On the H^1-L^1 boundedness of operators

TL;DR

This paper establishes conditions under which operators bounded on certain atomic spaces extend to bounded operators from H^1(R^n) to a Banach space, highlighting differences between atom types.

Contribution

It proves the extension of boundedness from atomic spaces to H^1(R^n) for (1,q)-atoms and continuous (1,)-atoms, clarifying limitations for (1,)-atoms.

Findings

Operators bounded on (1,q)-atoms extend to H^1(R^n)

Extension holds for continuous (1,)-atoms

Extension fails for (1,)-atoms in general

Abstract

We prove that if q is in (1,\infty), Y is a Banach space and T is a linear operator defined on the space of finite linear combinations of (1,q)-atoms in R^n which is uniformly bounded on (1,q)-atoms, then T admits a unique continuous extension to a bounded linear operator from H^1(R^n) to Y. We show that the same is true if we replace (1,q)-atoms with continuous (1,\infty)-atoms. This is known to be false for (1,\infty)-atoms.