Two-weight Hardy inequality on topological measure spaces

We consider a Hardy type integral operator $T$ associated with a family of open subsets $\Omega(t)$ of an open set $\Omega$ in a Hausdorff topological space $X$. In the inequality
$$ \left( \int_\Omega|Tf(x)|^q u(x)d\mu(x)\right)^{1/q} \leqslant C \left(\int_\Omega |f(x)|^p v(x) d\nu(x) \right)^{1/p}, $$
the measures $\mu$, $\nu$ are $\sigma$-additive Borel measures; the weights $u$, $v$ arepositive and finite almost everywhere, $1<p<\infty$, $0<q<\infty$, and $C>0$ is independent of $f$, $u$, $v$, $\mu$, $\nu$. We find necessary and sufficient conditions for the boundedness and compactness of the operator $T$ and obtain two-sided estimates for its approximation numbers. All results are proved using domain partitions, thus providing a roadmap for generalizing many one-dimensional results to a Hausdorff topological space.