Banach September 2009

banach@mathdept.okstate.edu

2 participants
14 discussions

Maurey-Schwartz Seminar Notes
by Dale Alspach 29 Sep '09

29 Sep '09

Perhaps you all already know this but just in case I just found out (perhaps not new info...) the Maurey Schwartz seminars have ALL been scanned and are on line at http://www.numdam.org/numdam-bin/recherche just select under journals seminaire d analyse fonctionnelle (also known as seminaire Maurey-Schwartz) so you can fill the gaps in your collection ! best gilles

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Abstract of a paper by Grigoris Paouris and Elisabeth M. Werner
by alspach＠fourier.math.okstate.edu 25 Sep '09

25 Sep '09

This is an announcement for the paper "Relative entropy of cone measures and $L_p$ centroid bodies" by Grigoris Paouris and Elisabeth M. Werner. Abstract: Let $K$ be a convex body in $\mathbb R^n$. We introduce a new affine invariant, which we call $\Omega_K$, that can be found in three different ways: as a limit of normalized $L_p$-affine surface areas, as the relative entropy of the cone measure of $K$ and the cone measure of $K^\circ$, as the limit of the volume difference of $K$ and $L_p$-centroid bodies. We investigate properties of $\Omega_K$ and of related new invariant quantities. In particular, we show new affine isoperimetric inequalities and we show a "information inequality" for convex bodies. Archive classification: math.FA Mathematics Subject Classification: 52A20, 53A15 The source file(s), PaourWern.tex: 116056 bytes, is(are) stored in gzipped form as 0909.4361.gz with size 27kb. The corresponding postcript file has gzipped size 188kb. 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/0909.4361 or http://arXiv.org/abs/0909.4361 or by email in unzipped form by transmitting an empty message with subject line uget 0909.4361 or in gzipped form by using subject line get 0909.4361 to: math(a)arXiv.org.

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Abstract of a paper by Lee-Ad Gottlieb and Robert Krauthgamer
by alspach＠fourier.math.okstate.edu 25 Sep '09

25 Sep '09

This is an announcement for the paper "A nonlinear approach to dimension reduction" by Lee-Ad Gottlieb and Robert Krauthgamer. Abstract: A powerful embedding theorem in the context of dimension reduction is the $\ell_2$ flattening lemma of Johnson and Lindenstrauss. It has been conjectured that improved dimension bounds may be achievable for some data sets by bounding the target dimension in terms of the intrinsic dimensionality of the data set (for example, the doubling dimension). One such problem was proposed by Lang and Plaut, and is still open. We pose another question in this line of work: Does the snowflake metric $d^{1/2}$ of a doubling set $S\subset\ell_2$ always embed with distortion O(1) into $\ell_2^D$, for dimension $D$ that depends solely on the doubling constant of the metric? We resolve this question in the affirmative, and furthermore obtain distortion arbitrarily close to 1. Moreover, our techniques are sufficiently robust to be applicable also to the more difficult spaces $\ell_1$ and $\ell_\infty$, although these extensions achieve dimension bounds that are quantitatively inferior than those for $\ell_2$. Archive classification: cs.CG cs.DS math.FA The source file(s), , is(are) stored in gzipped form as with size . The corresponding postcript file has gzipped size . Submitted from: adi(a)cs.nyu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0907.5477 or http://arXiv.org/abs/0907.5477 or by email in unzipped form by transmitting an empty message with subject line uget 0907.5477 or in gzipped form by using subject line get 0907.5477 to: math(a)arXiv.org.

