  We investigate quantitative properties of the nonnegative solutions
$u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u +
{\mathcal L} (u^m)=0$, posed in a bounded domain, $x\in\Omega\subset {\mathbb
R}^N$ with $m>1$ for $t>0$. As ${\mathcal L}$ we use one of the most common
definitions of the fractional Laplacian $(-\Delta)^s$, $0<s<1$, in a bounded
domain with zero Dirichlet boundary conditions. We consider a general class of
very weak solutions of the equation, and obtain a priori estimates in the form
of smoothing effects, absolute upper bounds, lower bounds, and Harnack
inequalities. We also investigate the boundary behaviour and we obtain sharp
estimates from above and below. The standard Laplacian case $s=1$ or the linear
case $m=1$ are recovered as limits. The method is quite general, suitable to be
applied to a number of similar problems.
