I am currently reading Gilbarg and Trudinger's (and Jost's PDE textbook alongside). In P.159 of GT (also Lemma 10.2.4 in Jost's PDE book), they defined an operator $V_\mu $ on $L^1(\Omega)$ as follows: $$V_\mu (f):=\int_\Omega |x-y|^{n(\mu-1)}f(y)~dy $$ for $0<\mu \leq 1$ then showed that $V_\mu 1\leq \mu^{-1}\omega_n^{1-n}|\Omega|^\mu$ (here $|\Omega|$ means the measure of $\Omega$). They have provided the steps as follow:
Choose a $R>0$ so that $|\Omega|=|B_R(x)|=\omega_nR^n$, then
\begin{align} V_\mu 1&=\int_\Omega |x-y|^{n(\mu-1)}~dy\\ &\leq \int_{B_R(x)} |x-y|^{n(\mu-1)}f(y)~dy\\ &=\color{red}{\mu^{-1}\omega_nR^{n\mu}}\\ &=\mu^{-1}\omega_n^{1-n}|\Omega|^\mu \end{align}
This may seem like an easy question, but the third line really confuses me as I do not know how did they derive that equality. I would be very grateful if anyone could help me clarifying this equality!!
The keypoint is to prove that: $$\int_{\Omega} |x-y|^{d\cdot(\mu-1)} dy \le \int_{B(x,R)} |x-y|^{d\cdot(\mu-1)} dy, $$ where $R$ is a ray such that $|B(x,R)|=|\Omega|=\omega_d\cdot R^d$. Now you notice that $$|\Omega\setminus (\Omega \cap B(x,R)|= |\Omega|-|\Omega \cap B(x,R)|=|B(x,R)|-|\Omega \cap B(x,R)| =|B(x,R)-(\Omega\cap B(x,R))|;$$ moreover, these two inequalities hold $$ |x-y|^{d\cdot(\mu-1)} \leq R^{d\cdot(\mu-1)} \text{for} \quad |x-y|\geq R,$$ and $$ |x-y|^{d\cdot(\mu-1)} \geq R^{d\cdot(\mu-1)} \text{for} \quad |x-y|\leq R.$$ Finally, you decompose the integral as a sum \begin{gather*} \int _{\Omega } |x-y|^{d\cdot ( \mu -1)} dy=\int _{\Omega \setminus ( \Omega \cap B( x,R))} |x-y|^{d\cdot ( \mu -1)} dy+\int _{\Omega \cap B( x,R)} |x-y|^{d\cdot ( \mu -1)} dy\\ \leq R^{d\cdot ( \mu -1)} \cdot ( |\Omega |-|\Omega \cap B( x,R) |) +\int _{\Omega \cap B( x,R)} |x-y|^{d\cdot ( \mu -1)} dy\\ =R^{d\cdot ( \mu -1)} \cdot ( |B( x,R) |-|\Omega \cap B( x,R) |) +\int _{\Omega \cap B( x,R)} |x-y|^{d\cdot ( \mu -1)} dy\\ =\int _{B( x,R) \setminus \Omega \cap B( x,R)} R^{d( \mu -1)} \ dy+\int _{\Omega \cap B( x,R)} |x-y|^{d\cdot ( \mu -1)} dy\\ \leq \int _{B( x,R) \setminus \Omega \cap B( x,R)} |x-y|^{d\cdot ( \mu -1)} \ dy+\int _{\Omega \cap B( x,R)} |x-y|^{d\cdot ( \mu -1)} dy\\ =\int _{B( x,R)} |x-y|^{d\cdot ( \mu -1)} \ dy\ =\frac{1}{\mu}\cdot \omega_d \cdot R^{d\cdot\mu}=\frac{1}{\mu}\cdot \omega_{d}^{1-\mu}\cdot |\Omega|^{\mu}. \end{gather*} For the last two equalities I have used respectively the integration of a radial function in polar coordinates, and the definition of $R$.