Banach November 2011

banach@mathdept.okstate.edu

2 participants
23 discussions

Meeting February 2012
by Gonzalez Ortiz, Manuel 09 Nov '11

09 Nov '11

ANNOUNCEMENT OF MEETING Convexity in Banach spaces -- An homage to Piero Papini Castro Urdiales (Cantabria, Spain), 20-24 February 2012 Convexity is a central topic with many applications in functional analysis, geometry, mathematical economy, control theory, etc. This edition of the Castro Urdiales Banach Space Meeting will be focused on the study of convexity; with special emphasis in its applications to Banach space theory. The Meeting is dedicated to celebrate Professor Piero Papini, whose work has always been close to the study of convexity in normed spaces, on the occasion of his retirement. The following mathematicians have accepted so far to deliver a plenary conference: Carlos Benitez (Univ. Extremadura) - Polarization constants in inner product spaces. Felix Cabello Sanchez (Univ. Extremadura) - Mathematical ping-pong. Vladimir Fonf (Ben Gurion Univ.) - Polyhedral spaces. Peter Gruber (Univ. Wien) - Great personalities of convex geometry from antiquity up to the present. - Normal bundles of convex bodies. Jose P. Moreno (Univ. Madrid) - Diametrically complete sets. Justo Puerto (Univ. Sevilla) - Location problems, solutions, algorithms and the like. David Yost (Univ. Ballarat) - Constants and parameters in Banach spaces. Additionally, those wishing to highlight some aspect of the career or research of Prof. Papini, or to present new results in convexity, will have the opportunity to deliver a short talk during the meeting. Please fill the corresponding request in the registration form. The meeting will be held in Castro Urdiales, a town by the sea in the north of Spain, about 20 Km from Bilbao, at the C.I.E.M. (Centro Internacional de Encuentros Matematicos) For registration and more information, please go to the web-site of the conference http://www.ciem.unican.es/encuentros/banach/2012/ Organizing Committee: Marco Baronti (Genova), Jesus M. F. Castillo (Badajoz) Manuel Gonz\'alez (Santander) and Clemente Zanco (Milano).

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Abstract of a paper by Karl-Mikael Perfekt
by alspach＠math.okstate.edu 01 Nov '11

01 Nov '11

This is an announcement for the paper "Duality and distance formulas in spaces defined by means of oscillation" by Karl-Mikael Perfekt. Abstract: For the classical space of functions with bounded mean oscillation, it is well known that VMO** = BMO and there are many characterizations of the distance from a function f in BMO to VMO. When considering the Bloch space, results in the same vein are available with respect to the little Bloch space. In this paper such duality results and distance formulas are obtained by pure functional analysis. Applications include general M\"obius invariant spaces such as Q_K-spaces, Lipschitz-H\"older spaces and rectangular BMO of several variables. Archive classification: math.FA math.CV Submitted from: perfekt(a)maths.lth.se The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1110.6766 or http://arXiv.org/abs/1110.6766

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Abstract of a paper by Gideon Schechtman
by alspach＠math.okstate.edu 01 Nov '11

01 Nov '11

This is an announcement for the paper "Dimension reduction in $L_p$, $0<p<2$" by Gideon Schechtman. Abstract: Complementing a recent observation of Newman and Rabinovich for $p=1$ we observe here that for all $0<p<2$ any $k$ points in $L_p$ embeds with distortion $(1+\e)$ into $\ell_p^n$ where $n$ is linear in $k$ (and polynomial in $\e^{-1}$). Archive classification: math.MG math.FA Mathematics Subject Classification: 46B85 Submitted from: gideon(a)weizmann.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1110.2148 or http://arXiv.org/abs/1110.2148

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Abstract of a paper by Gideon Schechtman
by alspach＠math.okstate.edu 01 Nov '11

