Hölder regularity for a class of doubly non linear PDEs

We prove local Hölder continuity for non negative, locally bounded, local weak solutions to the class of doubly nonlinear parabolic equations ${\mathrm{\partial }}_t(u_q)-\text{div}(\mathrm{\mid }Du{\mathrm{\mid }}^{p-2}Du)=0$ for $p>2$, $0<q<p-1$. The proof relies on expansion of positivity results combined with the study of an alternative (related to DeGiorgi-type lemmas) and an exponential shift which allows us to deal with the intrinsic geometry associated to the problem.