  We prove some weighted refined decoupling estimates. As an application, we
give an alternative proof of the following result on Falconer's distance set
problem by the authors in a companion work: if a compact set $E\subset
\mathbb{R}^d$ has Hausdorff dimension larger than
$\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$, where $d\geq 4$, then there is a
point $x\in E$ such that the pinned distance set $\Delta_x(E)$ has positive
Lebesgue measure. Aside from this application, the weighted refined decoupling
estimates may be of independent interest.
