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 $\mathcal H_p$ normalized in $L^p$ is a greedy basis of $L^p$, $1<p<\infty$. We study a question of when the Haar system $\mathcal H_p^w$ normalized in $L^p(w)$ is a greedy basis of $L^p(w)$, $1<p<\infty$. We prove that if $w$ is such that $\mathcal H_p^w$ is a Schauder basis of $L^p(w)$, then $\mathcal H_p^w$ is also a greedy basis of $L^p(w)$, $1<p<\infty$. 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.