Hardy's inequalities for monotone functions on partially ordered measure spaces

We characterize the weighted Hardy’s inequalities for monotone functions in ${\mathbb{R}}_{+}^n\mathrm{.}$ In dimension $n=1$, this recovers the classical theory of $B_p$ weights. For $n>1$, the result was only known for the case $p=1$. In fact, our main theorem is proved in the more general setting of partially ordered measure spaces.