  We investigate qualitative properties of local solutions $u(t,x)\ge 0$ to the
fast diffusion equation, $\partial_t u =\Delta (u^m)/m$ with $m<1$,
corresponding to general nonnegative initial data. Our main results are
quantitative positivity and boundedness estimates for locally defined solutions
in domains of the form $[0,T]\times\RR^d$. They combine into forms of new
Harnack inequalities that are typical of fast diffusion equations. Such results
are new for low $m$ in the so-called very fast diffusion range, precisely for
all $m\le m_c=(d-2)/d.$ The boundedness statements are true even for $m\le 0$,
while the positivity ones cannot be true in that range.
