Improved Poincare inequalities with weights

In this paper we prove that if $\Omega\in\mathbb{R}^n$ is a bounded John domain, the following weighted Poincare-type inequality holds: $$\inf_{a\in \mathbb{R}}\| (f(x)-a) w_1(x) \|_{L^q(\Omega)} \le C \|\nabla f(x) d(x)^\alpha w_2(x) \|_{L^p(\Omega)}$$ where $f$ is a locally Lipschitz function on $\Omega$, $d(x)$ denotes the distance of $x$ to the boundary of $\Omega$, the weights $w_1, w_2$ satisfy certain cube conditions, and $\alpha \in [0,1]$ depends on $p,q$ and $n$. This result generalizes previously known weighted inequalities, which can also be obtained with our approach.