Let $\alpha\in(0,1)$ and $g(x)=|x|^{-\alpha}$ for $x\in\mathbb{R}$. Prove that if $f\in L^{1}(\mathbb{R})$ is non-negative and $$K(x)=f*g(x)=\int_\mathbb{R}f(x-y)g(y)dy=\int_\mathbb{R}f(y)g(x-y)dy,$$ then $K$ is finite a.e. on $\mathbb{R}$.
I notice that $K\notin L^{p}$ for any $p\geq1$. So it seems that Holder inequality doesn't work. Moreover, it suffices to show that $\int_\mathbb{R}K(x)dx<\infty$. But how do we prove that?
Any help is appreciated.