How to show given function in $L^r(\Omega)$?

Question

I wanted to show that $h(x-y)=|x-y|^{n(\mu-1)}\in L^r(\Omega)$ where $\Omega $ is bounded .\ $\Omega\in B(x,R)$ for some $R$.

So I have to show that $\int_{B(x,R)}|h|^r\leq\infty $.

$\int_{B(x,R)}|h|^r=\int_0^R\int_{\partial B(x,k)}|k|^{nr(\mu-1)}dSdk=\int_0^R k^{nr(\mu-1)}nw_nk^{n-1}dk$

$=\int_0^R k^{nr(\mu-1)+n-1}nw_ndk=R^{nr(\mu-1)+n}nw_n<\infty$

Is this true ? How to make sure $nr(\mu-1)+n-1>0$.

Any Help will be appreciated

Answer 1

$\int_0^{R}k^{t}dt <\infty$ whenever $t >-1$. So you only need $nr(\mu -1)+n-1 >-1$ or $nr(\mu -1)+n >0$. This follows from the definition of $r$ and the fact $\mu >\frac 1 p-\frac 1 q$.