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Abstract of a paper by Tuomas P. Hytonen
by alspach＠fourier.math.okstate.edu 25 Sep '09

25 Sep '09

This is an announcement for the paper "New thoughts on the vector-valued Mihlin-H\"ormander multiplier theorem" by Tuomas P. Hytonen. Abstract: Let X be a UMD space with type t and cotype q, and let T be a Fourier multiplier operator with a scalar-valued symbol m. If the Mihlin multiplier estimate holds for all partial derivatives of m up to the order n/max(t,q')+1, then T is bounded on the X-valued Bochner spaces. For scalar-valued multipliers, this improves the theorem of Girardi and Weis (J. Funct. Anal., 2003) who required similar assumptions for derivatives up to the order n/r+1, where r is a Fourier-type of X. However, the present method does not apply to operator-valued multipliers, which are also covered by the Girardi-Weis theorem. Archive classification: math.FA Mathematics Subject Classification: 42B15; 46B09; 46B20 Remarks: 8 pages, submitted The source file(s), cotype-multipliers.bbl: 2535 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0909.3225 or http://arXiv.org/abs/0909.3225 or by email in unzipped form by transmitting an empty message with subject line uget 0909.3225 or in gzipped form by using subject line get 0909.3225 to: math(a)arXiv.org.

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Abstract of a paper by Heinz H. Bauschke, Xianfu Wang, and Liangjin Yao
by alspach＠fourier.math.okstate.edu 25 Sep '09

25 Sep '09

This is an announcement for the paper "Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter" by Heinz H. Bauschke, Xianfu Wang, and Liangjin Yao. Abstract: In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator $S$ on $\ell^{2}$ is skew. We show its domain is a proper subset of the domain of its adjoint $S^*$, and $-S^*$ is not maximal monotone. This gives a negative answer to a recent question posed by Svaiter. The second unbounded linear maximal monotone operator is the inverse Volterra operator $T$ on $L^{2}[0,1]$. We compare the domain of $T$ with the domain of its adjoint $T^*$ and show that the skew part of $T$ admits two distinct linear maximal monotone skew extensions. These unbounded linear maximal monotone operators show that the constraint qualification for the maximality of the sum of maximal monotone operators can not be significantly weakened, and they are simpler than the example given by Phelps-Simons. Interesting consequences on Fitzpatrick functions for sums of two maximal monotone operators are also given. Archive classification: math.FA math.OC Mathematics Subject Classification: 47A06; 47H05; 47A05; 47B65 The source file(s), arxiv.tex: 67090 bytes, is(are) stored in gzipped form as 0909.2675.gz with size 18kb. The corresponding postcript file has gzipped size 133kb. Submitted from: heinz.bauschke(a)ubc.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0909.2675 or http://arXiv.org/abs/0909.2675 or by email in unzipped form by transmitting an empty message with subject line uget 0909.2675 or in gzipped form by using subject line get 0909.2675 to: math(a)arXiv.org.

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Abstract of a paper by Volker Runde
by alspach＠fourier.math.okstate.edu 25 Sep '09

25 Sep '09

This is an announcement for the paper "(Non-)amenability of B(E)" by Volker Runde. Abstract: In 1972, the late B. E. Johnson introduced the notion of an amenable Banach algebra and asked whether the Banach algebra $B(E)$ of all bounded linear operators on a Banach space $E$ could ever be amenable if $\dim E = \infty$. Somewhat surprisingly, this question was answered positively only very recently as a by-product of the Argyros--Haydon result that solves the ``scalar plus compact problem'': there is an infinite-dimensional Banach space $E$, the dual of which is $\ell^1$, such that $B(E) = K(E)+ \mathbb{C} \, \id_E$. Still, $B(\ell^2)$ is not amenable, and in the past decade, $ B(\ell^p)$ was found to be non-amenable for $p=1,2,\infty$ thanks to the work of C. J. Read, G. Pisier, and N. Ozawa. We survey those results, and then---based on joint work with M. Daws---outline a proof that establishes the non-amenability of $B(\ell^p)$ for all $p \in [1,\infty]$. Archive classification: math.FA math.HO Mathematics Subject Classification: Primary 47L10; Secondary 46B07, 46B45, 46H20 Remarks: 16 pages; a survey article The source file(s), BE.tex: 42631 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0909.2628 or http://arXiv.org/abs/0909.2628 or by email in unzipped form by transmitting an empty message with subject line uget 0909.2628 or in gzipped form by using subject line get 0909.2628 to: math(a)arXiv.org.

