  In this paper, a class of anisotropic weights having the form of
$|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}$ in dimensions $n\geq2$
is considered, where $x=(x',x_{n})$ and $x'=(x_{1},...,x_{n-1})$. We first find
the optimal range of $(\theta_{1},\theta_{2},\theta_{3})$ such that this type
of weights belongs to the Muckenhoupt class $A_{q}$. Then we further study its
doubling property, which shows that it provides an example of a doubling
measure but is not in $A_{q}$. As a consequence, we obtain anisotropic weighted
Poincar\'{e} and Sobolev inequalities, which are used to study the local
behavior for solutions to non-homogeneous weighted $p$-Laplace equations.
