Equivalence of norms on finite linear combinations of atoms

TL;DR

This paper proves that on finite linear combinations of atoms in certain metric measure spaces, the H^1-norm and an atomic norm are equivalent, simplifying the understanding of these function spaces.

Contribution

It provides a simple function theoretic proof of norm equivalence on finite atomic combinations, extending to nondoubling metric measure spaces.

Findings

H^1-norm and atomic norm are equivalent on F^_{cont}(M)

The result applies to nondoubling metric measure spaces

Simplifies the analysis of atomic function spaces

Abstract

Let M be a space of homogeneous type and denote by F^\infty_{cont}(M) the space of finite linear combinations of continuous (1,\infty)-atoms. In this note we give a simple function theoretic proof of the equivalence on F^\infty_{cont}(M) of the H^1-norm and the norm defined in terms of finite linear combinations of atoms. The result holds also for the class of nondoubling metric measure spaces considered in previous works of A. Carbonaro and the authors.