Let $0<s<1<p<\infty$ and define $$ L^{p-1}_{ps}(\mathbb{R}^N):=\left\{v\in L^{p-1}_{\mathrm{loc}}(\mathbb{R}^N):\int_{\mathbb{R}^N}\frac{|v(y)|^{p-1}}{1+|y|^{N+ps}}\,dy<+\infty\right\}. $$ Also, for $x_0\in\mathbb{R}^N$, we define $$ \mathrm{Tail}(v;x_0,r)=\left(\int_{\mathbb{R}^N\setminus B_r(x_0)}\frac{|v(y)|^{p-1}}{|y-x_0|^{N+ps}}\,dy\right)^\frac{1}{p-1}. $$ Here $B_r(x_0):=\{x\in\mathbb{R}^N:|x-x_0|<r\}$ for $r>0$. Then if $v\in L^{p-1}_{ps}(\mathbb{R}^N)$, we have $\mathrm{Tail}(v;x_0,r)<\infty$ for any $x_0\in\mathbb{R}^N$ and $r>0$.
I tried in the following way: First let us assume that $x_0=0$ and let $v\in L^{p-1}_{ps}(\mathbb{R}^N)$. Then, for any $y\in\mathbb{R}^N\setminus B_r(0)$, that is if $|y|>r$, we have $$ 1+|y|^{N+ps}=r^{N+ps}r^{-N-ps}+|y|^{N+ps}<|y|^{N+ps}r^{-N-ps}+|y|^{N+ps}=(r^{-N-ps}+1)|y|^{N+ps}. $$ Therefore, we have $$ \int_{\mathbb{R}^N\setminus B_r(0)}\frac{|v(y)|^{p-1}}{|y|^{N+ps}}\,dy<\int_{\mathbb{R}^N\setminus B_r(0)}\frac{1}{(r^{-N-ps}+1)}\frac{|v(y)|^{p-1}}{1+|y|^{N+ps}}\,dy<\infty, $$ where the last inequality above holds due to the assumption that $v\in L^{p-1}_{ps}(\mathbb{R}^N)$. This concludes the proof for $x_0=0$. But I am unable to proceed when $x_0\neq 0$. Can someone please help me with the same? I guess for $x_0\neq 0$ also same type of inequality will hold.
Thanks.