01 Nov '11

This is an announcement for the paper "Euclidean sections of convex bodies, series of lectures" by Gideon Schechtman. Abstract: This is a somewhat expanded form of a four hours course given, with small variations, first at the educational workshop Probabilistic methods in Geometry, Bedlewo, Poland, July 6-12, 2008 and a few weeks later at the Summer school on Fourier analytic and probabilistic methods in geometric functional analysis and convexity, Kent, Ohio, August 13-20, 2008.\\ The main part of these notes gives yet another exposition of Dvoretzky's theorem on Euclidean sections of convex bodies with a proof based on Milman's. This material is by now quite standard. Towards the end of these notes we discuss issues related to fine estimates in Dvoretzky's theorem and there there are some results that didn't appear in print before. In particular there is an exposition of an unpublished result of Figiel (Claim \ref{claim:figiel}) which gives an upper bound on the possible dependence on $\e$ in Milman's theorem. We would like to thank Tadek Figiel for allowing us to include it here. There is also a better version of the proof of one of the results from \cite{sc2} giving a lower bound on the dependence on $\e$ in Dvoretzky's theorem. The improvement is in the statement and proof of Proposition \ref{prop:main} here which is a stronger version of the corresponding Corollary 1 in \cite{sc2}. Archive classification: math.FA math.MG Submitted from: gideon(a)weizmann.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1110.6401 or http://arXiv.org/abs/1110.6401

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Abstract of a paper by Tuomas P. Hytonen
by alspach＠math.okstate.edu 01 Nov '11

01 Nov '11

This is an announcement for the paper "Foundations of vector-valued singular integrals revisited---with random dyadic cubes" by Tuomas P. Hytonen. Abstract: The vector-valued $T(1)$ theorem due to Figiel, and a certain square function estimate of Bourgain for translations of functions with a limited frequency spectrum, are two cornerstones of harmonic analysis in UMD spaces. In this paper, a simplified approach to these results is presented, exploiting Nazarov, Treil and Volberg's method of random dyadic cubes, which allows to circumvent the most subtle parts of the original arguments. Archive classification: math.FA math.CA Mathematics Subject Classification: 42B20, 60G46 Remarks: 12 pages Submitted from: tuomas.hytonen(a)helsinki.fi The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1110.5826 or http://arXiv.org/abs/1110.5826

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Abstract of a paper by Pedro L. Kaufmann and Leonardo Pellegrini
by alspach＠math.okstate.edu 01 Nov '11

01 Nov '11

This is an announcement for the paper "Spaceability of sets of nowhere $L^q$ functions" by Pedro L. Kaufmann and Leonardo Pellegrini. Abstract: We say that a function $f:[0,1]\rightarrow \R$ is \emph{nowhere $L^q$} if, for each nonvoid open subset $U$ of $[0,1]$, the restriction $f|_U$ is not in $L^q(U)$. For a fixed $1\leq p <\infty$, we will show that the set $$ S_p\doteq \{f\in L^p[0,1]: f\mbox{ is nowhere $L^q$, for each }p<q\leq\infty\}, $$ united with $\{0\}$, contains an isometric and complemented copy of $\ell_p$. In particular, this improves a result from G. Botelho, V. F\'avaro, D. Pellegrino, and J. B. Seoane-Sep\'ulveda, $L_p[0,1]\setminus \cup_{q>p} L_q[0,1]$ is spaceable for every $p>0$, preprint, 2011., since $S_p$ turns out to be spaceable. In addition, our result is a generalization of one of the main results from S. G\l \c ab, P. L. Kaufmann, and L. Pellegrini, Spaceability and algebrability of sets of nowhere integrable functions, preprint, 2011. Archive classification: math.FA Mathematics Subject Classification: 26A30 Submitted from: leoime(a)yahoo.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/abs/1110.5774 or http://arXiv.org/abs/abs/1110.5774

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Abstract of a paper by Heinz H. Bauschke, Jonathan M. Borwein, Xianfu Wang, and Liangjin Yao
by alspach＠math.okstate.edu 01 Nov '11

01 Nov '11

This is an announcement for the paper "The Br\'ezis-Browder Theorem in a general Banach space" by Heinz H. Bauschke, Jonathan M. Borwein, Xianfu Wang, and Liangjin Yao. Abstract: During the 1970s Br\'ezis and Browder presented a now classical characterization of maximal monotonicity of monotone linear relations in reflexive spaces. In this paper, we extend and refine their result to a general Banach space. Archive classification: math.FA math.OC Mathematics Subject Classification: Primary 47A06, 47H05, Secondary 47B65, 47N10, 90C25 Remarks: 23 pages Submitted from: liangjinyao(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1110.5706 or http://arXiv.org/abs/1110.5706

