  We show in this paper that maximal $L^q$-regularity for time-dependent
viscous Hamilton-Jacobi equations with unbounded right-hand side and
superquadratic $\gamma$-growth in the gradient holds in the full range $ q >
(N+2)\frac{\gamma-1}\gamma$. Our approach is based on new
$\frac{\gamma-2}{\gamma-1}$-H\"older estimates, which are consequence of the
decay at small scales of suitable nonlinear space and time H\"older quotients.
This is obtained by proving suitable oscillation estimates, that also give in
turn some Liouville type results for entire solutions.
