Boundedness of operators on Hardy spaces via atomic decompositions

An example of a linear functional defined on a dense subspace of the Hardy space $H^1(\mathbb{R}^n)$ is constructed. It is shown that despite the fact that this functional is uniformly bounded on all atoms, it does not extend to a bounded functional on the whole $H^1$. Therefore, this shows that in general it is not enough to verify that an operator or a functional is bounded on atoms to conclude that it extends boundedly to the whole space. The construction is based on the fact due to Y. Meyer which states that quasi-norms corresponding to finite and infinite atomic decompositions in $H^p$, $0<p \le 1$, are not equivalent.