Banach August 2018

banach@mathdept.okstate.edu

1 participants
10 discussions

Informal Analysis Seminar February 24-25, 2018
by Artem Zvavitch 05 Sep '19

05 Sep '19

Dear Colleagues, The Analysis group at Kent State University is happy to announce a meeting of the Informal Analysis Seminar, which will be held at the Department of Mathematical Sciences at Kent State University, February 24-25. The seminar will feature plenary speakers Robert Connelly (Cornell University), and Peter Sternberg (Indiana University) Each speaker will deliver a four hour lecture series designed to be accessible for graduate students. Funding is available to cover the local and travel expenses of a limited number of participants. Graduate students, postdoctoral researchers, and members of underrepresented groups are particularly encouraged to apply for support. A poster session will be held for researchers to display their work. Graduate students are particularly encouraged to submit a poster. Posters can be submitted electronically in PDF format. Further information, and an online registration form, can be found online http://www.math.kent.edu/informal We encourage you to register as soon as possible, but to receive support and/or help with hotel reservation, please register before January 29, 2018. Finally, please feel free to forward this email to any colleagues or students who you think may be interested in attending. Best regards, The Kent State Analysis Group

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Abstract of a paper by Ivan S. Yaroslavtsev
by Bentuo Zheng (bzheng) 01 Aug '18

01 Aug '18

This is an announcement for the paper “Burkholder-Davis-Gundy inequalities in UMD Banach spaces” by Ivan S. Yaroslavtsev<https://arxiv.org/search/math?searchtype=author&query=Yaroslavtsev%2C+I+S>. Abstract: In this paper we prove Burkholder-Davis-Gundy inequalities for a general martingale $M$ with values in a UMD Banach space $X$. Assuming that $M_0=0$, we show that the following two-sided inequality holds for all $1\leq p<\infty$: \begin{align}\label{eq:main}\tag{{$\star$}} \mathbb E \sup_{0\leq s\leq t} \|M_s\|^p \eqsim_{p, X} \mathbb E \gamma([\![M]\!]_t)^p ,\;\;\; t\geq 0. \end{align} Here $ \gamma([\![M]\!]_t) $ is the $L^2$-norm of the unique Gaussian measure on $X$ having $[\![M]\!]_t(x^*,y^*):= [\langle M,x^*\rangle, \langle M,y^*\rangle]_t$ as its covariance bilinear form. This extends to general UMD spaces a recent result by Veraar and the author, where a pointwise version of \eqref{eq:main} was proved for UMD Banach functions spaces $X$. We show that for continuous martingales, \eqref{eq:main} holds for all $0<p<\infty$, and that for purely discontinuous martingales the right-hand side of \eqref{eq:main} can be expressed more explicitly in terms of the jumps of $M$. For martingales with independent increments, \eqref{eq:main} is shown to hold more generally in reflexive Banach spaces $X$ with finite cotype. In the converse direction, we show that the validity of \eqref{eq:main} for arbitrary martingales implies the UMD property for $X$. As an application we prove various It\^o isomorphisms for vector-valued stochastic integrals with respect to general martingales, which extends earlier results by van Neerven, Veraar, and Weis for vector-valued stochastic integrals with respect to a Brownian motion. We also provide It\^o isomorphisms for vector-valued stochastic integrals with respect to compensated Poisson and general random measures. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1807.05573

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Abstract of a paper by Cleon S. Barroso
by Bentuo Zheng (bzheng) 01 Aug '18

01 Aug '18

This is an announcement for the paper “On the fixed point property in Banach spaces isomorphic to $c_0$” by Cleon S. Barroso<https://arxiv.org/search/math?searchtype=author&query=Barroso%2C+C+S>. Abstract: We prove that every Banach space containing a subspace isomorphic to $\co$ fails the fixed point property. The proof is based on an amalgamation approach involving a suitable combination of known results and techniques, including James's distortion theorem, Ramsey's combinatorial theorem, Brunel-Sucheston spreading model techniques and Dowling, Lennard and Turett's fixed point methodology employed in their characterization of weak compactness in $\co$. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1807.11614

