Boundedness of operators on Hardy spaces via atomic decompositions

Source abstract

An example of a linear functional defined on a dense subspace of the Hardy space H 1 ( R n ) 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 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 H^p , 0 > p ≤ 1 0>p \le 1 , are not equivalent.