May 2014 - Banach - mathdept.okstate.edu

Abstract of a paper by Stanislaw Kwapien, Mark Veraar, and Lutz Weis
by alspach＠math.okstate.edu 02 May '14

02 May '14

This is an announcement for the paper "$R$-boundedness versus $\gamma$-boundedness" by Stanislaw Kwapien, Mark Veraar, and Lutz Weis. Abstract: It is well-known that in Banach spaces with finite cotype, the $R$-bounded and $\gamma$-bounded families of operators coincide. If in addition $X$ is a Banach lattice, then these notions can be expressed as square function estimates. It is also clear that $R$-boundedness implies $\gamma$-boundedness. In this note we show that all other possible inclusions fail. Furthermore, we will prove that $R$-boundedness is stable under taking adjoints if and only if the underlying space is $K$-convex. Archive classification: math.FA math.PR Mathematics Subject Classification: 47B99 (Primary) 46B09, 46B07, 47B10 (Secondary) Submitted from: m.c.veraar(a)tudelft.nl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1404.7328 or http://arXiv.org/abs/1404.7328

1 0

Abstract of a paper by Narutaka Ozawa and Gilles Pisier
by alspach＠math.okstate.edu 02 May '14

02 May '14

This is an announcement for the paper "A continuum of $\mathrm{C}^*$-norms on $\IB(H)\otimes \IB(H)$ and related tensor products" by Narutaka Ozawa and Gilles Pisier. Abstract: For any pair $M,N$ of von Neumann algebras such that the algebraic tensor product $M\otimes N$ admits more than one $\mathrm{C}^*$-norm, the cardinal of the set of $\mathrm{C}^*$-norms is at least $ {2^{\aleph_0}}$. Moreover there is a family with cardinality $ {2^{\aleph_0}}$ of injective tensor product functors for $\mathrm{C}^*$-algebras in Kirchberg's sense. Let $\IB=\prod_n M_{n}$. We also show that, for any non-nuclear von Neumann algebra $M\subset \IB(\ell_2)$, the set of $\mathrm{C}^*$-norms on $\IB \otimes M$ has cardinality equal to $2^{2^{\aleph_0}}$. Archive classification: math.OA Submitted from: pisier(a)math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1404.7088 or http://arXiv.org/abs/1404.7088

1 0

Abstract of a paper by Deping Ye
by alspach＠math.okstate.edu 02 May '14

02 May '14

This is an announcement for the paper "Dual Orlicz-Brunn-Minkowski theory: Orlicz $\varphi$-radial addition, Orlicz $L_{\phi}$-dual mixed volume and related inequalities" by Deping Ye. Abstract: This paper develops basic setting for the dual Orlicz-Brunn-Minkowski theory for star bodies. An Orlicz $\varphi$-radial addition of two or more star bodies is proposed and related dual Orlicz-Brunn-Minkowski inequality is established. Based on a linear Orlicz $\varphi$-radial addition of two star bodies, we derive a formula for the Orlicz $L_{\phi}$-dual mixed volume. Moreover, a dual Orlicz-Minkowski inequality for the Orlicz $L_{\phi}$-dual mixed volume, a dual Orlicz isoperimetric inequality for the Orlicz $L_{\phi}$-dual surface area and a dual Orlicz-Urysohn inequality for the Orlicz $L_{\phi}$-harmonic mean radius are proved. Archive classification: math.MG math.DG math.FA Mathematics Subject Classification: 52A20, 53A15 Submitted from: deping.ye(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1404.6991 or http://arXiv.org/abs/1404.6991