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Abstract of a paper by Eve Oja, Tauri Viil, Dirk Werner
by Bentuo Zheng (bzheng) 01 Aug '18

01 Aug '18

This is an announcement for the paper “Totally smooth renormings” by Eve Oja<https://arxiv.org/search/math?searchtype=author&query=Oja%2C+E>, Tauri Viil<https://arxiv.org/search/math?searchtype=author&query=Viil%2C+T>, Dirk Werner<https://arxiv.org/search/math?searchtype=author&query=Werner%2C+D>. Abstract: We study the problem of totally smooth renormings of Banach spaces and provide such renormings for spaces which are weakly compactly generated. We also consider renormings for $(a,B,c)$-ideals. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1807.07335

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Abstract of a paper by Mikhail I. Ostrovskii, Beata Randrianantoanina
by Bentuo Zheng (bzheng) 01 Aug '18

01 Aug '18

This is an announcement for the paper “A characterization of superreflexivity through embeddings of lamplighter groups” by Mikhail I. Ostrovskii<https://arxiv.org/search/math?searchtype=author&query=Ostrovskii%2C+M+I>, Beata Randrianantoanina<https://arxiv.org/search/math?searchtype=author&query=Randrianantoanina%2C+B>. Abstract: We prove that finite lamplighter groups $\{\mathbb{Z}_2\wr\mathbb{Z}_n\}_{n\ge 2}$ with a standard set of generators embed with uniformly bounded distortions into any non-superreflexive Banach space, and therefore form a set of test-spaces for superreflexivity. Our proof is inspired by the well known identification of Cayley graphs of infinite lamplighter groups with the horocyclic product of trees. We cover $\mathbb{Z}_2\wr\mathbb{Z}_n$ by three sets with a structure similar to a horocyclic product of trees, which enables us to construct well-controlled embeddings. https://arxiv.org/abs/1807.06692

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Abstract of a paper by William B. Johnson, Tomasz Kania
by Bentuo Zheng (bzheng) 01 Aug '18

01 Aug '18

This is an announcement for the paper “Embedding Banach spaces into the space of bounded functions with countable support” by William B. Johnson<https://arxiv.org/search/math?searchtype=author&query=Johnson%2C+W+B>, Tomasz Kania<https://arxiv.org/search/math?searchtype=author&query=Kania%2C+T>. Abstract: We prove that a WLD subspace of the space $\ell_\infty^c(\Gamma)$ consisting of all bounded, countably supported functions on a set $\Gamma$ embeds isomorphically into $\ell_\infty$ if and only if it does not contain isometric copies of $c_0(\omega_1)$. Moreover, a subspace of $\ell_\infty^c(\omega_1)$ is constructed that has an unconditional basis, does not embed into $\ell_\infty$, and whose every weakly compact subset is separable (in particular, it cannot contain any isomorphic copies of $c_0(\omega_1)$). The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1807.05239

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Abstract of a paper by Stephen J. Dilworth, Denka Kutzarova, Mikhail I. Ostrovskii
by Bentuo Zheng (bzheng) 01 Aug '18

01 Aug '18

This is an announcement for the paper “Lipschitz free spaces on finite metric spaces” by Stephen J. Dilworth<https://arxiv.org/search/math?searchtype=author&query=Dilworth%2C+S+J>, Denka Kutzarova<https://arxiv.org/search/math?searchtype=author&query=Kutzarova%2C+D>, Mikhail I. Ostrovskii<https://arxiv.org/search/math?searchtype=author&query=Ostrovskii%2C+M+I>. Abstract: Main results of the paper: (1) For any finite metric space $M$ the Lipschitz free space on $M$ contains a large well-complemented subspace which is close to $\ell_1^n$. (2) Lipschitz free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell_1^n$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs. Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of graphs which are not necessarily graph automorphisms; (b) In the case of such recursive families of graphs as Laakso graphs we use the well-known approach of Gr\"unbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1807.03814

