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25,954
result(s) for
"Banach spaces"
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Function Spaces of Logarithmic Smoothness: Embeddings and Characterizations
In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin).
We establish the following results:
The key tools behind our results
are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier
transforms for functions such that their Fourier transforms are of monotone type or lacunary series.
eBook
Local Banach-space dichotomies and ergodic spaces
We prove a local version of Gowers’ Ramsey-type theorem [Ann. of Math. 156 (2002)], as well as local versions both of the Banach space first dichotomy (the “unconditional/HI” dichotomy) of Gowers [Ann. of Math. 156 (2002)] and of the third dichotomy (the “minimal/tight” dichotomy) due to Ferenczi–Rosendal [J. Funct. Anal. 257 (2009)]. This means that we obtain versions of these dichotomies restricted to certain families of subspaces called D-families, of which several concrete examples are given. As a main example, non-Hilbertian spaces form D-families; therefore versions of the above properties for non-Hilbertian spaces appear in new Banach space dichotomies. As a consequence we obtain new information on the number of subspaces of non-Hilbertian Banach spaces, making some progress towards the “ergodic” conjecture of Ferenczi–Rosendal and towards a question of Johnson.
Journal Article
Embeddings of Decomposition Spaces
Many smoothness spaces in harmonic analysis are decomposition spaces. In this paper we ask: Given two such spaces, is there an
embedding between the two?
A decomposition space
We establish readily verifiable criteria which ensure the
existence of a continuous inclusion (“an embedding”)
In a nutshell, in order to apply the embedding results presented in this
article, no knowledge of Fourier analysis is required; instead, one only has to study the geometric properties of the involved
coverings, so that one can decide the finiteness of certain sequence space norms defined in terms of the coverings.
These
sufficient criteria are quite sharp: For almost arbitrary coverings and certain ranges of
We also prove a
The resulting embedding theory is illustrated by applications
to
eBook
Complex interpolation between Hilbert, Banach and operator spaces
Motivated by a question of Vincent Lafforgue, the author studies the Banach spaces X satisfying the following property: there is a function \\varepsilon\\to \\Delta_X(\\varepsilon) tending to zero with \\varepsilon0 such that every operator T\\colon \\ L_2\\to L_2 with \\|T\\|\\le \\varepsilon that is simultaneously contractive (i.e., of norm \\le 1) on L_1 and on L_\\infty must be of norm \\le \\Delta_X(\\varepsilon) on L_2(X). The author shows that \\Delta_X(\\varepsilon) \\in O(\\varepsilon^\\alpha) for some \\alpha0 if X is isomorphic to a quotient of a subspace of an ultraproduct of \\theta-Hilbertian spaces for some \\theta0 (see Corollary 6.7), where \\theta-Hilbertian is meant in a slightly more general sense than in the author's earlier paper (1979).
eBook
Higher moments of Banach space valued random variables
We define the
We study both the projective and injective
tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations:
Bochner integrals, Pettis integrals and Dunford integrals.
One of the problems studied is whether two random variables with the
same injective moments (of a given order) necessarily have the same projective moments; this is of interest in applications. We show
that this holds if the Banach space has the approximation property, but not in general.
Several chapters are devoted to results
in special Banach spaces, including Hilbert spaces,
One of the main motivations of this paper is the application to Zolotarev metrics and
their use in the contraction method. This is sketched in an appendix.
eBook
Recent trends in operator theory and applications: Workshop on Recent Trends in Operator Theory and Applications, May 3-5, 2018, the University of Memphis, Memphis, Tennessee
This volume contains the proceedings of the workshop on Recent Trends in Operator Theory and Applications (RTOTA 2018), held from May 3-5, 2018, at the University of Memphis, Memphis, Tennessee.The articles introduce topics from operator theory to graduate students and early career researchers. Each such article provides insightful references, selection of results with articulation to modern research and recent advances in the area.Topics addressed in this volume include: generalized numerical ranges and their application to study perturbation of operators, and connections to quantum error correction; a survey of results on Toeplitz operators, and applications of Toeplitz operators to the study of reproducing kernel functions; results on the 2-local reflexivity problem of a set of operators; topics from the theory of preservers; and recent trends on the study of quotients of tensor product spaces and tensor operators. It also includes research articles that present overviews of state-of-the-art techniques from operator theory as well as applications to recent research trends and open questions. A goal of all articles is to introduce topics within operator theory to the general public.
eBook
Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation
The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are
given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its highest curve, the
Airy
In this paper, we employ the Brownian Gibbs property to make a close
comparison between the Airy line ensemble’s curves after affine shift and Brownian bridge, proving the finiteness of a superpolynomially
growing moment bound on Radon-Nikodym derivatives.
We also determine the value of a natural exponent describing in Brownian last
passage percolation the decay in probability for the existence of several near geodesics that are disjoint except for their common
endpoints, where the notion of ‘near’ refers to a small deficit in scaled geodesic energy, with the parameter specifying this nearness
tending to zero.
To prove both results, we introduce a technique that may be useful elsewhere for finding upper bounds on
probabilities of events concerning random systems of curves enjoying the Brownian Gibbs property.
Several results in this article
play a fundamental role in a further study of Brownian last passage percolation in three companion papers (Hammond 2017a,b,c), in which
geodesic coalescence and geodesic energy profiles are investigated in scaled coordinates.
eBook
A Strong Convergence Theorem for Solving Pseudo-monotone Variational Inequalities Using Projection Methods
Several iterative methods have been proposed in the literature for solving the variational inequalities in Hilbert or Banach spaces, where the underlying operator A is monotone and Lipschitz continuous. However, there are very few methods known for solving the variational inequalities, when the Lipschitz continuity of A is dispensed with. In this article, we introduce a projection-type algorithm for finding a common solution of the variational inequalities and fixed point problem in a reflexive Banach space, where A is pseudo-monotone and not necessarily Lipschitz continuous. Also, we present an application of our result to approximating solution of pseudo-monotone equilibrium problem in a reflexive Banach space. Finally, we present some numerical examples to illustrate the performance of our method as well as comparing it with related method in the literature.
Journal Article
Approximating fixed points of enriched contractions in Banach spaces
We introduce a large class of contractive mappings, called enriched contractions, a class which includes, amongst many other contractive type mappings, the Picard–Banach contractions and some nonexpansive mappings. We show that any enriched contraction has a unique fixed point and that this fixed point can be approximated by means of an appropriate Krasnoselskij iterative scheme. Several important results in fixed point theory are shown to be corollaries or consequences of the main results of this paper. We also study the fixed points of local enriched contractions, asymptotic enriched contractions and Maia-type enriched contractions. Examples to illustrate the generality of our new concepts and the corresponding fixed point theorems are also given.
Journal Article