Banach January 2016

banach@mathdept.okstate.edu

2 participants
20 discussions

[Banach Space Bulletin Board] Abstract of a paper by Assaf Naor
by Bentuo Zheng (bzheng) 17 Jan '16

17 Jan '16

This is an announcement for the paper “Discrete Riesz transforms and sharp metric $X_p$ inequalities” by Assaf Naor. Abstract: For p∈[2,∞) the metric Xp inequality with sharp scaling parameter is proven here to hold true in Lp. The geometric consequences of this result include the following sharp statements about embeddings of Lq into Lp when 2<q<p<∞: the maximal θ∈(0,1] for which Lq admits a bi-θ-H\"older embedding into Lp equals q/p, and for m,n∈ℕ the smallest possible bi-Lipschitz distortion of any embedding into Lp of the grid {1,…,m}n⊆ℓnq is bounded above and below by constant multiples (depending only on p,q) of the quantity min{n(p−q)(q−2)/(q2(p−2)),m(q−2)/q}. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1601.03332 _______________________________________________ Banach mailing list Banach(a)mathdept.okstate.edu<mailto:Banach@mathdept.okstate.edu> http://cauchy.math.okstate.edu/cgi-bin/mailman/listinfo/banach Bentuo Zheng Associate Professor, Department of Mathematical Sciences, College of Arts and Sciences [UofM logo] The University of Memphis 359 Dunn Hall Memphis, TN 38152<http://www.memphis.edu/emailsignatures/emailsignaturemac.php#> 901.678<http://www.memphis.edu/emailsignatures/emailsignaturemac.php#>.3534 | memphis.edu<http://www.memphis.edu/> [UofM tw]<https://www.facebook.com/uofmemphis> [UofM tw] <https://twitter.com/uofmemphis> [UofM tw] <https://instagram.com/uofmemphis>

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[Banach Space Bulletin Board] Abstract of a paper by Lining Jiang, Zhenhua Ma
by Bentuo Zheng (bzheng) 17 Jan '16

17 Jan '16

This is an announcement for the paper “Closed subspaces and some basic topological properties of noncommutative Orlicz spaces” by Lining Jiang, Zhenhua Ma. Abstract: In this paper, we study the noncommutative Orlicz space Lφ(˜,τ), which generalizes the concept of noncommutative Lp space, where  is a von Neumann algebra, and φ is an Orlicz function. As a modular space, the space Lφ(˜,τ) possesses the Fatou property, and consequently, it is a Banach space. In addition, a new description of the subspace Eφ(˜,τ)=⋂Lφ(˜,τ)⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ in Lφ(˜,τ), which is closed under the norm topology and dense under the measure topology, is given. Moreover, if the Orlicz function φ satisfies the Δ2-condition, then Lφ(˜,τ) is uniformly monotone, and the convergence in the norm topology and measure topology coincide on the unit sphere. Hence, Eφ(˜,τ)=Lφ(˜,τ) if φ satisfies the Δ2-condition. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1601.02941 _______________________________________________ Banach mailing list Banach(a)mathdept.okstate.edu<mailto:Banach@mathdept.okstate.edu> http://cauchy.math.okstate.edu/cgi-bin/mailman/listinfo/banach Bentuo Zheng Associate Professor, Department of Mathematical Sciences, College of Arts and Sciences [UofM logo] The University of Memphis 359 Dunn Hall Memphis, TN 38152<http://www.memphis.edu/emailsignatures/emailsignaturemac.php#> 901.678<http://www.memphis.edu/emailsignatures/emailsignaturemac.php#>.3534 | memphis.edu<http://www.memphis.edu/> [UofM tw]<https://www.facebook.com/uofmemphis> [UofM tw] <https://twitter.com/uofmemphis> [UofM tw] <https://instagram.com/uofmemphis>

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Abstract of a paper by Karim Khanaki
by alspach＠math.okstate.edu 03 Jan '16

