Self-improving property of the fast diffusion equation

We show that the gradient of the $m$-power of a solution to a singular parabolic equation of porous medium-type (also known as fast diffusion equation), satisfies a reverse H"older inequality in suitable intrinsic cylinders. Relying on an intrinsic Calder'on-Zygmund covering argument, we are able to prove the local higher integrability of such a gradient for $minleft(rac{(n-2)_+}{n+2},1 ight)$. Our estimates are satisfied for a general class of growth assumptions on the non linearity. In this way, we extend the theory for $mgeq 1$ (see cite{GiaSch16} in the list of references) to the singular case. In particular, an intrinsic metric that depends on the solution itself is introduced for the singular regime.