A Note on Hardy Spaces and Bounded Operators

TL;DR

This paper investigates the atomic decomposition of functions in Hardy spaces Hp intersected with Ls, and characterizes bounded operators between these spaces based on their action on atoms.

Contribution

It establishes a new atomic decomposition for functions in Hp(Rn) intersected with Ls(Rn) and provides criteria for extending bounded operators between these spaces.

Findings

Atomic decomposition converges in Ls(Rn) for functions in Hp(Rn)∩Ls(Rn).

Bounded operators extend from Ls(Rn) to Hp(Rn) if bounded uniformly on atoms.

Characterizes bounded operators from Hp(Rn) to Lp(Rn) via atom behavior.

Abstract

In this note we show that if f belongs to Hp(Rn)\capLs(Rn), where 0 < p <= 1 < s < 1, then there exists a (p;infinite)-atomic decomposition which converges to f in Ls(Rn). From this fact, we prove that a bounded operator T on Ls(Rn) can be extended to a bounded operator from Hp(Rn) into Lp(Rn) if and only if T is bounded uniformly in Lp norm on all (p;infinite)-atoms. A similar result is also obtained from Hp(Rn) into Hp(Rn).