1 0

Abstract of a paper by Sean Li
by alspach＠math.okstate.edu 02 May '14

02 May '14

This is an announcement for the paper "Markov convexity and nonembeddability of the Heisenberg group" by Sean Li. Abstract: We compute the Markov convexity invariant of the continuous Heisenberg group $\mathbb{H}$ to show that it is Markov 4-convex and cannot be Markov $p$-convex for any $p < 4$. As Markov convexity is a biLipschitz invariant and Hilbert space is Markov 2-convex, this gives a different proof of the classical theorem of Pansu and Semmes that the Heisenberg group does not admit a biLipschitz embedding into any Euclidean space. The Markov convexity lower bound will follow from exhibiting an explicit embedding of Laakso graphs $G_n$ into $\mathbb{H}$ that has distortion at most $C n^{1/4} \sqrt{\log n}$. We use this to show that if $X$ is a Markov $p$-convex metric space, then balls of the discrete Heisenberg group $\mathbb{H}(\mathbb{Z})$ of radius $n$ embed into $X$ with distortion at least some constant multiple of $$\frac{(\log n)^{\frac{1}{p}-\frac{1}{4}}}{\sqrt{\log \log n}}.$$ Finally, we show somewhat unexpectedly that the optimal distortion of embeddings of binary trees $B_m$ into the infinite dimensional Heisenberg group is on the order of $\sqrt{\log m}$ Archive classification: math.MG math.FA Remarks: 20 pages Submitted from: seanli(a)math.uchicago.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1404.6751 or http://arXiv.org/abs/1404.6751

1 0

Abstract of a paper by Zakhar Kabluchko and Dmitry Zaporozhets
by alspach＠math.okstate.edu 02 May '14

02 May '14

This is an announcement for the paper "Intrinsic volumes of Sobolev balls" by Zakhar Kabluchko and Dmitry Zaporozhets. Abstract: A formula due to Sudakov relates the first intrinsic volume of a convex set in a Hilbert space to the maximum of the isonormal Gaussian process over this set. Using this formula we compute the first intrinsic volumes of infinite-dimensional convex compact sets including unit balls with respect to Sobolev-type seminorms and ellipsoids in the Hilbert space. We relate the distribution of the random one-dimensional projections of these sets to the distributions $S_1,S_2,C_1,C_2$ studied by Biane, Pitman, Yor [Bull. AMS 38 (2001)]. We show that the $k$-th intrinsic volume of the set of all functions on $[0,1]$ which have Lipschitz constant bounded by $1$ and which vanish at $0$ (respectively, which have vanishing integral) is given by $$ V_k = \frac{\pi^{k/2}}{\Gamma\left(\frac 32 k +1 \right)}, \text{ respectively } V_k = \frac{\pi^{(k+1)/2}}{2\Gamma\left(\frac 32 k +\frac 32\right)}. $$ This is related to the results of Gao and Vitale [Discrete Comput. Geom.} 26 (2001), Elect. Comm. Probab. 8 (2003)] who considered a similar question for functions with a restriction on the total variation instead of the Lipschitz constant. Using the results of Gao and Vitale we give a new proof of the formula for the expected volume of the convex hull of the $d$-dimensional Brownian motion which is due to Eldan [Elect. J. Probab., to appear]. Additionally, we prove an analogue of Eldan's result for the Brownian bridge. Similarly, we show that the results on the intrinsic volumes of the Lipschitz balls can be translated into formulae for the expected volumes of zonoids (Aumann integrals) generated by the Brownian motion and the Brownian bridge. Our proofs exploit Sudakov's and Tsirelson's theorems which establish a connection between the intrinsic volumes and the isonormal Gaussian process. Archive classification: math.PR math.FA math.MG Mathematics Subject Classification: Primary, 60D05, secondary, 60G15, 52A22 Remarks: 23 pages Submitted from: sachar.k(a)gmx.net The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1404.6113 or http://arXiv.org/abs/1404.6113

1 0

Abstract of a paper by Luis Rademacher
by alspach＠math.okstate.edu 02 May '14

02 May '14

This is an announcement for the paper "A simplicial polytope that maximizes the isotropic constant must be a simplex" by Luis Rademacher. Abstract: The isotropic constant $L_K$ is an affine-invariant measure of the spread of a convex body $K$. For a $d$-dimensional convex body $K$, $L_K$ can be defined by $L_K^{2d} = \det(A(K))/(\mathrm{vol}(K))^2$, where $A(K)$ is the covariance matrix of the uniform distribution on $K$. It is an outstanding open problem to find a tight asymptotic upper bound of the isotropic constant as a function of the dimension. It has been conjectured that there is a universal constant upper bound. The conjecture is known to be true for several families of bodies, in particular, highly symmetric bodies such as bodies having an unconditional basis. It is also known that maximizers cannot be smooth. In this work we study the gap between smooth bodies and highly symmetric bodies by showing progress towards reducing to a highly symmetric case among non-smooth bodies. More precisely, we study the set of maximizers among simplicial polytopes and we show that if a simplicial $d$-polytope $K$ is a maximizer of the isotropic constant among $d$-dimensional convex bodies, then when $K$ is put in isotropic position it is symmetric around any hyperplane spanned by a $(d-2)$-dimensional face and the origin. By a result of Campi, Colesanti and Gronchi, this implies that a simplicial polytope that maximizes the isotropic constant must be a simplex. Archive classification: math.FA math.MG math.PR Submitted from: lrademac(a)cse.ohio-state.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1404.5662 or http://arXiv.org/abs/1404.5662

