Hölder estimates for singular non-local parabolic equations

In this paper, we establish local Hölder estimate for non-negative solutions of the singular equation (M.P) below, for m in the range of exponents . Since we have trouble in finding the local energy inequality of v directly, we use the fact that the operator σ(−Δ) can be thought as the normal derivative of some extension v⁎ of v to the upper half space (Caffarelli and Silvestre, 2007 [5]), i.e., v is regarded as boundary value of v⁎ the solution of some local extension problem. Therefore, the local Hölder estimate of v can be obtained by the same regularity of v⁎. In addition, it enables us to describe the behavior of solution of non-local fast diffusion equation near their extinction time.