Leonhard Frerick

The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive

The Bohnenblust-Hille inequality says that the ‘ 2m m+1 -norm of the coefcients of an m-homogeneous polynomial P on C n is bounded by kPk1 times a constant independent of n, wherekk 1 denotes the supremum norm on the polydisc D n . The main result of this paper is that this inequality is hypercontractive, i.e., the constant can be taken to be C m for some C > 1. Combining this improved version of the Bohnenblust-Hille inequality with other results, we obtain the following: The Bohr radius for the polydisc D n behaves asymptotically as p (logn)=n modulo a factor bounded away from 0 and innity,

Transitive and Hypercyclic Operators on Locally Convex Spaces

Solutions are provided to several questions concerning topologically transitive and hypercyclic continuous linear operators on Hausdorff locally convex spaces that are not Frechet spaces. Among others, the following results are presented. (1) There exist transitive operators on the space ϕ of all finite sequences endowed with the finest locally convex topology (it was already known that there is no hypercyclic operator on ϕ. (2) The space of all test functions for distributions, which is also a complete direct sum of Frechet spaces, admits hypercyclic operators. (3) Every separable infinite-dimensional Frechet space contains a dense hyperplane that admits no transitive operator. 2000 Mathematics Subject Classification 47A16 (primary), 46A03, 46A04, 46A13, 37D45 (secondary).

A logarithmic lower bound for multi-dimensional bohr radii

We prove that the Bohr radiusKn of then-dimensional polydisc in ℂn is up to an absolute constant ≥ √logn/log logn/n. This improves a result of Boas and Khavinson.

Extension operators for spaces of infinite differentiable Whitney jets

Abstract In this article we present new results concerning the existence of extension operators for spaces of Whitney jets. Starting with the basic facts about Whitney functions on closed sets F we show that the existence of an extension operator is equivalent to the validity of certain simple interpolative inequalities on the space of Whitney functions. We give equivalent conditions in the spirit of the Markov type inequalities of Pawlucki, Plesniak, Bos, and Milman. Also simple sufficient geometric conditions are given.

Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables

Let

Polynomial approximation and maximal convergence on Faber sets

{\mathscr {H}}_\infty

Whitney extension operators without loss of derivatives

H∞ be the set of all ordinary Dirichlet series