Andreas Defant

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,

Variants of the Maurey–Rosenthal Theorem for Quasi Köthe Function Spaces

AbstractThe Maurey–Rosenthal theorem states that each bounded and linear operator T from a quasi normed space E into some Lp(ν)(0<p<r<∞) which satisfies a vector-valued norm inequality

A tensor norm preserving unconditionality for Lp-spaces

\left\| {\left( {\sum {\left| {Tx_k } \right|} ^r } \right)^{1/r} } \right\|_{L_p } \leqslant \left( {\sum {\left\| {x_k } \right\|_E^r } } \right)^{1/r} {\text{ for all }}x_1 , \ldots ,x_n \in E,

A logarithmic lower bound for multi-dimensional bohr radii

even allows a weighted norm inequality: there is a function 0≤w∈L0(ν) such that

Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables

\left( {\int {\frac{{\left| {Tx} \right|^r }}{w}dv} } \right)^{1/r} \leqslant \left\| x \right\|_E {\text{ for all }}x \in E.

Estimates for the first and second Bohr radii of Reinhardt domains

Continuing the work of Garcia-Cuerva and Rubio de Francia we give several scalar and vector-valued variants of this fundamental result within the framework of quasi Köthe function spaces X(ν) over measure spaces. They are all special cases of our main result (Theorem 2) which extends the Maurey–Rosenthal cycle of ideas to the case of homogeneous operators between vector spaces being homogeneously representable in quasi Köthe function spaces.