  We study fine global properties of nonnegative solutions to the Cauchy
Problem for the fast $p$-Laplacian evolution equation $u_t=\Delta_p u$ on the
whole Euclidean space, in the so-called "good fast diffusion range"
$\tfrac{2N}{N+1}<p<2$. It is well-known that non-negative solutions behave for
large times as $\mathcal{B}$, the Barenblatt (or fundamental) solution, which
has an explicit expression.
  We prove the so-called Global Harnack Principle (GHP), that is, precise
global pointwise upper and lower estimates of nonnegative solutions in terms of
$\mathcal{B}$. This can be considered the nonlinear counterpart of the
celebrated Gaussian estimates for the linear heat equation.
  We characterize the maximal (hence optimal) class of initial data such that
the GHP holds, by means of an integral tail condition, easy to check. The GHP
is then used as a tool to analyze the fine asymptotic behavior for large times.
For initial data that satisfy the same integral condition, we prove that the
corresponding solutions behave like the Barenblatt with the same mass,
uniformly in relative error.
  When the integral tail condition is not satisfied we show that both the GHP
and the uniform convergence in relative error, do not hold anymore, and we
provide also explicit counterexamples. We then prove a "generalized GHP", that
is, pointwise upper and lower bounds in terms of explicit profiles with a tail
different from $\mathcal{B}$. Finally, we derive sharp global quantitative
upper bounds of the modulus of the gradient of the solution, and, when data are
radially decreasing, we show uniform convergence in relative error for the
gradients.
  To the best of our knowledge, analogous issues for the linear heat equation
$p=2$, do not possess such clear answers, only partial results are known.
