Greedy bases in L p spaces

We consider a weighted L ^p space L ^p (w) with a weight function w. It is known that the Haar system H _p normalized in L ^p is a greedy basis of L ^p , 1 < p < ∞. We study a question of when the Haar system H _p ^w normalized in L ^p (w) is a greedy basis of L ^p (w), 1 < p < ∞. We prove that if w is such that H _p ^w is a Schauder basis of L ^p (w), then H _p ^w is also a greedy basis of L ^p (w), 1 < p < ∞. Moreover, we prove that a subsystem of the Haar system obtained by discarding finitely many elements from it is a Schauder basis in a weighted norm space L ^p (w); then it is a greedy basis.