Let $D$ be the closed unit ball in $\mathbb{R}^n$. Define $$(T x)(t)=\int_{D}\frac{x(s)}{|t-s|^\alpha}{\rm d}s$$ Where $0<\alpha<n$. Then $T$ is a continuous operator on $C(D)$. Is it compact?
I've been thinking using the Arzela-Ascoli theorem. For a bounded sequence $\{x_n\}$, it's easy to show $\{T x_n\}$ is uniformly bounded. But I failed to prove that they're equicontinuous, since $\nabla |t-s|^{-\alpha}$ is not integrable when $\alpha \geq n-1$.
Does anyone have an idea? Thanks.