03 Jan '16

This is an announcement for the paper "Correspondences between model theory and banach space theory" by Karim Khanaki. Abstract: In \cite{K3} we pointed out the correspondence between a result of Shelah in model theory, i.e. a theory is unstable if and only if it has IP or SOP, and the well known compactness theorem of Eberlein and \v{S}mulian in functional analysis. In this paper, we relate a {\em natural} Banach space $V$ to a formula $\phi(x,y)$, and show that $\phi$ is stable (resp NIP, NSOP) if and only if $V$ is reflexive (resp Rosenthal, weakly sequentially complete) Banach space. Also, we present a proof of the Eberlein-\v{S}mulian theorem by a model theoretic approach using Ramsey theorems which is illustrative to show some correspondences between model theory and Banach space theory. Archive classification: math.LO math.FA Mathematics Subject Classification: 03C45, 46E15, 46B50 Submitted from: khanaki(a)ipm.ir The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.08691 or http://arXiv.org/abs/1512.08691

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Abstract of a paper by Antonio Aviles, Antonio J. Guirao, Sebastian Lajara, Jose Rodriguez, and Pedro Tradacete
by alspach＠math.okstate.edu 03 Jan '16

03 Jan '16

This is an announcement for the paper "Weakly compactly generated Banach lattices" by Antonio Aviles, Antonio J. Guirao, Sebastian Lajara, Jose Rodriguez, and Pedro Tradacete. Abstract: We study the different ways in which a weakly compact set can generate a Banach lattice. Among other things, it is shown that in an order continuous Banach lattice $X$, the existence of a weakly compact set $K \subset X$ such that $X$ coincides with the band generated by $K$, implies that $X$ is WCG. Archive classification: math.FA Mathematics Subject Classification: 46B42, 46B50 Submitted from: avileslo(a)um.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.08628 or http://arXiv.org/abs/1512.08628

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Abstract of a paper by Giorgos Chasapis, Apostolos Giannopoulos and Dimitris-Marios Liakopoulos
by alspach＠math.okstate.edu 03 Jan '16

03 Jan '16

This is an announcement for the paper "Estimates for measures of lower dimensional sections of convex bodies" by Giorgos Chasapis, Apostolos Giannopoulos and Dimitris-Marios Liakopoulos. Abstract: We present an alternative approach to some results of Koldobsky on measures of sections of symmetric convex bodies, which allows us to extend them to the not necessarily symmetric setting. We prove that if $K$ is a convex body in ${\mathbb R}^n$ with $0\in {\rm int}(K)$ and if $\mu $ is a measure on ${\mathbb R}^n$ with a locally integrable non-negative density $g$ on ${\mathbb R}^n$, then \begin{equation*}\mu (K)\leq \left (c\sqrt{n-k}\right )^k\max_{F\in G_{n,n-k}}\mu (K\cap F)\cdot |K|^{\frac{k}{n}}\end{equation*} for every $1\leq k\leq n-1$. Also, if $\mu $ is even and log-concave, and if $K$ is a symmetric convex body in ${\mathbb R}^n$ and $D$ is a compact subset of ${\mathbb R}^n$ such that $\mu (K\cap F)\leq \mu (D\cap F)$ for all $F\in G_{n,n-k}$, then \begin{equation*}\mu (K)\leq \left (ckL_{n-k}\right )^{k}\mu (D),\end{equation*} where $L_s$ is the maximal isotropic constant of a convex body in ${\mathbb R}^s$. Our method employs a generalized Blaschke-Petkantschin formula and estimates for the dual affine quermassintegrals. Archive classification: math.MG math.FA Submitted from: gchasapis(a)math.uoa.gr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.08393 or http://arXiv.org/abs/1512.08393

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Abstract of a paper by Stephen Simons
by alspach＠math.okstate.edu 03 Jan '16

03 Jan '16

This is an announcement for the paper "Bootstrapping the Mazur--Orlicz--K\"onig theorem" by Stephen Simons. Abstract: In this paper, we give some extensions of K\"onig's extension of the Mazur-Orlicz theorem. These extensions include generalizations of a surprising recent result of Sun Chuanfeng, and generalizations to the product of more than two spaces of the ``Hahn-Banach-Lagrange'' theorem. Archive classification: math.FA Mathematics Subject Classification: 46A22, 46N10 Remarks: 9 pages Submitted from: stesim38(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.08020 or http://arXiv.org/abs/1512.08020

