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.