Atomic decompositions and operators on Hardy spaces

Journal article, 2009

This paper is essentially the second author's lecture at a CIMPA-UNESCO School. It summarises large parts of the three authors' paper "On the H1 - L1 boundedness of operators". Only one proof is given. In the setting of a Euclidean space, we consider operators defined and uniformly bounded on atoms of a Hardy space Hp. The question discussed is whether such an operator must be bounded on Hp. This leads to a study of the difference between countable and finite atomic decompositions in Hardy spaces.