  This article provides sharp constructive upper and lower bound estimates for
the non-linear Boltzmann collision operator with the full range of physical non
cut-off collision kernels ($\gamma > -n$ and $s\in (0,1)$) in the trilinear
$L^2(\R^n)$ energy $<\mathcal{Q}(g,f),f>$. These new estimates prove that, for
a very general class of $g(v)$, the global diffusive behavior (on $f$) in the
energy space is that of the geometric fractional derivative semi-norm
identified in the linearized context in our earlier works [2009, 2010, 2010
arXiv:1011.5441v1]. We further prove new global entropy production estimates
with the same anisotropic semi-norm. This resolves the longstanding, widespread
heuristic conjecture about the sharp diffusive nature of the non cut-off
Boltzmann collision operator in the energy space $L^2(\R^n)$.
