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A generalized Kahane-Khinchin inequality

S. Favorov (1998)

Studia Mathematica

The inequality ʃ l o g | a n e 2 π i φ n | d φ 1 d φ n C l o g ( | a n | 2 ) 1 / 2 with an absolute constant C, and similar ones, are extended to the case of a n belonging to an arbitrary normed space X and an arbitrary compact group of unitary operators on X instead of the operators of multiplication by e 2 π i φ .

A note on maximal estimates for stochastic convolutions

Mark Veraar, Lutz Weis (2011)

Czechoslovak Mathematical Journal

In stochastic partial differential equations it is important to have pathwise regularity properties of stochastic convolutions. In this note we present a new sufficient condition for the pathwise continuity of stochastic convolutions in Banach spaces.

A probabilistic version of the Frequent Hypercyclicity Criterion

Sophie Grivaux (2006)

Studia Mathematica

For a bounded operator T on a separable infinite-dimensional Banach space X, we give a "random" criterion not involving ergodic theory which implies that T is frequently hypercyclic: there exists a vector x such that for every non-empty open subset U of X, the set of integers n such that Tⁿx belongs to U, has positive lower density. This gives a connection between two different methods for obtaining the frequent hypercyclicity of operators.

Applications of spherical designs to Banach space theory

Hermann König (2004)

Banach Center Publications

Spherical designs constitute sets of points distributed on spheres in a regular way. They can be used to construct finite-dimensional normed spaces which are extreme in some sense: having large projection constants, big or small Banach-Mazur distance to Hilbert spaces or p -spaces. These examples provide concrete illustrations of results obtained by more powerful probabilistic techniques which, however, do not exhibit explicit examples. We give a survey of such constructions where the geometric invariants...

Between Paouris concentration inequality and variance conjecture

B. Fleury (2010)

Annales de l'I.H.P. Probabilités et statistiques

We prove an almost isometric reverse Hölder inequality for the euclidean norm on an isotropic generalized Orlicz ball which interpolates Paouris concentration inequality and variance conjecture. We study in this direction the case of isotropic convex bodies with an unconditional basis and the case of general convex bodies.

Chevet type inequality and norms of submatrices

Radosław Adamczak, Rafał Latała, Alexander E. Litvak, Alain Pajor, Nicole Tomczak-Jaegermann (2012)

Studia Mathematica

We prove a Chevet type inequality which gives an upper bound for the norm of an isotropic log-concave unconditional random matrix in terms of the expectation of the supremum of “symmetric exponential” processes, compared to the Gaussian ones in the Chevet inequality. This is used to give a sharp upper estimate for a quantity Γ k , m that controls uniformly the Euclidean operator norm of the submatrices with k rows and m columns of an isotropic log-concave unconditional random matrix. We apply these estimates...

Disjointification of martingale differences and conditionally independent random variables with some applications

Sergey Astashkin, Fedor Sukochev, Chin Pin Wong (2011)

Studia Mathematica

Disjointification inequalities are proven for arbitrary martingale difference sequences and conditionally independent random variables of the form f k ( s ) x k ( t ) k = 1 , where f k ’s are independent and xk’s are arbitrary random variables from a symmetric space X on [0,1]. The main results show that the form of these inequalities depends on which side of L₂ the space X lies on. The disjointification inequalities obtained allow us to compare norms of sums of martingale differences and non-negative random variables with...

Estimates for vector-valued holomorphic functions and Littlewood-Paley-Stein theory

Mark Veraar, Lutz Weis (2015)

Studia Mathematica

We consider generalized square function norms of holomorphic functions with values in a Banach space. One of the main results is a characterization of embeddings of the form L p ( X ) γ ( X ) L q ( X ) , in terms of the type p and cotype q of the Banach space X. As an application we prove L p -estimates for vector-valued Littlewood-Paley-Stein g-functions and derive an embedding result for real and complex interpolation spaces under type and cotype conditions.

Euclidean arrangements in Banach spaces

Daniel J. Fresen (2015)

Studia Mathematica

We study the way in which the Euclidean subspaces of a Banach space fit together, somewhat in the spirit of the Kashin decomposition. The main tool that we introduce is an estimate regarding the convex hull of a convex body in John's position with a Euclidean ball of a given radius, which leads to a new and simplified proof of the randomized isomorphic Dvoretzky theorem. Our results also include a characterization of spaces with nontrivial cotype in terms of arrangements of Euclidean subspaces.

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