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Abstract of a paper by Mark Veraar
by alspach＠fourier.math.okstate.edu 25 Sep '09

25 Sep '09

This is an announcement for the paper "On Khintchine inequalities with a weight" by Mark Veraar. Abstract: In this note we prove a weighted version of the Khintchine inequalities. Archive classification: math.PR math.FA Mathematics Subject Classification: 60E15; 60G50 The source file(s), Khintchine_arxiv.tex: 12141 bytes, is(are) stored in gzipped form as 0909.2586.gz with size 5kb. The corresponding postcript file has gzipped size 50kb. Submitted from: mark(a)profsonline.nl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0909.2586 or http://arXiv.org/abs/0909.2586 or by email in unzipped form by transmitting an empty message with subject line uget 0909.2586 or in gzipped form by using subject line get 0909.2586 to: math(a)arXiv.org.

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Abstract of a paper by Miguel Martin and T.S.S.R.K. Rao
by alspach＠fourier.math.okstate.edu 25 Sep '09

25 Sep '09

This is an announcement for the paper "On remotality for convex sets in Banach spaces" by Miguel Martin and T.S.S.R.K. Rao. Abstract: We show that every infinite dimensional Banach space has a closed and bounded convex set that is not remotal. Archive classification: math.FA Mathematics Subject Classification: Primary 46B20, Secondary 41A50 Remarks: 5 pages, to appear in the Journal of Approximation Theory The source file(s), Arxiv-2009-09-10-nonremotal.tex: 16101 bytes, is(are) stored in gzipped form as 0909.1992.gz with size 6kb. The corresponding postcript file has gzipped size 73kb. Submitted from: mmartins(a)ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0909.1992 or http://arXiv.org/abs/0909.1992 or by email in unzipped form by transmitting an empty message with subject line uget 0909.1992 or in gzipped form by using subject line get 0909.1992 to: math(a)arXiv.org.

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Abstract of a paper by Rafael Dahmen
by alspach＠fourier.math.okstate.edu 08 Sep '09

08 Sep '09

This is an announcement for the paper "Lie groups associated to H"older-continuous functions" by Rafael Dahmen. Abstract: We proof some basic tools about spaces of H"older-continuous functions between (in general infinite dimensional) Banach spaces and use them to construct new examples of infinite dimensional (LB)-Lie groups. Archive classification: math.FA The source file(s), hoelder.bbl: 1140 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0908.3843 or http://arXiv.org/abs/0908.3843 or by email in unzipped form by transmitting an empty message with subject line uget 0908.3843 or in gzipped form by using subject line get 0908.3843 to: math(a)arXiv.org.

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Abstract of a paper by Shannon Bishop
by alspach＠fourier.math.okstate.edu 08 Sep '09

08 Sep '09

This is an announcement for the paper "Mixed modulation spaces and their application to pseudodifferential operators" by Shannon Bishop. Abstract: This paper uses frame techniques to characterize the Schatten class properties of integral operators. The main result shows that if the coefficients of certain frame expansions of the kernel of an integral operator are in $ \ell^{2,p} $, then the operator is Schatten p-class. As a corollary, we conclude that if the kernel or Kohn-Nirenberg symbol of a pseudodifferential operator lies in a particular mixed modulation space, then the operator is Schatten p-class. Our corollary improves existing Schatten class results for pseudodifferential operators and the corollary is sharp in the sense that larger mixed modulation spaces yield operators that are not Schatten class. Archive classification: math.FA math.CA Mathematics Subject Classification: 35S05 (Primary) 42C15, 47B10 (Secondary) Remarks: To be published in Journal of Mathematical Analysis and Applications The source file(s), genmodshortD3.tex: 53295 bytes, is(are) stored in gzipped form as 0908.3420.gz with size 13kb. The corresponding postcript file has gzipped size 108kb. Submitted from: sbishop(a)math.gatech.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0908.3420 or http://arXiv.org/abs/0908.3420 or by email in unzipped form by transmitting an empty message with subject line uget 0908.3420 or in gzipped form by using subject line get 0908.3420 to: math(a)arXiv.org.

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Banach September 2009