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Hölder Estimates for Singular Non-local Parabolic Equations

In this paper, we establish local Hölder estimate for non-negative solutions of the singular equation \eqref{eq-nlocal-PME-1} below, for $m$ in the range of exponents $(\frac{n-2σ}{n+2σ},1)$. Since we have trouble in finding the local energy inequality of $v$ directly. we use the fact that the operator $(-\La)^σ$ 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 Hölder 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.