Duality in spaces of finite linear combinations of atoms

TL;DR

This paper characterizes the dual space and completion of finite linear combinations of $(p,ty)$-atoms for $0<p extless 1$ on ${f R}^n$, and demonstrates an extension property for operators bounded on these atoms.

Contribution

It provides a detailed description of the dual and completion of atomic spaces for $0<p extless 1$, and establishes an extension result for operators on these atomic spaces.

Findings

The dual space of finite atomic combinations is explicitly described.

The completion of the space is characterized.

Operators bounded on atoms extend to bounded operators on $H^p({f R}^n)$.

Abstract

In this note we describe the dual and the completion of the space of finite linear combinations of $(p,\infty)$-atoms, $0<p\leq 1$ on ${\mathbb R}^n$. As an application, we show an extension result for operators uniformly bounded on $(p,\infty)$-atoms, $0<p < 1$, whose analogue for $p=1$ is known to be false. Let $0 < p <1$ and let $T$ be a linear operator defined on the space of finite linear combinations of $(p,\infty)$-atoms, $0<p < 1 $, which takes values in a Banach space $B$. If $T$ is uniformly bounded on $(p,\infty)$-atoms, then $T$ extends to a bounded operator from $H^p({\mathbb R}^n)$ into $B$.