BOUNDEDNESS OF OPERATORS ON HARDY SPACES

In [1], the author provided an example which shows that there is a linear functional bounded uniformly on all atoms in H(1)(R(n)), and it can not be extended to a bounded functional on H(1)(R(n)). In this note, we first give a new atomic decomposition, where the decomposition converges in L(2)(R(n)) rather than only in the distribution sense. Then using this decomposition, we prove that for 0 < p <= 1, T is a linear operator which is bounded on L(2)(R(n)), then T can be extended to a bounded operator from H(p)(R(n)) to L(p)(R(n)) if and only if T is bounded uniformly on all (p, 2)-atoms in L(p)(R(n)). A similar result from H(p)(R(n)) to H(p)(R(n)) is also obtained. These results still hold for the product Hardy space and Hardy space on spaces of homogeneous type.