Lebesgue and Hardy spaces for symmetric norms I

TL;DR

This paper introduces a new class of symmetric norms on function spaces, extending classical Lebesgue and Hardy space results to these generalized spaces, including duality, factorization, and invariant subspace characterizations.

Contribution

It defines continuous rotationally symmetric norms on $L^{}(T)$ and extends fundamental theorems of analysis to the associated $L^{}$ and Hardy spaces, broadening the scope of classical harmonic analysis.

Findings

Defined the class $al R_c$ of symmetric norms on $L^{}(T)$.

Extended classical theorems such as dominated convergence, convolution, and duality to these new spaces.

Characterized invariant subspaces, inner-outer factorizations, and multipliers within the generalized framework.

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.