  This paper is devoted to studying the local behavior of non-negative weak
solutions to the doubly non-linear parabolic equation \begin{equation*}
  \partial_t u^q -
  \text{div}\big(|D u|^{p-2}D u\big)
  =
  0 \end{equation*} in a space-time cylinder. H\"older estimates are
established for the gradient of its weak solutions in the super-critical fast
diffusion regime $0<p-1< q<\frac{N(p-1)}{(N-p)_+}$ where $N$ is the space
dimension. Moreover, decay estimates are obtained for weak solutions and their
gradient in the vicinity of possible extinction time. Two main components
towards these regularity estimates are a time-insensitive Harnack inequality
that is particular about this regime, and Schauder estimates for the parabolic
$p$-Laplace equation.
