  We establish quantitative estimates for solutions $u(t,x)$ to the fractional
nonlinear diffusion equation, $\partial_t u +(-\Delta)^s (u^m)=0$ in the whole
range of exponents $m>0$, $0<s<1$. The equation is posed in the whole space
$x\in\mathbb{R}^d$. We first obtain weighted global integral estimates that
allow to establish existence of solutions for classes of large data. In the
core of the paper we obtain quantitative pointwise lower estimates of the
positivity of the solutions, depending only on the norm of the initial data in
a certain ball. The estimates take a different form in three exponent ranges:
slow diffusion, good range of fast diffusion, and very fast diffusion. Finally,
we show existence and uniqueness of initial traces.
