  We give new proofs of Hardy space estimates for fractional and singular
integral operators on weighted and variable exponent Hardy spaces. Our proofs
consist of several interlocking ideas: finite atomic decompositions in terms of
$L^\infty$ atoms, vector-valued inequalities for maximal and other operators,
and Rubio de Francia extrapolation. Many of these estimates are not new, but we
give new and substantially simpler proofs, which in turn significantly
simplifies the proofs of the Hardy spaces inequalities.