1 0

Abstract of a paper by Anna Kamont and Markus Passenbrunner
by alspach＠math.okstate.edu 02 May '14

02 May '14

This is an announcement for the paper "Unconditionality of orthogonal spline systems in $H^1$" by Anna Kamont and Markus Passenbrunner. Abstract: We give a simple geometric characterization of knot sequences for which the corresponding orthonormal spline system of arbitrary order $k$ is an unconditional basis in the atomic Hardy space $H^1[0,1]$. Archive classification: math.FA Remarks: 31 pages Submitted from: markus.passenbrunner(a)jku.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1404.5493 or http://arXiv.org/abs/1404.5493

1 0

Abstract of a paper by David Alonso-Gutierrez, Markus Passenbrunner and Joscha Prochno
by alspach＠math.okstate.edu 02 May '14

02 May '14

This is an announcement for the paper "Probabilistic estimates for tensor products of random vectors" by David Alonso-Gutierrez, Markus Passenbrunner and Joscha Prochno. Abstract: We prove some probabilistic estimates for tensor products of random vectors. As an application we obtain embeddings of certain matrix spaces into $L_1$. Archive classification: math.FA Mathematics Subject Classification: 46B09, 46B07, 46B28, 46B45 Remarks: 14 pages Submitted from: joscha.prochno(a)jku.at The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1404.5423 or http://arXiv.org/abs/1404.5423

1 0

Abstract of a paper by Grigoris Paouris and Petros Valettas
by alspach＠math.okstate.edu 02 May '14

02 May '14

This is an announcement for the paper "Neighborhoods on the Grasmannian of marginals with bounded isotropic constant" by Grigoris Paouris and Petros Valettas. Abstract: We show that for any isotropic log-concave probability measure $\mu$ on $\mathbb R^n$, for every $\varepsilon > 0$, every $1 \leq k \leq \sqrt{n}$ and any $E \in G_{n,k}$ there exists $F \in G_{n,k}$ with $d(E,F) < \varepsilon$ and $L_{\pi_F\mu} < C/\varepsilon$. Archive classification: math.FA Submitted from: petvalet(a)math.tamu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1404.4988 or http://arXiv.org/abs/1404.4988

1 0

Abstract of a paper by N. Albuquerque, D. Nunez-Alarcon, J. Santos and D. M. Serrano-Rodriguez
by alspach＠math.okstate.edu 02 May '14

02 May '14

This is an announcement for the paper "Absolutely summing multilinear operators via interpolation" by N. Albuquerque, D. Nunez-Alarcon, J. Santos and D. M. Serrano-Rodriguez. Abstract: We use an interpolative technique from \cite{abps} to introduce the notion of multiple $N$-separately summing operators. Our approach extends and unifies some recent results; for instance we recover the best known estimates of the multilinear Bohnenblust-Hille constants due to F. Bayart, D. Pellegrino and J. Seoane-Sep\'ulveda. More precisely, as a consequence of our main result, for $1\leq t<2$ and $m\in \mathbb{N}$ we prove that $$ \left( \sum_{i_{1},\dots ,i_{m}=1}^{\infty }\left\vert U\left(e_{i_{1}},\dots ,e_{i_{m}}\right) \right\vert^{\frac{2tm}{2+(m-1)t}}\right)^{\frac{2+(m-1)t}{2tm}} \leq \left[\prod_{j=2}^{m}\Gamma \left( 2-\frac{2-t}{jt-2t+2}\right) ^{\frac{t(j-2)+2}{2t-2jt}}\right] \left\Vert U\right\Vert $$ for all complex $m$-linear forms $U:c_{0}\times \cdot \cdot \cdot \times c_{0}\rightarrow \mathbb{C}$. Archive classification: math.FA Submitted from: ngalbqrq(a)gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1404.4949 or http://arXiv.org/abs/1404.4949

1 0