Atomic decomposition and boundedness of operators on weighted hardy spaces

In this article, we establish a new atomic decomposition for f ε L ² _w ∩ H ^p _w , where the decomposition converges in L ² _w-norm rather than in the distribution sense. As applications of this decomposition, assuming that T is a linear operator bounded on L ² _w and 0 < p ≤ 1, we obtain (i) if T is uniformly bounded in L ^p _w -norm for all w-p-atoms, then T can be extended to be bounded from HL ^p _w to L ^p _w ; (ii) if T is uniformly bounded in H ^p _w -norm for all w-p-atoms, then T can be extended to be bounded on H ^p _w ; (iii) if T is bounded on H ^p _w , then T can be extended to be bounded from H ^p _w to L ^p _w.