Assume that $\Omega$ is a domain in $\mathbf{R}^n$ with the same area as a ball $B(x,r)$ and let $\alpha\in[1,0)$. I need the reference for the following inequality $$\int\_{\Omega} |x-y|^{\alpha-n} dy\le \int\_{B(x,r)} |x-y|^{\alpha-n} dy.$$



Riesz potential inequality
$$\int\_{\Omega}|x-y|^{\alpha-n}dy$$
$$=\int\_{\Omega\cap B(x,r)}|x-y|^{\alpha-n}dy+\int\_{\Omega\cap B(x,r)^c}|x-y|^{\alpha-n}dy$$
$$\leq\int\_{\Omega\cap B(x,r)} |x-y|^{\alpha-n}dy+\int\_{\Omega\cap B(x,r)^c}|r|^{\alpha-n}dy$$
$$=\int\_{\Omega\cap B(x,r)} |x-y|^{\alpha-n}dy+\int\_{\Omega^c\cap B(x,r)}|r|^{\alpha-n}dy$$
$$\leq\int\_{\Omega\cap B(x,r)} |x-y|^{\alpha-n}dy+\int\_{\Omega^c\cap B(x,r)}|x-y|^{\alpha-n}dy$$
$$=\int\_{B(x,r)} |x-y|^{\alpha-n}dy$$



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> **Attention** The second equality is by the same area.
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