  It is well-known that solutions to the basic problem in the calculus of
variations may fail to be Lipschitz continuous when the Lagrangian depends on
t. Similarly, for viscosity solutions to time-dependent Hamilton-Jacobi
equations one cannot expect Lipschitz bounds to hold uniformly with respect to
the regularity of coefficients. This phenomenon raises the question whether
such solutions satisfy uniform estimates in some weaker norm. We will show that
this is the case for a suitable H\"older norm, obtaining uniform estimates in
(x,t) for solutions to first and second order Hamilton-Jacobi equations. Our
results apply to degenerate parabolic equations and require superlinear growth
at infinity, in the gradient variables, of the Hamiltonian. Proofs are based on
comparison arguments and representation formulas for viscosity solutions, as
well as weak reverse H\"older inequalities.