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Abstract of a paper by S. Artstein-Avidan, B. Klartag, C. Schuett and E. Werner
by alspach＠math.okstate.edu 01 Nov '11

01 Nov '11

This is an announcement for the paper "Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality" by S. Artstein-Avidan, B. Klartag, C. Schuett and E. Werner. Abstract: We give a functional version of the affine isoperimetric inequality for log-concave functions which may be interpreted as an inverse form of a logarithmic Sobolev inequality inequality for entropy. A linearization of this inequality gives an inverse inequality to the Poincar'e inequality for the Gaussian measure. Archive classification: math.FA Mathematics Subject Classification: 52A20 Submitted from: elisabeth.werner(a)case.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1110.5551 or http://arXiv.org/abs/1110.5551

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Abstract of a paper by S.Waleed Noor and Dan Timotin
by alspach＠math.okstate.edu 01 Nov '11

01 Nov '11

This is an announcement for the paper "Embeddings of M\"{u}ntz spaces: the Hilbertian case" by S.Waleed Noor and Dan Timotin. Abstract: Given a strictly increasing sequence $\Lambda=(\lambda_n)$ of nonegative real numbers, with $\sum_{n=1}^\infty \frac{1}{\lambda_n}<\infty$, the M\"untz spaces $M_\Lambda^p$ are defined as the closure in $L^p([0,1])$ of the monomials $x^{\lambda_n}$. We discuss properties of the embedding $M_\Lambda^p\subset L^p(\mu)$, where $\mu$ is a finite positive Borel measure on the interval $[0,1]$. Most of the results are obtained for the Hilbertian case $p=2$, in which we give conditions for the embedding to be bounded, compact, or to belong to the Schatten--von Neumann ideals. Archive classification: math.FA math.CA Mathematics Subject Classification: 46E15, 46E20, 46E35 Submitted from: dtimotin(a)yahoo.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1110.5422 or http://arXiv.org/abs/1110.5422

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Abstract of a paper by Ohad Giladi, Assaf Naor, and Gideon Schechtman
by alspach＠math.okstate.edu 01 Nov '11

01 Nov '11

This is an announcement for the paper "Bourgain's discretization theorem" by Ohad Giladi, Assaf Naor, and Gideon Schechtman. Abstract: Bourgain's discretization theorem asserts that there exists a universal constant $C\in (0,\infty)$ with the following property. Let $X,Y$ be Banach spaces with $\dim X=n$. Fix $D\in (1,\infty)$ and set $\d= e^{-n^{Cn}}$. Assume that $\mathcal N$ is a $\d$-net in the unit ball of $X$ and that $\mathcal N$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $D$. Then the entire space $X$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $CD$. This mostly expository article is devoted to a detailed presentation of a proof of Bourgain's theorem. We also obtain an improvement of Bourgain's theorem in the important case when $Y=L_p$ for some $p\in [1,\infty)$: in this case it suffices to take $\delta= C^{-1}n^{-5/2}$ for the same conclusion to hold true. The case $p=1$ of this improved discretization result has the following consequence. For arbitrarily large $n\in \N$ there exists a family $\mathscr Y$ of $n$-point subsets of $\{1,\ldots,n\}^2\subseteq \R^2$ such that if we write $|\mathscr Y|= N$ then any $L_1$ embedding of $\mathscr Y$, equipped with the Earthmover metric (a.k.a. transportation cost metric or minimumum weight matching metric) incurs distortion at least a constant multiple of $\sqrt{\log\log N}$; the previously best known lower bound for this problem was a constant multiple of $\sqrt{\log\log \log N}$. Archive classification: math.FA math.MG Submitted from: naor(a)cims.nyu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1110.5368 or http://arXiv.org/abs/1110.5368

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Banach November 2011