The problem is:
Show that $(\mathbb{R}^d,d_2,(1+\|x\|^\alpha)dx)$ is a space of homogeneous type for any $\alpha \in \mathbb{R}$, where $d_2$ is the Euclidean metric.
The definition of space of homogeneous type is:
Definition: We say that $(E,d,\mu)$ is a space of homogeneous type if the followings hold:
- $(E,d)$ is a quasi-metric space;
- there exists some $C \in [1,+\infty[$ such that for any $x \in E$ and any $r>0$, $\mu(B(x,2r)) \leq C\mu(B(x,r)) < +\infty$.
In this case, the goal is to show that for any $x \in \mathbb{R}^d$ and any $r>0$, \[ \int_{B(x,2r)} (1+\|y\|)^\alpha dy \leq C \int_{B(x,r)} (1+\|y\|)^\alpha dy \] Obviously, the main work is to approximate the integration on a ball not centered at the origin and get the constant free of $x$ and $r$ (either one is easy, while controlling both of them requires rather hard effort).
Could anyone give a proper approximation? Your help is greatly appreciated in advance!