarse Lipschitz maps between Banach spaces” by Aude Dalet<https://arxiv.org/find/math/1/au:+Dalet_A/0/1/0/all/0/1>, Gilles Lancien<https://arxiv.org/find/math/1/au:+Lancien_G/0/1/0/all/0/1>. Abstract: We study the structure of the space of coarse Lipschitz maps between Banach spaces. In particular we introduce the notion of norm attaining coarse Lipschitz maps. We extend to the case of norm attaining coarse Lipschitz equivalences, a result of G. Godefroy on Lipschitz equivalences. This leads us to include the non separable versions of classical results on the stability of the existence of asymptotically uniformly smooth norms under Lipschitz or coarse Lipschitz equivalences. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.01364

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Abstract of a paper by Dan Freeman, Thomas Schlumprecht, Andras Zsak
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “Closed Ideals of Operators between the Classical Sequence Spaces” by Dan Freeman<https://arxiv.org/find/math/1/au:+Freeman_D/0/1/0/all/0/1>, Thomas Schlumprecht<https://arxiv.org/find/math/1/au:+Schlumprecht_T/0/1/0/all/0/1>, Andras Zsak<https://arxiv.org/find/math/1/au:+Zsak_A/0/1/0/all/0/1>. Abstract: We prove that the spaces $\mathcal{L}(\ell_p, c_0), \mathcal{L}(\ell_p, \ell_{\inty})$ and $\mathcal{L}(\ell_1, \ell_q)$ of operators with $1<p, q<\infty$ have continuum many closed ideals. This extends and improves earlier works by Schlumprecht and Zs\'ak, by Wallis, and by Sirotkin and Wallis. Several open problems remain. Key to our construction of closed ideals are matrices with the Restricted Isometry Property that come from Compressed Sensing. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.01153

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Abstract of a paper by Ben Wallis
by Bentuo Zheng (bzheng) 01 Jan '17

01 Jan '17

This is an announcement for the paper “Garling sequence spaces” by Ben Wallis<https://arxiv.org/find/math/1/au:+Wallis_B/0/1/0/all/0/1>. Abstract: By generalizing a construction of Garling, for each $1\leq p<\infty$ and each normalized, nonincreasing sequence of positive numbers $w\in c_0-\ell_1$ we exhibit an $\ell_p$-saturated, complementably homogeneous Banach space $g(w,p)$ related to the Lorentz sequence space $d(w,p)$. Using methods originally developed for studying $d(w,p)$, we show that $g(w,p)$ admits a unique (up to equivalence) subsymmetric basis, although when $w=(n^{-\theta})_{n=1}^{\infty}$ for some $0<\theta<1$, it does not admit a symmetric basis. We then discuss some additional properties of $g(w,p)$ related to uniform convexity and superreflexivity. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.01145

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Banach January 2017