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Abstract of a paper by Sofiya Ostrovska and Mikhail I. Ostrovskii
by alspach＠math.okstate.edu 03 Jan '16

03 Jan '16

This is an announcement for the paper "Nonexistence of embeddings with uniformly bounded distortions of Laakso graphs into diamond graphs" by Sofiya Ostrovska and Mikhail I. Ostrovskii. Abstract: Diamond graphs and Laakso graphs are important examples in the theory of metric embeddings. Many results for these families of graphs are similar to each other. In this connection, it is natural to ask whether one of these families admits uniformly bilipschitz embeddings into the other. The well-known fact that Laakso graphs are uniformly doubling but diamond graphs are not, immediately implies that diamond graphs do not admit uniformly bilipschitz embeddings into Laakso graphs. The main goal of this paper is to prove that Laakso graphs do not admit uniformly bilipschitz embeddings into diamond graphs. Archive classification: math.MG math.CO math.FA Mathematics Subject Classification: 05C12, 30L05, 46B85 Submitted from: ostrovsm(a)stjohns.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.06439 or http://arXiv.org/abs/1512.06439

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Abstract of a paper by Siu Lam Leung, Sarah Nelson, Sofiya Ostrovska, and Mikhail Ostrovskii
by alspach＠math.okstate.edu 03 Jan '16

03 Jan '16

This is an announcement for the paper "Distortion of embeddings of binary trees into diamond graphs" by Siu Lam Leung, Sarah Nelson, Sofiya Ostrovska, and Mikhail Ostrovskii. Abstract: Diamond graphs and binary trees are important examples in the theory of metric embeddings and also in the theory of metric characterizations of Banach spaces. Some results for these families of graphs are parallel to each other, for example superreflexivity of Banach spaces can be characterized both in terms of binary trees (Bourgain, 1986) and diamond graphs (Johnson-Schechtman, 2009). In this connection, it is natural to ask whether one of these families admits uniformly bilipschitz embeddings into the other. This question was answered in the negative by Ostrovskii (2014), who left it open to determine the order of growth of the distortions. The main purpose of this paper is to get a sharp-up-to-a-logarithmic-factor estimate for the distortions of embeddings of binary trees into diamond graphs. Archive classification: math.MG math.CO math.FA Mathematics Subject Classification: 05C12, 30L05, 46B85 Submitted from: ostrovsm(a)stjohns.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.06438 or http://arXiv.org/abs/1512.06438

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Abstract of a paper by Guillermo P. Curbera and Werner J. Ricker
by alspach＠math.okstate.edu 03 Jan '16

03 Jan '16

This is an announcement for the paper "The weak Banach-Saks property for function spaces" by Guillermo P. Curbera and Werner J. Ricker. Abstract: We establish the weak Banach-Saks property for function spaces arising as the optimal domain of an operator. Archive classification: math.FA Mathematics Subject Classification: 46E30, 46B20, 46G10 Submitted from: curbera(a)us.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.05728 or http://arXiv.org/abs/1512.05728

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Abstract of a paper by Jesus A. Jaramillo, Raquel Gonzalo and Diego Yanez
by alspach＠math.okstate.edu 03 Jan '16

03 Jan '16

This is an announcement for the paper "Asymptotic Smoothness, Convex Envelopes and Polynomial Norms" by Jesus A. Jaramillo, Raquel Gonzalo and Diego Yanez. Abstract: We introduce a suitable notion of asymptotic smoothness on infinite dimensional Banach spaces, and we prove that, under some structural restrictions on the space, the convex envelope of an asymptotically smooth function is asymptotically smooth. Furthermore, we study convexity and smoothness properties of polynomial norms, and we obtain that a polynomial norm of degree N has modulus of convexity of power type N. Archive classification: math.FA Submitted from: jaramil(a)mat.ucm.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.05407 or http://arXiv.org/abs/1512.05407

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Banach January 2016