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Abstract of a paper by Bernardo Cascales, Rafa Chiclana, Luis García-Lirola, Miguel Martín, Abraham Rueda Zoca
by Bentuo Zheng (bzheng) 01 Aug '18

01 Aug '18

This is an announcement for the paper “On strongly norm attaining Lipschitz operators” by Bernardo Cascales<https://arxiv.org/search/math?searchtype=author&query=Cascales%2C+B>, Rafa Chiclana<https://arxiv.org/search/math?searchtype=author&query=Chiclana%2C+R>, Luis García-Lirola<https://arxiv.org/search/math?searchtype=author&query=Garc%C3%ADa-Lirola%2C…>, Miguel Martín<https://arxiv.org/search/math?searchtype=author&query=Mart%C3%ADn%2C+M>, Abraham Rueda Zoca<https://arxiv.org/search/math?searchtype=author&query=Zoca%2C+A+R>. Abstract: We study the set $\SA(M,Y)$ of those Lipschitz operators from a (complete pointed) metric space $M$ to a Banach space $Y$ which (strongly) attain their Lipschitz norm (i.e.\ the supremum defining the Lipschitz norm is a maximum). Extending previous results, we prove that this set is not norm dense when $M$ is length (or local) or when $M$ is a closed subset of $\R$ with positive Lebesgue measure, providing new example which have very different topological properties than the previously known ones. On the other hand, we study the linear properties which are sufficient to get Lindenstrauss property A for the Lipschitz-free space $\mathcal{F}(M)$ over $M$, and show that all of them actually provide the norm density of $\SA(M,Y)$ in the space of all Lipschitz operators from $M$ to any Banach space $Y$. Next, we prove that $\SA(M,\R)$ is weak sequentially dense in the space of all Lipschitz functions for all metric spaces $M$. Finally, we show that the norm of the bidual space to $\mathcal{F}(M)$ is octahedral provided the metric space $M$ is discrete but not uniformly discrete or $M'$ is infinite. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1807.03363

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Abstract of a paper by Christopher Boyd, Raymond A. Ryan, Nina Snigireva
by Bentuo Zheng (bzheng) 01 Aug '18

01 Aug '18

This is an announcement for the paper “Geometry of Spaces of Orthogonally Additive Polynomials on C(K)” by Christopher Boyd<https://arxiv.org/search/math?searchtype=author&query=Boyd%2C+C>, Raymond A. Ryan<https://arxiv.org/search/math?searchtype=author&query=Ryan%2C+R+A>, Nina Snigireva<https://arxiv.org/search/math?searchtype=author&query=Snigireva%2C+N>. Abstract: We study the space of orthogonally additive $n$-homogeneous polynomials on $C(K)$. There are two natural norms on this space. First, there is the usual supremum norm of uniform convergence on the closed unit ball. As every orthogonally additive $n$-homogeneous polynomial is regular with respect to the Banach lattice structure, there is also the regular norm. These norms are equivalent, but have significantly different geometric properties. We characterise the extreme points of the unit ball for both norms, with different results for even and odd degrees. As an application, we prove a Banach-Stone theorem. We conclude with a classification of the exposed points. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1807.02713

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Abstract of a paper by Markus Passenbrunner
by Bentuo Zheng (bzheng) 01 Aug '18

01 Aug '18

This is an announcement for the paper “Spline Characterizations of the Radon-Nikodým property” by Markus Passenbrunner<https://arxiv.org/search/math?searchtype=author&query=Passenbrunner%2C+M>. Abstract: We give necessary and sufficient conditions for a Banach space $X$ having the Radon-Nikod\'{y}m property in terms of polynomial spline sequences. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1807.01861

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Banach August 2018