On the H^1-L^1-boundedness of operators

We prove that if q is in (1,∞), Y is a Banach space, and T is a linear operator defined on the space of finite linear combinations of (1, q)-atoms in R^n with the property that sup{TaY : a is a (1, q)-atom} < ∞, then T admits a (unique) continuous extension to a bounded linear operator from H^1(R^n) to Y . We show that the same is true if we replace (1, q)-atoms by continuous (1,∞)-atoms. This is known to be false for (1,∞)-atoms.