This is an announcement for the paper "Lebesgue and Hardy spaces for symmetric norms I" by Yanni Chen.

Abstract: In this paper, we define and study a class $\mathcal{R}_{c}$ of norms on $L^{\infty}\left( \mathbb{T}\right) $, called $continuous\ rotationally\ symmetric \ norms$, which properly contains the class $\left { \left \Vert \cdot \right \Vert _{p}:1\leq p<\infty \right } .$ For $\alpha \in \mathcal{R}% _{c}$ we define $L^{\alpha}\left( \mathbb{T}\right) $ and the Hardy space $H^{\alpha}\left( \mathbb{T}\right) $, and we extend many of the classical results, including the dominated convergence theorem, convolution theorems, dual spaces, Beurling-type invariant spaces, inner-outer factorizations, characterizing the multipliers and the closed densely-defined operators commuting with multiplication by $z$. We also prove a duality theorem for a version of $L^{\alpha}$ in the setting of von Neumann algebras.

Archive classification: math.OA

Submitted from: yet2@wildcats.unh.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1407.7920

or

http://arXiv.org/abs/1407.7920