Intrinsic Harnack estimates for singular doubly non-linear equations

We establish forward, backward and elliptic Harnack inequalities for non-negative solutions to a class of doubly non-linear parabolic partial differential equations. These Harnack estimates are established in a proper range of parameters p and q. Such a range is shown to be optimal for a Harnack estimate to hold. Quantitative boundedness estimates for solutions and an expansion of positivity result for non-negative super-solutions are instrumental in the proof.