Conjugate Hardy's inequalities with decreasing weights

We prove that for a decreasing weight $\omega$ on $\mathbf{R}^+$, the conjugate Hardy transform is bounded on $L_p(\omega)$ ($1\leq p<\infty$) if and only if it is bounded on the cone of all decreasing functions of $L_p(\omega)$. This property does not depend on $p$.