  In this paper, we establish local H\"older estimate for non-negative
solutions of the singular equation \eqref{eq-nlocal-PME-1} below, for $m$ in
the range of exponents $(\frac{n-2\sigma}{n+2\sigma},1)$. Since we have trouble
in finding the local energy inequality of $v$ directly. we use the fact that
the operator $(-\La)^{\sigma}$ can be thought as the normal derivative of some
extension $v^{\ast}$ of $v$ to the upper half space, \cite{CS}, i.e., $v$ is
regarded as boundary value of $v^{\ast}$ the solution of some local extension
problem. Therefore, the local H\"older estimate of $v$ can be obtained by the
same regularity of $v^{\ast}$. In addition, it enables us to describe the
behaviour of solution of non-local fast diffusion equation near their
extinction time.

Analysis of PDEs